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Zongbo Shi, Tuan Vu, Simone Kotthaus, Roy M. Harrison, Sue Grimmond, Siyao Yue, Tong Zhu, James Lee, Yiqun Han, Matthias Demuzere, Rachel E. Dunmore, Lujie Ren, Di Liu, Yuanlin Wang, Oliver Wild, James Allan, W. Joe Acton, Janet Barlow, Benjamin Barratt, David Beddows, William J. Bloss, Giulia Calzolai, David Carruthers, David C. Carslaw, Queenie Chan, Lia Chatzidiakou, Yang Chen, Leigh Crilley, Hugh Coe, Tie Dai, Ruth Doherty, Fengkui Duan, Pingqing Fu, Baozhu Ge, Maofa Ge, Daobo Guan, Jacqueline F. Hamilton, Kebin He, Mathew Heal, Dwayne Heard, C. Nicholas Hewitt, Michael Hollaway, Min Hu, Dongsheng Ji, Xujiang Jiang, Rod Jones, Markus Kalberer, Frank J. Kelly, Louisa Kramer, Ben Langford, Chun Lin, Alastair C. Lewis, Jie Li, Weijun Li, Huan Liu, Junfeng Liu, Miranda Loh, Keding Lu, Franco Lucarelli, Graham Mann, Gordon McFiggans, Mark R. Miller, Graham Mills, Paul Monk, Eiko Nemitz, Fionna O'Connor, Bin Ouyang, Paul I. Palmer, Carl Percival, Olalekan Popoola, Claire Reeves, Andrew R. Rickard, Longyi Shao, Guangyu Shi, Dominick Spracklen, David Stevenson, Yele Sun, Zhiwei Sun, Shu Tao, Shengrui Tong, Qingqing Wang, Wenhua Wang, Xinming Wang, Xuejun Wang, Zifang Wang, Lianfang Wei, Lisa Whalley, Xuefang Wu, Zhijun Wu, Pinhua Xie, Fumo Yang, Qiang Zhang, Yanli Zhang, Yuanhang Zhang, and Mei Zheng
Atmos. Chem. Phys., 19, 7519–7546, https://doi.org/10.5194/acp-19-7519-2019,https://doi.org/10.5194/acp-19-7519-2019, 2019
Short summary
Brain-derived neurotrophic factor (BDNF) is critically involved in the pathophysiology of chronic pain. However, the mechanisms of BDNF action on specific neuronal populations in the spinal superficial dorsal horn (SDH) requires further study. We used chronic BDNF treatment (200 ng/ml, 5–6 days) of defined-medium, serum-free spinal organotypic cultures to study intracellular calcium ([Ca2+]i) fluctuations. A detailed quantitative analysis of these fluctuations using the Frequency-independent biological signal identification (FIBSI) program revealed that BDNF simultaneously depressed activity in some SDH neurons while it unmasked a particular subpopulation of ‘silent’ neurons causing them to become spontaneously active. Blockade of gap junctions disinhibited a subpopulation of SDH neurons and reduced BDNF-induced synchrony in BDNF-treated cultures. BDNF reduced neuronal excitability assessed by measuring spontaneous excitatory postsynaptic currents. This was similar to the depressive effect of BDNF on the [Ca2+]i fluctuations. This study reveals novel regulatory mechanisms of SDH neuronal excitability in response to BDNF.
Injury to, or disease of, the somatosensory system frequently generates chronic and sometimes intractable neuropathic pain1, 2. In experimental animals, peripheral nerve damage, such as that generated by chronic constriction or section of the sciatic nerve, induces pain-related behaviours that serve as a model for human neuropathic pain3, 4. Seven or more days of sciatic nerve injury promote an enduring increase in the excitability of first order primary afferent neurons5,6,7,8. These become chronically active and release a variety of mediators (cytokines, chemokines, neuropeptides, ATP and growth factors) that predispose spinal microglia to a more ‘activated’ state9,10,11,12,13,14. These in turn, release further mediators, including brain derived neurotrophic factor (BDNF) that promote a slowly developing, but persistent increase in excitability of second order neurons in the spinal dorsal horn. This ‘central sensitization’ is thought to be responsible for the allodynia, hyperalgesia, spontaneous pain and causalgia that characterize neuropathic pain3, 15. Spinal actions of BDNF involve alteration in Cl- gradients such that the normally inhibitory actions of GABA become excitatory16, 17. There is also increased excitatory synaptic drive to putative excitatory neurons9, 18. This results in an overall increase in excitability as monitored by confocal Ca2+ imaging in organotypic cultures of rat or mouse spinal cord9, 18,19,20. Despite this, the long-term effects of upregulated BDNF on neuronal plasticity in the superficial dorsal horn (SDH) are not fully understood.
In our previous work, we developed and used a model to study neuropathic pain whereby defined-medium spinal organotypic cultures are treated long-term (5–6 days) with BDNF (200 ng/ml)9, 20,21,22,23. BDNF treatment is administered at day 20 or more after organotypic cultures are established to allow neurons to mature to a more ‘adult-like’ state. We demonstrated that lamina II neurons in these organotypic cultures are similar to those in acute slices from young adult rats21. We also showed that the changes that occur in spinal organotypic dorsal horn neurons ‘mirror’ those observed in dorsal horn neurons from acute slices obtained from chronic constriction injury (CCI) rats at peak mechanical hypersensitivity levels20. Naïve cultures display spontaneous oscillatory changes in intracellular Ca2+ levels [Ca2+]i and we have previously shown that the amplitude and frequency of these changes in [Ca2+]i are profoundly increased following 5–6 days treatment with BDNF20.
We used this model and a new Frequency-independent biological signal identification (FIBSI)24 program to quantitatively measure the fluctuations of [Ca2+]i and examine their synchronicity. Unexpectedly, we observed two opposite effects of BDNF that appeared to occur simultaneously. First, BDNF caused an overall decrease in fluctuation size. Second, we noticed a particular population of SDH neurons in naïve cultures that did not display typical, marked fluctuations of [Ca2+]i. This population of ‘silent’ neurons was never seen in BDNF-treated cultures, suggesting that BDNF unmasks these ‘silent’ neurons and causes them to become spontaneously active. We next investigated the role of gap junctions in mediating the [Ca2+]i fluctuations; application of the gap junction blocker octanol to chronic BDNF-treated neurons revealed a subpopulation of neurons that generate low-frequency, large [Ca2+]i fluctuations. Further pharmacological experiments indicated the [Ca2+]i fluctuations in active neurons are regulated by diverse mechanisms including voltage-gated calcium and sodium channels as well as GABA and NMDA receptors. Finally, we used FIBSI to reanalyze a previous dataset of spontaneous excitatory postsynaptic current (sEPSC) recordings from BDNF-treated dorsal horn neurons in order to qualitatively compare the effects of BDNF on the sEPSCs and [Ca2+]i fluctuations in SDH neurons. The depressive effects of BDNF on the sEPSCs in delay and tonic-firing SDH neurons were consistent with the effects on the [Ca2+]i fluctuations. This study reveals novel mechanisms of BDNF regulation of dorsal horn excitability, which has implications for the study of chronic neuropathic pain physiology.
In previous work we identified that chronic BDNF treatment (200 ng/ml, 5–6 days) induces Ca2+ fluctuations compared to naïve cultures20, 25 (Supplementary Video 1A, B). In this study, we performed a detailed quantitative analysis of the fluctuations in [Ca2+]i using Fluo-4 AM Ca2+ imaging and the FIBSI analysis program24 to further establish the nature of BDNF-induced changes in lamina II neurons of the SDH. Refer to Fig. 1A1–A3 for an example of [Ca2+]i fluctuations detected by the FIBSI program. Inspection of the FIBSI-processed recordings revealed a subset of naïve neurons, which were relatively ‘silent’ (Fig. 1B, left; raw recordings in Supplemental Fig. 1) displaying only small amplitude fluctuations for the duration of the recording (‘silent’ = 1.7 ± 0.04 AU; naïve = 53.7 ± 1.6 AU; BDNF = 25.5 ± 0.7 AU). No ‘silent’ neurons were found in the BDNF treatment group (Fig. 1B, right). Refer to Table 1 for detailed statistical information. This stark contrast suggested BDNF may be unmasking, or activating, these ‘silent’ neurons. We grouped the fluctuations in the ‘silent’ neurons together for further comparison to the fluctuations in the other active naïve neurons and BDNF-treated neurons. Log-transformed fluctuation amplitudes (Fig. 1C) and area under the curve (AUC; Fig. 1D) were larger in the active naïve population and in BDNF treated neurons compared to ‘silent’ neurons. Unexpectedly, both measures were significantly decreased in the BDNF condition compared to naive. A plausible explanation for this effect was that chronic BDNF induced a concomitant enhancement of [Ca2+]i fluctuations in some neurons and depression in others. Indeed, the effects of BDNF on the relative frequencies (%) for the fluctuation amplitudes (amplitudes were normalized to the largest fluctuation in each neuron) revealed a biphasic effect; frequencies of the smaller- and larger-amplitude fluctuations were greater in the BDNF-treated neurons compared to the naïve and ‘silent’ neurons (Fig. 1E). We next compared the mean parameters between neurons in order to control for differences in sampling periods. A < 10% maximal fluctuation amplitude cut-off was applied to each neuron to reduce the effects of low-amplitude noise. As expected, nonparametric comparisons indicated that fluctuation amplitude (Fig. 1F) and AUC (Fig. 1G) were both greater in the naïve and BDNF-treated neurons compared to the ‘silent’ neurons, but were not significantly different between naïve and BDNF-treated neurons. Further analysis of the fluctuation kinetics indicated no effect on duration (Fig. 1H), a modest, but significant decrease in rise time (i.e., peak time is more negative) compared to the ‘silent’ neurons (Fig. 1I), and no effect on frequency (Fig. 1J).
BDNF-induced fluctuations of [Ca2+]I in superficial dorsal horn neurons. (A1–A3) Example recording [Ca2+]i fluctuations sampled in a SDH neuron and the processing steps used by the FIBSI event-detection program. The running median (red line in A1, window = ~ 5 s) was calculated based on the raw amplitude (AU) and time coordinate data, then the peaks below the running median were traced to form a reference line (red line in A2), and then the Ramer–Douglas–Peucker algorithm detected waveforms above the reference (red x at event peaks). (B) Left: Examples of [Ca2+]i activity detected by FIBSI in 3 different neurons. Right: The proportion of ‘silent’ neurons in the BDNF-treatment condition was significantly reduced compared to the naïve condition. Proportions were compared using a Fisher exact test. (C, D) Violin plots of the log-transformed values for fluctuation amplitude and AUC. Fluctuation amplitudes and AUC were significantly increased in naïve and BDNF-treated neurons compared to the ‘silent’ neurons, while both measures in the BDNF-treated neurons were significantly reduced compared to naïve. Neurons sampled: ‘silent’ n = 20 (849 fluctuations), naïve n = 78 (2651 fluctuations), and BDNF n = 57 (1555 fluctuations). Median with quartiles shown. Comparisons made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. (E) Relative frequency (%) of the fluctuations binned by amplitude. Amplitude values were normalized to the largest fluctuation in each neuron. (F–G) The mean fluctuation amplitude and AUC in the naïve and BDNF-treated neurons were significantly increased compared to the ‘silent’ neurons. (H–J) Further analysis revealed little to no effect of BDNF on fluctuation duration or frequency, but the fluctuations in ‘silent’ neurons exhibited longer rise times. For scatterplots F–J, the fluctuations that were < 10% the maximal fluctuation amplitude in each neuron were omitted to reduce the effects of low-amplitude noise. Scatterplots in F–J show the median with the interquartile range. All comparisons of medians in F–K were made using Kruskal–Wallis tests and Dunn’s post-hoc test. **P < 0.01, ***P < 0.001, ****P < 0.0001.
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It has been shown that gap junctions have regulatory control over network activity in the substantia gelatinosa26,27,28,29. Therefore, gap junctions may play a role in regulating the properties of [Ca2+]i across the network, but the response of individual neurons to the chronic BDNF in the culture may be masked by the complex responses of the different types of neurons to BDNF20, 30. To address this and determine the potential role of gap junctions in mediating the BDNF-induced fluctuations, we recorded [Ca2+]i from SDH neurons exposed to chronic BDNF and then applied a non-specific gap junction blocker octanol at 1 mM for ≥ 3 min. We again performed a detailed analysis using the FIBSI program and a < 10% maximal fluctuation amplitude cut-off was applied to each neuron. Inspection of the FIBSI-processed recordings revealed octanol disproportionately affected some neurons compared to others (i.e., large increases in amplitude and AUC in some neurons vs minor decreases in others). Neurons were grouped based on whether the octanol treatment caused a negative (decrease, n = 39) or positive (increase, n = 40) fold change in mean fluctuation amplitude compared to the BDNF (control) condition (Fig. 2A). Paired analyses indicated blocking gap junctions with octanol decreased the frequency of the BDNF-induced fluctuations (Fig. 2B) while increasing their duration (Fig. 2C) only in the neurons that had a positive fold change in mean fluctuation amplitude. Fluctuation rise time was also significantly decreased in the same group of neurons (Fig. 2D). These data suggest that blocking gap junctions with octanol selectively disinhibited the response to chronic BDNF in a subpopulation of SDH neurons.
Effect of gap junction blocker octanol on BDNF-induced [Ca2+]I fluctuations. (A) Scatterplot summarizing the effects of octanol on mean fluctuation amplitude and examples of FIBSI-processed recordings. The two bottom recordings show a decrease in fluctuation amplitude and the two top show an increase. (B–D) Neurons were sorted based on whether they exhibited a positive or negative fold change in mean fluctuation amplitude in response to octanol. Paired analyses showed octanol selectively and significantly decreased the mean fluctuation frequency, increased the mean fluctuation duration, and decreased the mean fluctuation rise time (i.e., more negative peak time) in the neurons with positive fold changes in mean fluctuation amplitude. Control vs octanol paired comparisons for the groups in (B–D) were made using two-way repeated measures ANOVAs and Sidak’s post-hoc test. **P < 0.01, ****P < 0.0001.
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Previous work has shown BDNF-treated SDH neurons generate synchronous [Ca2+]i fluctuations20. We verified that finding by using the FIBSI-processed recordings and comparing the degree of synchrony between neurons imaged together in naïve and BDNF-treated cultures. Adjacency matrixes were generated using Pearson r coefficients; the recording of each neuron was correlated with every other neuron imaged within the same naïve or BDNF-treated culture using a Pearson correlation matrix. The adjacency matrixes for the BDNF-treated cultures showed the neurons were more positively correlated with Pearson r coefficients closer to 1 (Fig. 3A), suggesting they exhibited more synchronous [Ca2+]i fluctuations than the naïve cultures. Indeed, we observed synchronous spikes in [Ca2+]i across multiple BDNF-treated neurons compared to the naïve neurons (Fig. 3B1–B2). Raster plots depicting the [Ca2+]i fluctuation peaks revealed the presence of multiple waves of synchronous activity in BDNF-treated neurons (Fig. 3C1–C2). To further confirm this prediction, the mean coefficient for each neuron was calculated (i.e., the mean correlation of each neuron with every other neuron in its respective culture; refer to Fig. 3A for Fisher Z transformation steps). Comparison of the mean Pearson r coefficients revealed BDNF-treated neurons were more positively correlated with other neurons in the culture than naïve neurons (Fig. 3D). Considering the role gap junctions may play in regulating network activity and synchronous activity, this result led us to anticipate that blocking gap junctions may reduce BDNF-induced synchrony within the culture. We used the same approach to measure synchrony among neurons within the same cultures treated with BDNF prior to application of octanol, and then we asked whether blocking gap junctions caused a decrease in the mean Pearson r coefficients. Indeed, octanol caused a net decrease in the mean Pearson r coefficient (Fig. 3E1,E2). Closer inspection revealed octanol decreased the mean Pearson r coefficient in 100% (25/25) of the neurons in the group with decreased fluctuation amplitudes, while only 63% (22/35) of the neurons in the group with increased fluctuation amplitudes exhibited a decreased mean Pearson r coefficient. This suggests that blocking gap junctions may actually increase [Ca2+]i fluctuation synchrony in some neurons.
Synchrony of [Ca2 +]i fluctuations in BDNF-treated cultures and the effect of octanol. (A) Example adjacency matrix constructed from Pearson r coefficients corresponding to the correlation in activity between neurons in a naïve or BDNF-treated culture, each with 8 sampled neurons. Steps to calculate the mean Pearson r for each neuron within a culture using the Fisher Z transformation are also depicted. (B1,B2) Example recordings of asynchronous [Ca2+]i fluctuations in naïve neurons and synchronous [Ca2+]i fluctuations in BDNF-treated neurons. Recordings shown were processed by FIBSI for event detection and each one was normalized (0 to 1 scale) before calculation of the adjacency matrix in A. (C1,C2) Raster plots depicting the fluctuation peak times detected by FIBSI for the recordings in B1-B2. Red stars in C2 denote the presence of synchronous waves of [Ca2+]i activity. (D) Neurons sampled from BDNF-treated cultures exhibited significantly greater mean Pearson r coefficients compared to neurons sampled from naïve cultures. Neurons were imaged from 4 naïve cultures and 3 BDNF-treated cultures, each with ≥ 8 neurons imaged. Medians shown with the 95% confidence interval. Medians were compared using a Mann–Whitney test. (E1,E2) Treatment with octanol significantly decreased the mean Pearson r in neurons sampled from cultures exposed to chronic BDNF, and this treatment effect impacted both subsets of neurons binned based on their initial response to octanol. Neurons were imaged from 4 cultures. The control vs octanol paired comparison in C1 was made using a Wilcoxon rank sum test. Control vs octanol paired comparisons for the groups in C2 were made using a measures two-way repeated measures ANOVA and Sidak’s post-hoc test. *P < 0.05, **P < 0.01, ****P < 0.0001.
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In order to further characterize the properties of the [Ca2+]i fluctuations, we used pharmacological treatments to block a variety of voltage-gated ion channels, glutamate receptors, and GABA receptors. Comparisons between the pharmacologically-treated neurons and the BDNF-treated (control) neurons revealed the [Ca2+]i fluctuation amplitudes (Fig. 4A) and frequency (Fig. 4B) are controlled by diverse mechanisms (example recordings and data available in Supplemental Fig. 2). The fluctuations were ablated when extracellular Ca2+ was removed, when all voltage-gated calcium channels (VGCCs) were blocked with Cd2+, when all TTX-sensitive voltage-gated sodium channels (VGSCs) were blocked with tetrodotoxin (TTX), when AMPA or kainite glutamate receptors were blocked with CNQX/NBQX, and when GABA was applied to the cultures. Interestingly, the fluctuations remained when we blocked T-type (with Ni2+), L-type (with nitrendipine), or N-type (with ω-conotoxin) Ca2+ channels, although the frequency was significantly reduced with no effect on amplitude. Riluzole, which blocks glutamate release and VGSCs, only affected the fluctuation frequency but not amplitude. The NMDA blocker D-AP5 was the only blocker to significantly reduce fluctuation amplitudes, but it had no effect on frequency. We also noted that applying GABA-A blockers bicuculline (10 µM) and strychnine (1 µM) to BDNF-treated slices (Supplementary Fig. 2P) abolishes higher frequency smaller amplitude fluctuations, leaving inducing the larger amplitude low frequency fluctuations that occur in naive cultures (Supplementary Fig. 2P). These data suggest the BDNF-induced [Ca2+]i fluctuations are controlled by a combination of VGCCs, TTX-sensitive VGSCs, GABA and NMDA receptors.
Pharmacology of [Ca2+]i fluctuations from BDNF-treated cultures. Summary of the effects of various pharmacological treatments on the (A) amplitude and (B) frequency of the [Ca2+]i fluctuations in BDNF-treated cultures compared to before drug treatment. Bars show the mean + standard error of the mean. *P < 0.001, paired t-test, n = 5–20 cells per condition.
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It is well established that BDNF release in the SDH is involved in neuropathic pain10, 16, 17. However, to our knowledge our findings are the first to show chronic BDNF treatment unmasks [Ca2+]i fluctuations in a subpopulation of SDH neurons while depressing activity in others. We had previously shown that sEPSCs recorded from SDH neurons are perturbed in BDNF-treated spinal cord organotypic cultures20. It is plausible that the newly discovered unmasked BDNF-induced [Ca2+]i fluctuations underlie changes in network excitability. We decided to readdress the effects of BDNF on network excitability by using the FIBSI program to quantitatively analyze a previous subset of sEPSC recordings in delay and tonic-firing SDH neurons (example recordings and low-pass filtering/matching methods are available in Supplemental Fig. 3). Inspection of the sEPSC interevent interval data indicated there might be natural clustering between single “small” sEPSCs and larger summated sEPSCs. We used partitioning around medoids (a medoid is like a centroid, but is restricted to an actual observation in the dataset) to cluster the sEPSCs in each neuron based on 3 sEPSC parameters: interevent interval (ms), amplitude (pA), and charge transfer (Q; essentially area under the curve). Optimal clustering (k medoids = 3–5 per neuron) identified 3 main clusters of sEPSCs that were present in all delay (Fig. 5A1,A2) and tonic (Fig. 5B1,B2) neurons in both control and BDNF treatment conditions (clusters named based on amplitude; interevent interval): (1) small; short, (2) small; long, and (3) large. Two other clusters were identified (small; mid and medium), but they were not present in all neurons or treatment conditions and were not analyzed for this study. We were mainly interested in the effects of BDNF on the cluster of large-amplitude sEPSCs in both neuron types (refer to Supplemental Fig. 4 for the “small” sEPSC analysis). Chronic BDNF treatment significantly decreased sEPSC amplitudes (Fig. 5C1,C2) and charge transfer (Fig. 5D1,D2) in the tonic-firing neurons. We did not observe a significant effect on sEPSC amplitudes in the delay neurons, but charge transfer was significantly decreased. The sEPSCs in the BDNF-treated delay neurons were narrower (control duration = 83.6 ± 1.2 ms; BDNF duration = 49.9 ± 0.4 ms). The sEPSC interevent intervals were significantly shorter in both neuron types (Fig. 5E1,E2), and peak time measurements indicated BDNF significantly increased activation of the sEPSCs in the tonic neuron (Fig. 5F1,F2).
Cluster analysis of spontaneous EPSCs in delay and tonic-firing superficial dorsal horn neurons. Three dimensional representations of the sEPSCs in (A1,A2) delay neurons and (B1,B2) tonic neurons plotted based on interevent interval (x-axis), amplitude (y-axis), and charge transfer (z-axis). Mapping of the clustering results and treatment shown for the total sample of sEPSCs. The sEPSCs in each neuron were clustered independently from the other neurons. Optimal partitioning around medoids identified 3 primary clusters present in all neurons in both treatment conditions (naming based on amplitude; interevent interval): small; short, small; long, and large. Two additional clusters were identified in some neurons, but were not present in all neurons in both conditions: small; mid and medium. Neurons sampled: 19 delay (5 control, 14 BDNF); 9 tonic (5 control, 4 BDNF). Total number of sEPSCs: 33,206 in delay, control; 108,261 in delay, BDNF; 40,658 in tonic, control; 36,521 in tonic, BDNF. The effects of BDNF on sEPSC amplitude, charge transfer (i.e., area under the curve), and peak time were assessed for the 3 primary clusters (large cluster shown, refer to Supplemental Fig. 3 for the two small-amplitude clusters). (C1,C2) Treatment with BDNF significantly decreased sEPSC amplitudes in tonic neurons, but a significant effect was not observed in the delay neurons. Cumulative distributions of the measured amplitudes shown (left) alongside the means ± standard error of the means (right). (D1,D2) The BDNF treatment significantly reduced sEPSC charge transfer measures in both delay and tonic neurons. (E1,E2) Interevent intervals for the sEPSCs within the large cluster were significantly decreased in the BDNF-treated neurons. (F1, F2) The sEPSC activation kinetics were significantly faster in the BDNF-treated neurons, but no significant effect was observed for the sEPSCs in the delay neurons. Large sEPSC cluster sample sizes: delay neurons control n = 753, BDNF n = 4539; tonic neurons control n = 2348, BDNF n = 2367. Comparisons between means were made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. **P < 0.01, ****P < 0.0001.
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The present study was undertaken to better understand mechanisms underlying synchronous activity among SDH neurons and the influence BDNF may have on network excitability in naïve, uninjured organotypic cultures. Our experiments indicate long-term exposure to BDNF produces a complex set of neuron-dependent changes in network excitability in the SDH. First, BDNF causes a concomitant activation of Ca2+ signaling in a subpopulation of ‘silent’ SDH neurons (~ 20% sampled) and depression of Ca2+ signaling in many others. This finding was largely unexpected as we have previously shown chronic BDNF increases the size and frequency of [Ca2+]i fluctuations20. It is plausible that the presence of the ‘silent’ neurons can greatly influence data analysis and interpretation depending on how they are grouped together, or discarded. Another possibility is that resolution differences between the two event-detection programs used (FIBSI in the current study, Mini Analysis Software (Synaptosoft, NJ)) in the previous study) could influence analysis of the [Ca2+]i fluctuations. However, the goal of the present study was not to compare/contrast the two programs. The second significant finding was that gap junction signaling may play a crucial role in regulating BDNF-induced changes in Ca2+ signaling in the SDH. Blockade of gap junctions with octanol unveiled a subpopulation of neurons that respond to BDNF with large-amplitude, low-frequency [Ca2+]i fluctuations. This finding suggests that gap junctions modulate global network activity, either by directly or indirectly inhibiting those particular neurons with increased Ca2+ signaling. Indeed, this is supported by the third major finding that BDNF profoundly increases synchrony of the [Ca2 +]i fluctuations, and blocking gap junctions with octanol reverses BDNF-induced synchrony.
It is known that delay neurons are mostly excitatory glutamatergic neurons expressing vesicular glutamate transporter 2 (VGLUT2) and that tonic neurons are mostly GABA- or glycinergic inhibitory neurons31. We also know from the pioneering studies of Perl and others that tonic neurons mainly possess an islet morphology and that delay neurons possess a radial morphology32. While the physiological correlates of the BDNF-related [Ca2+]i fluctuations are unclear, we do not think they are being driven by delay or tonic-firing neurons in the SDH. Unlike our analysis of the [Ca2+]i fluctuations where we describe the ‘silent’ neurons, we did not observe any unmasking effect(s) of BDNF on the sEPSCs in the delay or tonic neurons. All neurons sampled in the control (naïve) condition generated visually appreciable, large-amplitude sEPSCs. The main depressive effect of BDNF on the sEPSCs (refer also to Supplemental Fig. 4) mirrored the main effect of BDNF on comparable properties for the [Ca2+]i fluctuations. A summary of each dataset (Fig. 6, table) shows BDNF reduced the amplitude and area under the curve parameters for the [Ca2+]i fluctuations and large sEPSCs. More nuanced effects were observed for frequency (reciprocal of interevent interval) and activation kinetics: BDNF increased the frequency of the large sEPSCs but had little to no effect on the frequency of the [Ca2+]i fluctuations; BDNF increased activation of the large sEPSCs in the tonic neurons and had little to no effect on the delay neurons or [Ca2+]i fluctuations. However, these data collectively suggest chronic BDNF may have reduced excitability in the delay and tonic SDH neurons, and this may provide a physiological basis for the depressive effect of chronic BDNF on the [Ca2+]i fluctuations. On the contrary, the ‘silent’ neurons are predicted to have responded to chronic BDNF with [Ca2+]i fluctuations of greater amplitude, total area, frequency, and presumably activation. Our working model (Fig. 6, bottom) posits that under naïve conditions many delay and tonic SDH neurons in organotypic spinal cord cultures are generally in an active state, with some gap junction coupling with neighboring neurons. Long-term exposure to BDNF causes a homeostatic shift that pushes the SDH into a dampened state marked by synchronous, low-level activity spread across the network via increased gap junction coupling. This synchronous activity in turn mitigates the BDNF-activated ‘silent’ neurons. From our pharmacology experiments (Fig. 4 and Supplementary Fig. 1), we also suggest that activity of GABA- and glycinergic inhibitory receptor systems are necessary to support BDNF-induced Ca2+ fluctuations. Given the role of GABA and glycine in sensory processing associated with chronic pain33, it is possible that BDNF-induced fluctuations are important for pathophysiology of neuropathic pain.
Working model summarizing the effects of chronic BDNF on SDH neurons in this study.
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Several types of oscillatory and/or rhythmic bursting activity have been previously observed in spinal cord organotypic cultures34, 35 and in ex vivo slice preparations26, 36,37,38. BDNF has also been shown to induce long term potentiation (LTP) in the spinal dorsal horn39 and is critically involved in mechanisms of synaptic plasticity, which may induce synchronous activity in different pain states. Synchronous activation of groups of spinal nociceptive neurons might contribute to the ‘electric shock’-like sensations experienced by some neuropathic pain patients40. We hypothesize that [Ca2+]i fluctuations leading to synchronous network activity may be attributed to the “shooting pains” that chronic pain patients may experience. In particular, “shooting pain” is very commonly experienced by patients with radicular lower back pain (sciatica) or trigeminal neuralgia. In these patients, stimulus movement elicits so-called “traveling waves” in which neural activity sweeps across the body-part representation in somatosensory maps41. The mechanisms of this have been attributed in part to Ca2+ waves and gap junctions 41, 42. Therefore, given the properties of the [Ca2+]i fluctuations we have identified, they would make a good candidate for the neural substrate of shooting pain. In a cohort of chemotherapy-induced peripheral neuropathy (CIPN) patients, some of whom reported shooting or burning pain in the hands or feet, serum nerve growth factor (NGF) levels were much higher compared with patients with painless or absent CIPN43. Therefore, there is a precedent for the correlation of neurotrophic factors related to BDNF to the incidence of shooting pain. Indeed it has been shown that NGF regulates the expression of BDNF44. It is has also been demonstrated that BDNF overexpression induces spasticity in rodent models of spinal cord injury, which may also explain the effect of BDNF on influencing network excitability in the dorsal horn45. With regards to the involvement of the dorsal horn in shooting pain, it has been shown that dorsal root entry zone lesioning (DREZotomy) successfully reduces shooting pain caused by brachial plexus root avulsion (BPRA)46. Also, in one patient with shooting pain caused by osteoid osteoma, a partial laminectomy was able to significantly relieve pain symptoms47. Ideally however, more experiments using human ex vivo tissue are needed to determine whether these particular [Ca2+]i fluctuations correlate with shooting pain in patients.
All experimental protocols were approved by the University of Alberta Animal Care and Use Committee, which is charged with maintaining standards set forth by the Canadian Council on Animal Care.
All procedures complied with the guidelines of the Canadian Council for Animal Care and the University of Alberta Health Sciences Laboratory Animal Services Welfare Committee. Spinal cords were isolated from embryonic (E13–14) rats and transverse slices (300 ± 25 μm) were cultured using the roller-tube technique20, 22. Since tissue is obtained from embryos, sex could not be determined. Serum-free conditions were established after 5 days in vitro. Medium was exchanged with freshly prepared medium every 3–4 days. Slices were treated after 15–21 days in vitro for a period of 5–6 days with 50–200 ng/ml in serum-free medium as described previously20. Age-matched, untreated DMOTC slices served as controls.
Each organotypic slice was incubated for 1 h prior to imaging with the fluorescent Ca2+-indicator dye Fluo-4-AM (Molecular Probes, Invitrogen, Carlsbad, CA, USA). The conditions for incubating the dye were standardized across different slices to avoid uneven dye loading. After dye loading, the slice was transferred to a recording chamber and superfused with external solution containing (in mM): 131 NaCl, 2.5 KCl, 1.2 NaH2PO4, 1.3 MgSO4, 26 NaHCO3, 25 d-glucose, and 2.5 CaCl2 (20 °C, flow rate 4 ml/min). Regions of interest (ROI) corresponding to individual cell bodies of neurons were identified based on morphology and size. Changes in Ca2+-fluorescence intensity evoked by a high K+ solution (20, 35, or 50 mM, 90 s application) or other pharmacological agents, were measured in dorsal horn neurons with a confocal microscope equipped with an argon (488 nm) laser and filters (20 × XLUMPlanF1-NA-0.95 objective; Olympus FV300, Markham, Ontario, Canada). Full frame images (512 × 512 pixels) in a fixed xy plane were acquired at a scanning time of 0.8–1.08 s/frame48. In some experiments, images were cropped to accommodate faster scan rates. Selected regions of interest were drawn around distinct cell bodies depicting neurons based on morphology as described previously49 and fluorescence intensity traces were generated with FluoView v.4.3 (Olympus).
Whole cell patch-clamp recordings were obtained from neurons in organotypic slice cultures under infrared differential interference contrast optics. Neurons selected for recording were located 250–800 μm from the dorsal edge of the cultures in an area presumed to reflect the substantia gelatinosa and up to a depth of 100 μm from the surface. Neurons were categorized according to their firing pattern in response to depolarizing current commands as tonic, delay, phasic, irregular, or transient30. Recordings were obtained with an NPI SEC-05LX amplifier (ALA Scientific Instruments, Westbury, NY, USA) in bridge balance or in discontinuous, single electrode, current or voltage-clamp mode. Neurons were sampled at 2 k Hz for 180 s when measuring sEPSCs. For recording, slices were superfused at room temperature (∼ 22 °C) with 95% O2-5% CO2-saturated aCSF that contained (in mM) 127 NaCl, 2.5 KCl, 1.2 NaH2PO4, 26 NaHCO3, 1.3 MgSO4, 2.5 CaCl2, and 25 d-glucose, pH 7.4. Patch pipettes were pulled from thin-walled borosilicate glass (1.5/1.12 mm OD/ID; WPI, Sarasota, FL) to 5- to 10-MΩ resistances when filled with an internal solution containing (in mM) 130 potassium gluconate, 1 MgCl2, 2 CaCl2, 10 HEPES, 10 EGTA, 4 Mg-ATP, and 0.3 Na-GTP, pH 7.2, 290–300 mosM.
Unless otherwise stated, all chemicals were from Sigma (St. Louis, MO, USA). Fluo-4 AM dye was dissolved in a mixture of dimethyl sulfoxide (DMSO) and 20% pluronic acid (Invitrogen, Burlington, Ontario, Canada) to a 0.5 mM stock solution and kept frozen until used. The dye was thawed and sonicated thoroughly before incubating with a DMOTC slice. TTX was dissolved in distilled water as a 1 mM stock solution and stored at – 20 °C until use. TTX was diluted to a final desired concentration of 1 µM in external recording solution on the day of the experiment. Strychnine was prepared in a similar manner to TTX, and bicuculline (Tocris, Ballwin, MO, USA) was dissolved in DMSO as a 10 mM stock solution. 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX, Tocris), 2,3-dihydroxy-6-nitro-7-sulfamoyl-benzo[f]quinoxaline-2,3-dione (NBQX, Tocris) and N,N,H,-Trimethyl-5-[(tricyclo[3.3.1.13,7]dec-1-ylmethyl)amino]-1-pentanaminiumbromide hydrobromide (IEM-1460, Tocris) were prepared as 10 mM stocks dissolved in distilled water, and d(-)-2-Amino-5-phosphonopentanoic acid (d-AP5, Tocris) was prepared as 50 mM stocks dissolved in 30% 1 M NaOH. Riluzole was prepared as a 10 mM stock, kyneurinic acid and GABA as 1 M stocks, and nitrendipine as a 1 mM stock made up in distilled water. These drugs were used at a 1:1000 dilution prepared freshly with external recording solution immediately prior to the start of experiments.
Raw time-lapse calcium fluorescence intensity and whole-cell voltage-clamp recordings were analyzed using the Frequency-Independent Biological Signal Identification (FIBSI) program24 written using the Anaconda v2019.7.0.0 (Anaconda, Inc, Austin, TX) distribution of Python v3.5.2 and the NumPy and matplotlib.pyplot libraries. The FIBSI program incorporates the Ramer-Douglas-Peucker algorithm to detect significant waveforms against a time-dependent generated reference line using the least number of total points to represent said waveform, and provides quantitative measurements for each detected waveform. The calcium fluorescence traces were fit using a sliding median with a window size between 5 and 25 s, and peaks below the sliding median line were traced together to form the reference line. The voltage-clamp traces were fit using a sliding median with a window of 50 ms, and peaks above the sliding median were traced together to form the reference line. A custom Python script was used to match sEPSCs detected in filtered voltage-clamp recordings with the unfiltered recordings. The detected sEPSCs for each neuron were clustered based on their amplitude, charge transfer, and interevent interval using the partitioning around medoids function (default settings, manhattan distance used) in the cluster v2.1.0 package in R v4.0.2 (2020-06-22; R Core Team, The R Foundation for Statistical Computing, Vienna, Austria). Each neuron was clustered independently. The average silhouette width method was used to select the optimal number of clusters for each neuron. Clusters were graphed using the plotly v4.9.2.1 package with dependency on ggplot2. The FIBSI source code, custom Python matching script, and a tutorial for using FIBSI are available on a GitHub repository (https://github.com/rmcassidy/FIBSI_program).
Statistical analysis of the calcium fluctuations and sEPSCs were performed using Prism v8.2.1 (GraphPad Software, Inc, La Jolla, CA). The comparison between the proportion of ‘silent’ neurons in the naïve and BDNF treatment conditions was made using Fisher’s exact test. The Brown–Forsythe and Welch ANOVA tests and Games–Howell post-hoc test (for n > 50) were used to compare the effects of BDNF when the calcium fluctuations from all neurons were grouped together. To control for differences in recording times, the means for each fluctuation parameter (amplitude, AUC, etc.) were calculated for each neuron (fluctuations < 10% the maximal amplitude in each neuron were omitted to remove the effects of low-amplitude noise). The amplitude and AUC means were normalized across all neurons to permit direct comparisons. Normality was assessed using the D’Agostino & Pearson omnibus test, and nonparametric comparisons between group medians were made using the Kruskal–Wallis test and Dunn’s post-hoc test. Two-way repeated measures ANOVAs and Sidak’s post-hoc tests were used to compare the pre- and post-treatment effects of octanol on the calcium fluctuations. Adjacency matrixes for neurons imaged together from the same organotypic culture were constructed using Pearson correlation matrixes; the FIBSI-processed calcium fluctuation recordings were used as input (each recording was normalized on a 0 to 1 scale). Next, the mean Pearson r coefficient for each neuron within a culture was calculated using the Fisher Z transformation (steps shown in Fig. 3A). The mean Pearson r coefficients in the naïve and BDNF conditions were not normally distributed, so the comparison between the two was made using a Mann–Whitney test. Comparisons between the pre- and post-treatment effects of octanol were first made using a Wilcoxon matched-pairs signed rank test, and then using a two-way repeated measures ANOVA and Sidak’s post-hoc test. Paired t-tests were used for pharmacology experiments. Brown-Forsythe and Welch ANOVA tests and Games-Howell post-hoc test were used to compare the effects of BDNF on the sEPSC clusters. Statistical significant was set to P < 0.05 and all reported P values are two-tailed. Details for the statistical analyses can be found in Table 1.
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We thank Edgar Walters for useful discussions. We thank Klaus Ballanyi and Araya Ruangkittisakul for use of and technical assistance with confocal imaging equipment.
We acknowledge funding from the Canadian Institutes for Health Research. We acknowledge funding from the research endowment fund of the Department of Anesthesiology and Critical Care Medicine, University of New Mexico Health Sciences Center.
These authors contributed equally: Sascha R. A. Alles and Max A. Odem.
Department of Anesthesiology & Critical Care Medicine, University of New Mexico Health Sciences Center, MSC 10-6000, 2211 Lomas Blvd. NE, ACM 200, Albuquerque, NM, 87106, USA
Sascha R. A. Alles
Neuroscience and Mental Health Institute & Department of Pharmacology, University of Alberta, Edmonton, AB, Canada
Sascha R. A. Alles, Van B. Lu & Peter A. Smith
Department of Microbiology and Molecular Genetics, McGovern Medical School at UTHealth, Houston, TX, USA
Max A. Odem
Wellcome-MRC Institute of Metabolic Science, University of Cambridge, Cambridge, UK
Van B. Lu
Department of Psychiatry and Behavioral Sciences, Vanderbilt University Medical Center, Nashville, TN, USA
Ryan M. Cassidy
Open Access
Peer-reviewed
Roles Formal analysis, Investigation, Software, Validation, Visualization, Writing – original draft, Writing – review & editing
Current address: Department of Electrical and Electronic Engineering, Faculty of Engineering, University of Peradeniya, Peradeniya, Sri Lanka
Affiliations Biological Cybernetics, Faculty of Biology, Bielefeld University, Bielefeld, Germany, Cognitive Interaction Technology—Center of Excellence (CITEC), Bielefeld University, Bielefeld, Germany
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In computational modelling of sensory-motor control, the dynamics of muscle contraction is an important determinant of movement timing and joint stiffness. This is particularly so in animals with many slow muscles, as is the case in insects—many of which are important models for sensory-motor control. A muscle model is generally used to transform motoneuronal input into muscle force. Although standard models exist for vertebrate muscle innervated by many motoneurons, there is no agreement on a parametric model for single motoneuron stimulation of invertebrate muscle. Although several different models have been proposed, they have never been evaluated using a common experimental data set. We evaluate five models for isometric force production of a well-studied model system: the locust hind leg tibial extensor muscle. The response of this muscle to motoneuron spikes is best modelled as a non-linear low-pass system. Linear first-order models can approximate isometric force time courses well at high spike rates, but they cannot account for appropriate force time courses at low spike rates. A linear third-order model performs better, but only non-linear models can account for frequency-dependent change of decay time and force potentiation at intermediate stimulus frequencies. Some of the differences among published models are due to differences among experimental data sets. We developed a comprehensive toolbox for modelling muscle activation dynamics, and optimised model parameters using one data set. The “Hatze-Zakotnik model” that emphasizes an accurate single-twitch time course and uses frequency-dependent modulation of the twitch for force potentiation performs best for the slow motoneuron. Frequency-dependent modulation of a single twitch works less well for the fast motoneuron. The non-linear “Wilson” model that optimises parameters to all data set parts simultaneously performs better here. Our open-access toolbox provides powerful tools for researchers to fit appropriate models to a range of insect muscles.
Insects are important study organisms in the neuroscience of sensory-motor systems. Since the dynamics of muscle contraction and associated changes in force, torque or stiffness are central to our understanding of sensory-motor systems in general, the choice of the most appropriate model for insect muscle matters. Computational modelling of muscle properties typically separates activation dynamics from contraction dynamics. The former models the development of muscle force in response to motoneuron activity, whereas the latter describes how this force is affected by the current physical state of the muscle: its length and contraction velocity. We evaluate five published activation dynamics models for insect muscle. We explain differences between them, suggest how to decide which one to use, and provide an open-source toolbox for activation dynamics modelling. We further show that non-linear models are the best choice if: (i) the time course of a single muscle twitch is slow, (ii) the spike frequency ranges between one and thirty spikes per second, or (iii) the sensory-motor system tends to execute movements in a similar manner even if the demand on joint torque or stiffness changes.
Citation: Harischandra N, Clare AJ, Zakotnik J, Blackburn LML, Matheson T, Dürr V (2019) Evaluation of linear and non-linear activation dynamics models for insect muscle. PLoS Comput Biol 15(10): e1007437. https://doi.org/10.1371/journal.pcbi.1007437
Editor: Joseph Ayers, Northeastern University, UNITED STATES
Received: May 3, 2019; Accepted: September 25, 2019; Published: October 14, 2019
Copyright: © 2019 Harischandra et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The Matlab Toolbox IMADSim vs 1.0.4 is available on Matlab Central (https://de.mathworks.com/matlabcentral/fileexchange/72487-insect-muscle-activation-dynamics-simulator-imadsim). The experimental data are available from Bielefeld University under http://doi.org/10.4119/unibi/2937068
Funding: Supported by a grant from the Studienstiftung des Deutschen Volkes to JZ, project grants of the cluster of excellence EXC277 of the German Research Council (DFG) to NH and VD, a Biotechnology and Biological Sciences Research Council (UK) studentship to LMLB, a BBSRC research grant (BB/H014047/1) and a BBSRC Research Development Fellowship (BB/I019065/1) to TM. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The small number of motoneurons per muscle in insects has permitted detailed analyses of context-dependent adaptation of motor patterns and behaviour in several model species. Counter-intuitively, however, this small number of motoneurons poses a problem for the modelling of sensory-motor control, and the computational neuroscience of behaviour in general. This is because it necessitates accurate modelling of the spike-by-spike activation dynamics of the muscle, rather than simpler modelling based on average motor spike rate. Several muscle activation models have been proposed for insect muscle [1–4], but these have very different properties. Furthermore, although their published parameters were obtained from fits to experimental data on the same leg muscle (extensor tibiae) and, in three of four cases, the same leg and species (the metathoracic jumping leg of the locust Schistocerca gregaria [1, 3, 4]), the models yield very different force time courses when driven with identical inputs. This calls for a comparative evaluation. To identify the most reliable and computationally sound model of insect muscle activation dynamics, we provide a comprehensive muscle activation model toolbox for Matlab, comprising three linear and two non-linear models. We use this toolbox to identify differences caused by the models themselves, as opposed to effects caused by differences in the sample data used to tune the published versions of the models. To this end, we use a data set [5] comprising isometric force contraction time courses in response to metathoracic fast extensor tibiae (FETi) and slow extensor tibiae (SETi) motoneuron stimulation in the locust. In particular, we compare the model predictions of frequency-dependencies of maximum force, rise- and decay times using our own and published experimental data. Understanding arthropod muscle and being able to model its actions is important, because insects, crustaceans and spiders are widely and increasingly being used as models for biomimetic and robotic applications (e.g., flight: [6]; walking: [7, 8]; jumping: [9]). The active and passive properties of arthropod muscle provide remarkable solutions to the conflicting demands of flexibility and stability of movement, driven by relatively simple nervous control systems. Engineers struggle to manufacture equally versatile actuators, in part because the properties of muscles and their patterns of activation and modulation are still not fully understood.
Muscle models come in two types: ‘physiological’ and ‘conceptual’. Physiological models describe the molecular mechanisms underlying muscle contraction, e.g., by formal description of calcium release dynamics, binding-unbinding kinetics and diffusion processes. Such models can predict muscle forces accurately and reproduce effects such as potentiation and depression [10,11], but often contain many parameters that are unknown or difficult to determine (indirectly) from force measurements or behavioural data. Conceptual models, on the other hand, estimate the transfer function from motoneuron spike activity (input) to muscle contraction (output). Generally, they have fewer parameters because they do not model details of the underlying physiological mechanisms. In some cases, they may not model any physiological mechanism at all but instead implement an abstract transfer function [12].
The generation of muscle force is typically modelled as a two-stage process that separates the activation dynamics of muscle from its contraction dynamics [13]. Contraction dynamics describe how much force is produced given the active state and the current muscle length and any length change, taking into account the force-length-velocity characteristics of the muscle, and the elastic properties of muscle and tendon. In essence, the equations describing contraction dynamics scale the isometric force predicted by the activation dynamics model. Activation dynamics describe the transform of the neural signal to the active state of the muscle, which corresponds to the number of actin-myosin cross-bridges. This determines the muscular tension produced without muscle shortening, i.e., the isometric force. Given the finding that the force-length dependency of contraction dynamics shifts with calcium concentration [14] which, in turn, governs activation, the computational separation of the activation and contraction dynamics may be challenged. For example, Rockenfeller and Günther recently proposed a model where muscle activation depends on the volumetric density of calcium binding sites, which in turn is a function of muscle fibre length [15,16].
The present study focuses on conceptual models of muscle activation dynamics for typical isometric force measurements (varying motoneuron firing frequency at constant muscle length). A simple and very widely used model was proposed by Zajac [13]. It applies a first-order low-pass filter to transform a rectified electromyogram (EMG) signal into a force output. The “Zajac model” is sufficient for many studies in vertebrate muscle, where muscle activation is largely determined by the number of motoneurons recruited. Other first-order models have been applied to the smooth muscle of Aplysia [17,18], or to the stomatogastric system of crustaceans [19]. Variants of these first-order models have also been applied to the mesothoracic extensor tibiae muscle of the stick insect Carausius morosus [2].
In contrast to these first-order models, models that use higher-order differential equations can create a delayed response to a pulsed input such as an action potential [20,21]. The “Hatze model” uses a pair of non-linear second-order differential equations, based on theoretical assumptions regarding the transformation of the neural signal to force [20]. Zakotnik [1] introduced a variant of this model for locust extensor tibiae muscle that includes a frequency-dependent modulation of twitch shape (Fig 1). More recently, Wilson and co-workers applied a set of different models to experimental data from the same insect muscle, finding that a linear third-order model captured well the activation dynamics following slow motoneuron stimulation [3]. Subsequently, they found that a non-linear fourth-order model was still better-suited to modelling the responses to different—slow and fast—motoneuron types [4].
Fig 1. Twitch shape parameters of the Hatze-Zakotnik model.
Black to grey curves show changes in single-twitch shape with variation of parameters θ3 (A) and θ4 (B) in Eq 2. Parameter θ3 modulates the twitch peak force and decay rate at a constant area under the twitch, whereas θ4 jointly modulates peak force and single-twitch duration. Zakotnik’s extension to Hatze’s original model includes a spike-frequency-dependent modulation of parameter θ4 (see Eq 5).
https://doi.org/10.1371/journal.pcbi.1007437.g001
Three of the five models compared in the present study, including our own former work [1] and that of Wilson [3,4], have focussed on the extensor tibiae muscle of the locust hind leg, so we have used exemplar data from that muscle for this comparative analysis. The physiology and the underlying neuronal control of this muscle have been well studied [22–25]. Time constants for force production, and descriptions of history-dependent effects such as potentiation are available [26]. The muscle consists of slow, intermediate and fast fibre types and is innervated by two excitatory motoneurons, one inhibitory motoneuron, and a neuromodulatory dorsal unpaired median (DUM) neuron [26–28]. The fast extensor tibiae motoneuron (FETi) usually fires 12 to 17 spikes at up to 70 s-1 as part of a stereotyped motor sequence during jumping and kicking [29,30], but is generally not active in walking [31]. It fires up to 9 spikes during each cycle of a scratching movement [32]. In contrast, the slow extensor tibiae motoneuron (SETi) is active during both fast (jumping and kicking) and slower behaviours: for example only 2 to 3 SETi spikes per step allow the muscle to produce forces sufficient for propulsion [31]. During rhythmical scratching, SETi fires bursts of up to 30 spikes at a median instantaneous rate of 96 sp s-1, with transient peak frequencies up to approx. 400 s-1 [33]. The locust metathoracic extensor tibiae muscle is thus used in a range of natural behaviours, including walking, scratching, kicking and jumping. While the hind leg is adapted for jumping, this adaptation has more to do with muscle volume, biomechanical specialisation of the joint and motor patterns than with muscle fibre physiology. Its innervation by fast, slow and common inhibitory motor neurons is the same as that of the stick insect mesothoracic extensor tibiae muscle modelled by Blümel et al. [2].
The activation dynamics model of a muscle becomes critical when the time course of force generation is not a simple function of average motoneuron spike rate. This is the case not only for the hind leg extensor muscle of locusts: but also for insect muscles in general, which are typically innervated by only a small number of motoneurons, and where single muscle potentials can generate considerable force. We present a detailed comparative evaluation of three linear and two non-linear activation dynamics models that is important for the use of insect model systems in neuroscience. We explain why non-linear models are clearly superior whenever low and intermediate spike rates up to 30 Hz are of interest; and we demonstrate how suitable the two available non-linear models are in accounting for the properties of different motoneuron and muscle types. Moreover, we present an open-source muscle activation dynamics toolbox for research and education.
Modelling realistic time courses of muscle contraction in insects requires appropriate consideration of the characteristic properties of insect muscle, including the importance of single twitches, the long-lasting twitch force, and non-linear force potentiation. As these properties may differ considerably among particular muscle activation dynamics models, we present a comparative evaluation of five models that have been proposed in the literature. We do so in four steps. First, we focus on the time course of the single twitch because it can be considered the elementary muscle contraction event. Second, we relate the properties of the single twitch to those of isometric muscle contractions in response to arbitrary motoneuron spike rates. Third, we compare and evaluate the properties of five published models. To illustrate the differences of the models in exactly the way that they were proposed originally, we initially adhere to the published parameter sets. Fourth, we evaluate the properties of the two most accurate models after optimising their parameters to one exemplar experimental data set.
Single twitch contractions of insect muscle are generally fairly slow: for both slow and fast motoneurons, a single action potential may lead to a change in isometric muscle force for 200 ms or longer (Fig 2). In response to an SETi action potential (Fig 2A and 2B), the time course of force development in the extensor tibiae muscle was of sigmoid shape with maximum rising velocity at approximately 25 ms. The time to peak force was on average 67 ms (SD: 9 ms) so force development was always faster than force decay, which could take up to 1 s. In response to an FETi action potential (Fig 2C and 2D) the overall shape of the isometric force time course was similar to that following an SETi action potential, but peak force was considerably higher. The time to peak force was on average 61 ms (SD: 6 ms). In some experiments on FETi stimulation, a single stimulus caused additional weak contractions of unknown origin that followed the initial strong twitch (Fig 2C and 2D).
Fig 2. Modelling a single twitch.
Top row shows an isometric twitch response to a single SETi spike (blue) and corresponding model simulation results for that single twitch (solid black) using the Hatze-Zakotnik model (A) and non-linear Wilson model (B). Force is normalised to the maximum SETi-induced force of the extensor tibiae muscle of this particular animal. The spike onset is at time t = 0. The grey lines in A and C show the force responses predicted by Zajac’s first-order model, where the rise time is much shorter than in a real twitch. C, D show an isometric twitch response to a single FETi spike (red) and corresponding simulation results (black) for the same two models. Force is normalised to the maximum FETi-induced force of the extensor tibiae muscle of this particular animal. Note that for this figure, model parameters were optimised to fit the single twitch response only. In the Hatze-Zakotnik model, there were four parameters only (parameters K1 and K2 were kept constant). In the Wilson non-linear model, there were six.
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In Fig 2, both the Hatze-Zakotnik and the non-linear Wilson models were fitted to experimental recordings of single twitches. In the case of the Hatze-Zakotnik model, this involved only four parameters (θ1-θ4) because no parameters were needed to model force potentiation (see Eqs 1, 4 and 6: c(f) = 1 for f = 1, irrespective of K1 and K2). In contrast, Wilson's model requires optimisation of six parameters. Both models achieved very good fits. For example, the Hatze-Zakotnik model accurately captured single twitch force time courses with an average root mean square error (RMSE) of 1.1% of peak twitch force for SETi stimulation, and 6.6% for FETi stimulation (solid black lines in Fig 2). For comparison, modelling the twitch response with a first-order model (e.g., [13]) leads to an exceedingly short rise time and lacks the rounded peak observed in experiments (dashed lines in Fig 2). The third-order, linear Wilson model can reproduce the delayed, sigmoid increase and rounded peak [3].
Modelling force potentiation (i.e., the supra-linear increase of peak force with increasing stimulation frequency) requires a non-linearity. One way to incorporate such a non-linearity is to alter the shape of a single twitch in a frequency-dependent manner. In the case of Hatze's model (Eqs 1 and 2), this can be achieved by varying parameters that affect not only the peak twitch force, but also the rates of increase and decay (θ3) or the duration of the twitch (θ4, see Fig 1). In the Hatze-Zakotnik model, θ4 is modified by the frequency-dependent factor c(f)). As a result, summation of twitches is adjusted so as to capture frequency-dependent force potentiation.
The effect of non-linear force potentiation is best observed in the Hatze-Zakotnik model when increasing the motoneuron spike rate from 10 Hz to 50 Hz, for both slow (SETi) and fast (FETi) motoneuron stimulation (Fig 3). The increase in peak force was much larger when the stimulus frequency was changed from 10 Hz to 20 Hz than when it was changed from 20 Hz to 50 Hz. At stimulus frequencies above 20 Hz, single twitches fused into a smooth tetanic contraction for both SETi (Fig 3A) and FETi (Fig 3B).
Fig 3. Frequency-dependent modulation of single twitch shape can explain non-linear force potentiation.
Measured isometric force traces in response to SETi (blue, A) and FETi stimulation (red, B) and corresponding simulation using Hatze-Zakotnik model (black). Stimulation frequencies (Hz) are indicated to the right. Parameters θ1 - θ4 were optimised for the single twitch (as in Fig 2A and 2C) and then θ4 was optimized separately for each spike frequency. Each SETi time course could be fitted extremely well except for the first three twitches at 10 and 12.5 Hz. For FETi, the model slightly overestimates the rise time at high frequencies and underestimates the rise time at low frequencies.
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Two extensions of Hatze’s model have been proposed to account for non-linear force potentiation. For vertebrate muscle, van Zandwijk and colleagues [34] suggested scaling the peak twitch force by a sigmoid function. However, application of this Hatze-van-Zandwijk model to the insect data shown in Fig 3 proves insufficient. S2 Fig shows the result after optimising the sigmoid function parameters A and γ0 of Eq 3 by fitting force traces of all stimulation frequencies with constant coefficients θ1 to θ4. While the Hatze-van-Zandwijk model performs reasonably well for low stimulation frequencies (S2 Fig), it fails to replicate the experimental data for higher stimulation frequencies. The time taken to reach maximum tetanic force is shorter in the model than in experiments. For example, a full tetanus for SETi stimulation at 50 Hz is generated after approximately 300 ms in the model, but only after 600 ms in the experiment. In addition, force values in the model are too low at SETi frequencies above 10 Hz. Furthermore, the tetanus fuses only at frequencies above 25 Hz.
A much better fit to experimental data was obtained when the sigmoid relationship was replaced with a non-linear, frequency-dependent force potentiation. Fig 3 shows a direct comparison of measured and simulated force profiles for a representative SETi stimulation experiment, with a constant set of parameters θ1 to θ3 (optimised to single twitch) and frequency-specific optimisation of parameter θ4 (see Eq 4). The resulting model predictions match the experimental data extremely well across all stimulation frequencies. Fig 3 thus demonstrates the concept of the Hatze-Zakotnik model that scales parameter θ4 by the frequency-dependent factor c(f) (Eq 5). Table 1 lists the improvement achieved by optimisation of θ4. The fit quality improved in all cases tested. Although additional optimisation of θ3 improved fit quality substantially in one test data set, it led to much less improvement or even lead to worse results in other cases. To show how factor θ4 needs to be modulated with spike frequency, Fig 4 plots the optimal values of c(f) relative to the inter-spike interval 1/f, superimposed on the time course of a single twitch. Potentiation factor c(f) is 1 for a single twitch (i.e., no potentiation) and less than 1 for higher stimulus frequencies. For both SETi and FETi stimulation, c(f) was lowest at an inter-spike interval of 0.05 s, equivalent to f = 20 Hz (dotted line in Fig 4). The overall shape of the function graph c(1/f) resembles the shape of an inverted twitch, with its minimum approximately 15 ms before peak twitch force. During a spike train, force potentiation is thus largest if a spike occurs just prior to the maximum twitch force caused by the preceding spike. In the experimental data, this effect was consistent across all preparations.
Fig 4. Frequency-dependent scaling of single twitch force.
Values of function c(t) of Eq 6 for different SETi stimulation frequencies. Black circles and solid vertical lines indicate medians and inter-quartile ranges. Smaller values indicate a stronger potentiation of twitches, with a minimum at 20 Hz (dashed line). The smaller this value, the stronger is force potentiation (see Fig 1). As a reference, the values are superimposed on a single twitch (black curve and shaded area indicate mean and inter-quartile range of experimental data, n = 5). Maximum potentiation was achieved when a spike occurred approximately 15 ms before peak twitch force generated by the preceding spike (displacement of dashed and dotted lines). The lower scale relates frequency to inter-spike interval because in Eq 5 factor c is a function of frequency, whereas in Eq 6 it is a function of time.
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Table 1. RMS error values for three kinds of optimization runs for the Hatze-Zakotnik model prediction for SETi data (11 frequencies of 1 s duration from each animal).
Optimising θ4 (middle column) always improves fit quality compared to the reference parameters of Zakotnik (2006). Additional optimization of θ3 (right column) may or may not further improve fit quality.
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The frequency-dependent potentiation factor c(f) in the Hatze-Zakotnik model is implemented as a Michaelis-Menten function (Eq 6). The two parameters of this function were obtained by least-square fits to the experimental data from four preparations (two SETi, and two FETi stimulation experiments). As shown in Fig 5, Eq 6 approximated well the determined values for c: it was 1 for a single twitch (equivalent to t = 1 s) and reached a minimum at about t = 0.04s (24.9 Hz). Mean parameter values for SETi were K1 = 1.46·10−2 (SD: 2.3·10−3) and K2 = 3.9·10−4 (SD: 2.6·10−4). For FETi, mean parameter values were K1 = 7.8·10−2 (SD: 2.6·10−2) and K2 = 8.0·10−4 (SD: 1.8·10−4). For an overview of all model parameters see Table 2.
Fig 5. Values of force potentiation factor c as a function of inter-spike interval.
Both SETi (A) and FETi (B) panels show data from two animals (different symbols for different animals). For each symbol, factor c was calculated after frequency-specific optimisation of θ4 to measured force traces, as shown in Fig 3. Fits are Michaelis-Menten-type functions according to Eq 6 (solid and dashed lines). Each function fit was weighted by the stimulus frequency, improving fit quality at small inter-spike intervals.
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Table 2. Estimated model parameters for six locusts when fitting to SETi data (animals A, B, and D) or FETi data (animals 2, 3, and 4), using both the Hatze-Zakotnik and Wilson non-linear models.
Published model parameters are shown for comparison.
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To quantitatively compare the properties of available muscle activation models “as published”, we examined their responses to constant-frequency stimulation of the slow motoneuron SETi (Fig 6, normalised to single twitch force), the corresponding responses to more natural random spike trains (Fig 7), the dependence of maximum isometric force on stimulation frequency (Fig 8A), and the rise and decay rates in response to a step onset or offset of stimulation (Fig 9A and 9C). Additionally, we compared the properties of the two non-linear models with parameter sets for a fast motoneuron (constant-frequency stimulation: S4 Fig; random stimulation: S5 Fig; peak force: Fig 8B; rise and decay rates: Fig 9B). Note that these comparisons were based on the parameter sets that were published in the original model descriptions. Accordingly, differences may, to some extent, depend on the particular properties of the experimental data sets that had been used to obtain these parameters sets in the first place.
Fig 6. Constant frequency responses of five muscle activation models “as published”.
A, B show the time course of isometric SETi contractions at different stimulation frequencies (1–50 Hz) for two kinds of second-order, non-linear models: the Hatze-Zakotnik model [65] and the non-linear Wilson model ([4]. Note that, for immediate comparison, model output was normalised to maximum force of the single-twitch. This was set to 0.1. C-E show corresponding time courses of three published linear models: (C) [13], (D) [2] (both first order), and (E) [3] (third order). Constant frequency stimulation started at t = 0 s and persisted for 2 s. For comparison with FETi contractions see S4 Fig.
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Fig 7. Response to random activation.
Comparison of simulated isometric SETi contraction forces in response to two Poisson spike trains with mean frequencies of 20 Hz (A) and 5 Hz (B). The same models and model parameters are used as in Fig 6. Differences between models are most prominent where force potentiation is strongest, i.e., when inter-spike intervals are approximately 50 ms. The time courses predicted by the Hatze-Zakotnik and linear Wilson models are relatively similar, as are those of the two first-order models. The non-linear Wilson model deviates most strongly from the others. For comparison with FETi contractions see S5 Fig.
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Fig 8. Frequency dependence of peak isometric force.
As a summary of Fig 6 (A: SETi) and S4 Fig (B: FETi), normalised peak isometric force was plotted as a function of spike frequency for constant stimulation. Data were normalised to the peak force for a stimulation frequency of 50 Hz. For linear models, peak force linearly depends on stimulation frequency. With the published parameter sets, the Hatze-Zakotnik model has a saturating, supra-linear non-linearity for SETi stimulation, whereas the non-linear Wilson model has a sub-linear non-linearity. For FETi stimulation, both models are supra-linear, with stronger saturation for the Hatze-Zakotnik model.
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Fig 9. Half-maximal rise and decay times.
Rise time to 50% of peak force at a given constant stimulation frequency for SETi (A) and FETi (B) is shown in the top row. C, D: Decay time from peak to 50% of peak force for SETi and FETi, respectively. The results were derived for the same models as used in Figs 6–8. Linear models show no frequency-dependence of decay time, and only the third-order Wilson linear model has frequency-dependent rise time. The two non-linear models differ most strongly with regard to their decay time, particularly for FETi stimulation.
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The two first-order linear models (Zajac and Blümel) stand out in three ways: first, they cannot replicate the typical rounded peaks of muscle twitches (Fig 6C and 6D; Fig 7); second, they imply perfectly linear force potentiation with increasing spike frequency (Fig 8A); third, they show very little frequency-dependence in rise time to half-maximal force (Fig 9A)—and none at all for decay time (Fig 9C). With regard to all of these properties, they clearly differ from the properties of muscle. Examining the properties of the third-order linear Wilson model shows that some of these short-comings are a consequence of the first-order dynamics of the Blümel and Zajac models, whereas others are a consequence of linearity. For example, the linear Wilson model produces natural looking rounded twitch peaks (Fig 6E; Fig 7) and shows a strong change of rise time for stimulation frequencies between 10 and 20 Hz (Fig 9A). Both of these properties are related to the third-order dynamics of the model. In contrast, the linear Wilson model behaves the same as the other linear models in that it has perfectly linear force potentiation (Fig 8A) and lacks frequency-dependence of decay time (Fig 9C). Only the non-linear muscle activation models can capture the latter two properties of insect muscle.
With their parameter settings “as published”, the two non-linear models have considerably different properties. For example, the tetanic force for 50 Hz stimulation of a slow motoneuron reached more than thirty times the peak force of a single twitch in the non-linear Wilson model (Fig 6B), whereas it was less than ten times peak twitch force in the Hatze-Zakotnik model (Fig 6A). Differences between models are most prominent where force potentiation is strongest, i.e., when the spike frequency is around 20 Hz (Fig 7A). This is very similar for both the SETi stimulation (Fig 6) and the FETi stimulation (S4 Fig). The difference is reflected in a supra-linear (saturating) frequency dependence of maximum isometric force for the Hatze-Zakotnik model, and a sub-linear dependence for the non-linear Wilson model (Fig 8). Note that in the linear force-potentiation range (i.e., at low stimulation frequencies), the linear Wilson model behaves in a similar way to the non-linear Hatze-Zakotnik model (Fig 7A). For random FETi stimulation, the overall time courses of the two non-linear models were similar, though with considerably higher peak force for the Hatze-Zakotnik model (S5 Fig).
Although all of the higher-order models have a very similar frequency-dependence of half-maximal rise time (Fig 9A), they differ strongly in their frequency-dependence of decay time. At the end of a spike train of either SETi or FETi, measured extensor tibiae muscle force decayed in an exponential manner (S6 Fig). Similarly, all muscle activation models show an exponential decay after stimulus offset (Fig 6). The half-maximal decay time of the Hatze-Zakotnik model is strongly frequency-dependent (Fig 9C and 9D), whereas in the non-linear Wilson model it is hardly frequency-dependent in the case of SETi stimulation (Fig 9C), and shows nearly opposite behaviour to the Hatze-Zakotnik model for FETi stimulation (Fig 9D). In the latter case, decay time peaked at 10 Hz and then decreased with increasing frequency in the Hatze-Zakotnik model, whereas it reached a minimum at 5 Hz and then increased with increasing frequency in the non-linear Wilson model.
Finally, the tested muscle activation models differ strongly with regard to computational efficiency (Table 3). Whereas the two linear-first order models clearly outperform all other models, the third most efficient model is the iterative solution of the Hatze-Zakotnik model. It is only fifteen times slower than the linear first-order models, more than ten times faster than the linear Wilson model and almost 50 times faster than the non-linear Wilson model.
Table 3. Comparison of maximum CPU-time for simulating a 40 Hz spike train of 2 s followed by a relaxation time of 1 s (n = 10).
The iterative version of the Hatze-Zakotnik model is only fifteen times slower than the linear first-order models, and nearly 50 times faster than the non-linear Wilson model.
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Given the considerable differences among the published models, it was important to determine the extent to which these differences are due to the computational properties of the models, or rather due to differences in the experimental data for which the published parameter sets had been optimised. To resolve this issue, we fitted the two non-linear models to the same experimental data set, obtaining three particular solutions per model for both SETi and FETi stimulation (see Table 2 for model parameters and performance measures). Owing to the limitations of the linear models, as mentioned in the previous section, we did not consider them further.
The overall fit quality of the two models is shown in Fig 10. Both models can simulate isometric force time courses similarly well for SETi stimulation (Fig 10A and 10B). The core of the Hatze-Zakotnik model is an accurate model of a single twitch response that is then modulated in a frequency-dependent manner, so the model’s fit quality is best for low stimulation frequencies. With increasing frequency it tends to overestimate the half-maximal rise time, such that force build-up is slightly slower than in the experimental data (Fig 10A). In contrast, parameters of the non-linear Wilson model are optimised across all spike frequencies simultaneously, such that fit quality is similar for all stimulation frequencies. As a consequence, the fits to the single twitch and to responses to low-frequency stimulation are less good than those of the Hatze-Zakotnik model. For intermediate SETi stimulation frequencies, the non-linear Wilson model tends to underestimate rise time (Fig 10B). With regard to FETi stimulation, the Hatze-Zakotnik model performs less well than for SETi stimulation. It underestimates both the rise time and the maximum force for intermediate stimulation frequencies (Fig 10C). In comparison, the non-linear Wilson model achieves a better fit of the maximum force, while slightly overestimating rise time (Fig 10D). When comparing fit quality across experimental data sets, both non-linear models prove to capture the frequency-dependent force potentiation equally well for SETi stimulation (Fig 11A). Regarding FETi stimulation, the non-linear Wilson model achieves better fits (Fig 11B, compare model results with shaded area of experimental data). Despite the fact that the Hatze-Zakotnik model could replicate the saturating force potentiation curves well for SETi, optimisation did not lead to a similarly good match for FETi stimulation.
Fig 10. Comparison of non-linear models optimised to the same experimental data.
Model fits (black) to experimental data sets for SETi (A, B, blue) and FETi (C, D, red) stimulation. Both models have six free parameters. Since the non-linear Wilson model optimises the complete parameter set for all force traces simultaneously, its single-twitch fit is worse than that of the Hatze-Zakotnik model. In the latter, four parameters are optimised for the single twitch, and the remaining two describe the frequency-dependent modulation of the single-twitch time course. Constant stimulation frequencies used were: 1, 7, 10, 12.5, 15, 20, 25, 30, 40 and 50 Hz for SETi, and 1, 10, 20, 30 and 50 Hz for FETi.
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Fig 11. Frequency-dependent force potentiation.
Both non-linear models were fit to data from six preparations, three for SETi (A) and three for FETi (B). The experimental data range (N = 3) is shown in grey. Values were normalised to peak force obtained for stimulation at 50 Hz.
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As described above for the models ‘as published’ (Fig 9), frequency-dependent change of half-maximal rise time is similar in both optimised models (Fig 12A: SETi, 12B: FETi). For low frequency stimulation of two SETi data sets, the non-linear Wilson model overestimated rise time by more than 100 ms (Fig 12A). The Hatze-Zakotnik model overestimated rise time to a similar degree for one FETi data set (Fig 12B). Concerning the half-maximal decay time, the two models differ in much the same way as described for Fig 9. Whereas the Hatze-Zakotnik model shows the longest decay times for low to medium stimulation frequencies, with faster decay after high-frequency stimulation (Fig 12C and 12D), five parameter sets of the non-linear Wilson model lead to increasingly slower decay with increasing stimulation frequency. When comparing the models with experimental data, the Hatze-Zakotnik model underestimated decay time at high frequencies, whereas the non-linear Wilson model underestimated decay at intermediate frequencies of SETi stimulation (Fig 12C). S6 Fig compares the decay time courses of the Hatze-Zakotnik model with experimental data. For SETi stimulation, the model fit is very good for low frequencies (5 and 10 Hz). At higher stimulation frequencies (20 and 40 Hz) the main difference between model and experimental data was the delayed onset of decay in the experiment. As a consequence, the decay time course of the model looks very similar to the experimental time course, but the former leads the latter by approximately 70 ms. For FETi stimulation, the model differed from the experimental data in two ways: for stimulation at 50 Hz, the experimental decay again lagged the model decay; the decay of the experimental data was also considerably slower than in the model for stimulation at 10 and 20 Hz (S6 Fig).
Fig 12. Half-maximal rise and decay times.
A, B: Rise time to 50% peak force at a given constant stimulation frequency for SETi (A) and FETi (B). C, D: Decay time from peak to 50% peak force for SETi (C) and FETi (D). Both models were fit to the same experimental data sets as used for Fig 11. The experimental data range is shown in grey (N = 3). No experimental data are available for frequency dependence of FETi decay. Continuous and dashed colour lines depict best-fit results for the Hatze-Zakotnik and non-linear Wilson models, respectively.
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In summary, the non-linear models are similarly capable of replicating the properties of slow motoneuron (SETi) induced isometric contraction, with the Hatze-Zakotnik model being slightly superior with regard to the half-maximal decay time. For stimulation of the fast motoneuron (FETi), the non-linear Wilson model achieves better fits, particularly with regard to frequency-dependent force potentiation.
The choice of an appropriate muscle activation model is important whenever dynamic properties of movement sequences are of interest. In the simplest case, muscle activation is a function of the number of motoneurons recruited and can be related to the overall instantaneous firing rate. This is the rule in vertebrates, where Zajac’s linear first-order model of muscle activation [13] is usually appropriate, and isometric muscle force can be considered a low-pass transform of the rectified EMG. Although first-order muscle activation models have been applied also to muscles of molluscs, crustaceans and insects [2,17–19], they can capture neither the delayed slow rise and rounded peak of a single twitch, nor the non-linear force potentiation observed in the frequency range between 10 and 30 Hz. Our model comparison (Figs 6–9) illustrates that rounded twitch peaks and the frequency-dependent increase in half-maximal rise time typical of invertebrate muscle require the activation model to have higher-order dynamics. On the other hand, frequency-dependent potentiation and change in decay time require a non-linearity. The two models that fulfil these criteria require three times more parameters than first-order linear models, and may require considerably more computation time (Table 3). For this reason, it is worth considering when the properties of higher-order non-linear models are of particular relevance, which one of the models is preferable, and what other aspects require consideration when modelling insect muscle. The following sections address these three questions when, which and what:
Given the strong, sustained force of a single twitch (Fig 2), and the pronounced non-linear force potentiation for stimulation frequencies between 10 and 30 Hz (Fig 3), one may argue that a muscle activation model should account for these properties whenever the system to be modelled commonly experiences motoneuron frequencies below 30 Hz. On the other hand, even under these conditions, model-dependent differences in isometric force may be small compared to the strong attenuating effects of changes in muscle length and contraction velocity [13]. Thus, in movement sequences where limb kinematics require the use of a wide range of muscle lengths and contraction velocities, variation in muscle force is likely to be governed more by muscle contraction dynamics than by activation dynamics. However, animals often execute the same movement sequence with very similar kinematics, despite marked changes of the mechanical demand. For example, insects can compensate for changes in load without significant changes in kinematics. In this case, length- and velocity-dependent changes in force as described by a muscle contraction dynamics model cannot counter the altered mechanical demand. Similarly, passive muscle properties, which have a strong effect on the dynamics of limb movements in insects [1,35–38] cannot account for the compensation of altered load without an associated change in kinematics. In other words, if load compensation occurs without a corresponding change in kinematics, an appropriate change of muscle activation is strictly necessary. For example, scratching locusts can compensate for substantial additional loads to the hind leg with essentially no change in joint angle time courses [39]. They control the target-specific movement by appropriate activation of antagonist muscles [32]. In response to a change in load, the associated change in motoneuronal spike number per burst is very small, but this causes a substantial change in co-contraction of antagonist muscles and, as a consequence, a change in net joint torque [1].
In walking or climbing insects, changes in load occur naturally whenever the animal changes its body attitude or encounters a change in inclination of the substrate. For example, during steady-state walking on upward and downward slopes (±45°), stick insects neither change their average speed nor do they show strong changes in leg kinematics. Nevertheless, at the same time, joint torques change drastically with the change in body weight distribution among legs [40]. As for scratching locusts, walking stick insects alter the relative activation of antagonist muscles during the early stance phase, suggesting that they maintain similar kinematics by regulating joint stiffness rather than joint angle or velocity. In both cases, antagonist muscles tend to be activated by alternating bursts of motoneuron activity; but the onset and offset of bursts is not always well defined, and a burst of motoneuron activity in one muscle may be opposed by only a few, or sometimes even a single spike of an antagonist motoneuron. Moreover, owing to the strong and long-lasting time course of a single twitch force, antagonist muscle co-activation can arise even without co-activity of antagonist motoneurons. Thus, even if intermittent spikes between “typical bursts” are rare, the effect of non-linear force potentiation may be substantial because the decay of force after a motoneuron burst may last well into the force build-up cause by an antagonist motoneuron burst. As a consequence, the choice of an appropriate muscle activation model is particularly relevant if the motor behaviour involves co-activation of antagonist muscles, for example for context-dependent regulation of joint stiffness or net torque.
The first decision to make when selecting an activation dynamics model should be whether or not a linear model is sufficient for the purpose of modelling. As outlined in the previous section, this decision should depend on whether or not low to medium motoneuron spike rates (1–30 Hz) commonly occur in the movement being modelled. If so, this would call for appropriate consideration of non-linear force potentiation that is most pronounced in this frequency range. On the other hand, if antagonistic motoneurons are consistently firing bursts in alternation and at high frequency (>50 Hz), and temporal separation of antagonistic bursts suggests little or no overlap in the activation time courses of antagonistic muscles, neither non-linear force potentiation nor non-linear effects on force decay would be expected and a linear muscle activation model is likely to be sufficient. Whenever context-dependent modulation of antagonist co-activation occurs (see examples on load compensation above), non-linear force potentiation will be an issue.
Comparing the properties of the two non-linear models, Table 3 shows that the Hatze-Zakotnik model is much more efficient computationally than the non-linear Wilson model. Figs 10 and 11 suggest that modulating a single parameter of the single-twitch model works very well for activation by the slow motoneuron SETi, while revealing considerable mismatch for activation by the fast motoneuron FETi (Table 2; S9 Fig). Apparently, the transition from single twitch to tetanus does not follow the same physiological principles for SETi and FETi activation. Since the non-linear Wilson model does not depend on the modulation of a single twitch, it is possible to find suitable parameter sets for both SETi and FETi activation.
For the slow motoneuron, the Hatze-Zakotnik model has the advantage of being based on a physiologically plausible underlying concept. Twitches are maximally potentiated at a stimulus frequency of 20 Hz (Fig 4), which corresponds approximately to the time from twitch onset to twitch maximum; i.e. maximal potentiation occurs when the following twitch is triggered at the maximum of the preceding twitch. This can also be observed in cat gastrocnemius muscle [40, 41]. Here, potentiation occurs if an additional spike is triggered during the twitch-falling phase, due to reaction kinetics of calcium dynamics, and this leads to increased twitch amplitude and slower force decay. Studies of calcium dynamics in single barnacle muscle fibres show that enhanced release of calcium is the main reason for twitch potentiation [41].
In the Hatze-Zakotnik model, calcium release is included in the second differential equation (Eq 2), in which a single parameter is modulated through a bi-sigmoid Michaelis-Menten type equation (Eq 6). As a formal description of calcium release and removal, it is an intuitive conceptual model of the process underlying the potentiation of twitch force. Frequency-dependent modulation of twitch shape introduces only two additional parameters, and the low complexity makes optimisation of the model parameters to experimental data feasible. This leads to non-redundant solutions, which can be directly compared for different muscles or motoneurons. An extension to the model of muscle force which includes more detail and additional observations can be implemented by including twitch potentiation state as a more complex model parameter. For example, a framework for decomposition of tetanic forces into a series of individual twitches of different sizes was proposed by [42].
As the present study focuses on properties of isometric force time courses under the assumption of no change in muscle fibre length, it is important to note that muscle activation dynamics gets more complicated as soon as muscle is allowed to shorten, or if isometric force time courses are to be compared for different muscle lengths. This is because the force-length dependency of contraction dynamics depends on the level of activation (e.g., [14]). In a recent study, Rockenfeller and Günther discussed a range of activation dynamics models that include length-dependency [16] that transform the normalised calcium concentration γ(t) (Eq 4) into a length-dependent active state q(γ, l), where l denotes the relative fibre length. In their own model and in all of Hatze’s own variants [20,43,44], the nonlinear transform q contains the product of γ and a lever function ϖ(l) (see Table 2 and Eqs 2 and 3 of [16]). Thus, under the assumption of constant fibre length, all of these length-dependent activation dynamics models scale the normalised calcium concentration γ with a constant factor ϖ fix. Since none of the nonlinear transforms used in the three model variants discussed here includes length-dependency (Hatze-Zandwijk, Eq 3; Hatze-Zakonik, Eq 5; Wilson non-linear, Eq 8) they can be related to other models by setting ϖ (l) = ϖ fix = 1. Given the non-linear, frequency-dependent force facilitation shown in Fig 11 or S7 Fig, a non-linear transform q(γ) is necessary even without any change in muscle length. Future experiments will need to elucidate the potential interaction of motoneuron firing frequency on the one hand and muscle fibre length on the other.
Both non-linear muscle activation models can account for frequency-dependent, non-linear force potentiation (Fig 10), however, they do not include long-term potentiation of twitches. For example, Brown and colleagues [46] modelled the sag effect, i.e., a slow decay in muscle force with long stimulation times, by including increased calcium removal at different stimulus frequencies. The effect of sag on twitches (cf. Fig 4 in [46]) resembles twitch modulation in the Hatze-Zakotnik model, so future work could lead to an incorporation of long-term potentiation into this model.
Another effect not included in current models is long term potentiation of force after a break [26] and the catch-effect, i.e., a prolonged increase in force production resulting from as few as one spike in an otherwise constant frequency stimulus sequence (p. 238 in [47]). Catch-like tension is thought to be based on complex calcium dynamics in the muscle [10].
A critical aspect of muscle models in general is that even an optimal set of model parameters can only account for the most typical behaviour in the face of very strong inter-individual variation of muscle properties [48]. Blümel and colleagues showed that for stick insect extensor tibiae muscle contraction dynamics, the use of individually fitted model parameters can halve the error of model estimates [2].
Our own experimental data also suggest that inter-individual variation can result in a considerable range of parameters of activation dynamics (see parameter ranges of the three per-animal fits in Table 2). As a consequence, individually optimised parameter sets lead to different half-maximal rise and decay times (Fig 11), whereas force potentiation appears to be affected less (at least for a given type of model, Fig 10). The fact that the Hatze-Zakotnik and non-linear Wilson models differ substantially more when using the parameters “as published” (Figs 6 to 9; S4 and S5 Figs) than after parameter optimisation to the same data set (Figs 10–12) also indicates that variation among experimental data is substantial. To illustrate this further, we compared the peak forces and half-maximal rise and decay times among three experimental data sets obtained from the same muscle in the same leg of the same insect species, i.e., the hind leg extensor tibiae muscle of the locust Schistocerca gregaria ([4,26] and our own data). Frequency-dependent potentiation of peak force was different in our experimental data compared to that of [4,26] for both SETi and FETi (S7 Fig). Our data showed stronger force potentiation at low frequencies. The situation for half-maximal rise time was quite different however: here, the data set of [26] stood out, with up to ten times slower rise and decay of force for low and medium frequencies compared to the data of [4] and our own (S8A Fig). Variation among data sets was smaller in the case of half-maximal rise and decay times during FETi stimulation (S8B Fig).
Physiologically meaningful differences in experimental data both within and between studies may be attributed to a number of factors including inter-individual or genetic strain differences in neuromuscular function, diurnal changes in physiology, the effects of circulating neuromodulators, or direct muscle inhibition. Differences in methodological procedures in different studies could markedly affect modulatory effects in particular. For example, levels of the insect neuromodulator octopamine are elevated following stress [49,50] which may vary among animals and between studies. Octopamine is also released peripherally from Dorsal Unpaired Median (DUM) neurons during particular types of behaviour such as flight or kicking [51,52]. Octopamine modulates both neuromuscular transmission and muscle contractile properties [53]. Moreover, Common Inhibitor motoneurons (for review see [54]) release γ-aminobutyric acid (GABA) onto insect skeletal muscles, which influences muscle contraction and relaxation dynamics [55–57]. In the locust, Common Inhibitor activation reduces extensor muscle relaxation times [58] and may reduce the force generated by excitatory motor spikes, thus facilitating fast cyclical leg movements [56,57,59]. For pharmacological induction of muscle relaxation related to inhibitory innervation see [60].
In summary, much of the difference between the two published non-linear activation dynamics models is due to considerable differences between the experimental data sets, but this inter-individual variation is an important aspect of muscle physiology. In the face of this inter-individual variation, a test for generality of any computational model involving muscle properties will require systematic variation of model parameters within the documented parameter ranges. Our muscle activation toolbox for Matlab [61] and the corresponding experimental data set [5] will facilitate the comprehensive and computationally efficient use of different activation dynamics models, and will also help us to learn more about variation of muscle properties in different preparations, types of muscle, and species.
Experiments were carried out on adult male and female Schistocerca gregaria, from crowded colonies at the Department of Zoology, University of Cambridge, UK or Department of Biology, University of Leicester, UK. Locusts were fixed in modelling clay, ventral side uppermost. The right hind leg was immobilised with dental cement (Protemp, ESPE) with the femur at right angles to the body, and the femoro-tibial angle initially set at 140° or 90°, for SETi and FETi experiments respectively to facilitate the subsequent attachment of the extensor muscle to the force transducer. For SETi experiments, a window was cut in the distal end of femur, and the accessory flexor muscle, overlying trachea and air sacs were removed. The end of the apodeme of the extensor muscle was grasped with a pair of forceps attached to a force transducer (see below) and the apodeme was cut distal to the forceps. Stimulation of FETi causes high extensor muscle forces that can fracture the extensor muscle apodeme at the point where it is grasped by forceps. To avoid this in FETi experiments, the proximal tibia was first braced in an aluminium sleeve which was then tightly attached to the force transducer with suture thread, such that the attachment point was aligned with the axis of pull of the extensor muscle. The distal femoral cuticle of the femoro-tibial joint was dissected away, so that the extensor muscle remained attached solely to the transducer through its natural point of attachment to the dorsal proximal tibia. For both SETi and FETi experiments, the extensor muscle length was adjusted to correspond to 90° femur-tibia angle before stimulation.
A window was cut into the ventral thoracic cuticle, and overlying air sacs were removed. Contractions in the extensor tibiae muscle were elicited by stimulation of Nerve 3b (N3b) or Nerve 5 (N5) in the thorax, for SETi and FETi respectively, through a pair of 50 μm silver hook electrodes placed under the nerve and insulated with petroleum jelly. Stimulation strength was set just above the threshold for eliciting a single twitch reliably. The axon of common inhibitor motoneuron CI1 runs in the same nerve as that of SETi, but it was not activated in these experiments. This was confirmed by making intracellular recordings from slow extensor tibiae muscle fibres during N3b stimulation. Such recordings revealed excitatory junctional potentials (from SETi) but no inhibitory potentials [62].
Stimuli were generated using a Master 8 stimulator (AMPI, Jerusalem, Israel). Series of pulses at various frequencies were delivered for 10 s or 1 s, for SETi and FETi respectively, with a 60 s gap between each stimulus train to allow the muscle to recover. FETi stimulus trains were restricted to 1 s to prevent damage to the apodeme insertion point. Muscle forces were measured using an isometric force transducer (Model 305C, Aurora Scientific, Canada), digitised at 5 kHz using a micro1401 interface and Spike 2 software (both Cambridge Electronic Design, Cambridge, UK).
To focus on the time course of muscle activation, measured forces were normalised to the interval [0,1], where 1 corresponds to maximum measured force per animal. They were not filtered. In the model, the stimulus protocol was shifted by a fixed delay (e.g. 10 ms) to account for the time taken by neural signals to be conducted to the muscle from the point of nerve stimulation.
Either constrained Levenberg-Marquardt or trust-region-reflective optimisation algorithms, implemented in Matlab (Mathworks Inc, Natick, USA), were used to fit the model parameters. These algorithms minimised a least squares error function that captured the distance between the measured and stimulated forces. The fitting procedure was repeatedly initialised with randomly distributed values to avoid local minima. Both algorithms were applied to the same experimental data set, and the one with the better performance (lower error) used for further analysis. This is mentioned in the data structures of the optimised data provided with the supplementary MATLAB IMADSim toolbox [61]. In the following sections, we introduce the five muscle activation models compared in this study.
Hatze [20, 44] proposed a second-order model comprising a pair of coupled second-order inhomogeneous differential equations, to capture two stages of signal processing (Eqs 16 and 17 in [20]; Eqs 3.21 and 3.22 with ρ*(ξ) = 1 in [44]). The first stage describes the transformation of neural activity α(t) at the motor endplate to membrane potential β(t) in the T-tubular system of a muscle. α(t) is zero except for 1 ms periods in which each motoneuron spike is modelled as a half sine wave, idealising the depolarised portion of an action potential (S1 Fig). When entering the T-tubular system, the action potential α(t) is transformed to signal β(t) according to the differential Eq 1: (Eq 1)
The second stage of the model transforms β(t) into the intracellular free ionic calcium concentration [Ca2+]i and therefore models calcium release and re-uptake by the sarcoplasmic reticulum. Differential Eq 2 determines the calcium concentration γ(t): (Eq 2)
Both processes can be viewed as over-critically damped, second-order systems, each with a different parameter set [20]. See Part I of the Supplementary Appendix S10 for a numerical solution for γ(t).
The relationship between the concentration of released calcium and the active state of the muscle is typically described by a sigmoid function as measured in [63] and [64]. Hatze [20] proposed that the active state also depended on muscle fibre length (his Eq 14), and varied this dependency in different versions (for a detailed treatise of this length dependence and its significance, see [16]). Given the lack of experimental data with systematic co-variation of both motoneuron frequency and muscle fibre length, the present study focuses on variants of the Hatze model that assume muscle fibre length to remain constant. For example, van Zandwijk and colleagues [45] expanded the Hatze model by including a sigmoid relationship between the calcium concentration, i.e., γ(t), and the force-producing active state: (Eq 3) where A and γ0 describe the slope and offset of the sigmoid, respectively. Owing to an unsatisfactory fit of the Hatze-van-Zandwijk model to isometric force measurements in insect muscle (S2 Fig), Zakotnik [65] replaced van Zandwijk’s sigmoid (Eq 3) with a frequency-dependent potentiation factor c(f) (hence the name Hatze-Zakotnik model). Reordering of Eq 2 shows that the calcium concentration γ is divided by the parameter value θ4:
(Eq 4)
Therefore, if value θ4 is decreased, the twitch force γ increases and has a slower decay. In contrast, parameter θ3 controls the twitch shape such that tetanic force does not change (Fig 1). To model force potentiation in the Hatze-Zakotnik model, θ4 is multiplied by a factor c(f) that depends on the instantaneous motor spike frequency f: (Eq 5)
Assuming that there is no potentiation for a single twitch (f = 1, c(1) = 1), values for c(f) can be determined by optimising parameter θ4 for each stimulation frequency independently and dividing this value by the measured θ4 at single twitch stimulation. Note that, for this procedure, the other three parameters, θ1 to θ3, remain fixed.
Because calcium kinetics are thought to be the main reason for force potentiation, a Michaelis-Menten-type equation is used for c(f). It is related to a [Ca2+]i reaction process that potentiates a twitch, and a calcium pump that reduces both [Ca2+]i and twitch size. The value for c(t) depending on the time since the previous stimulus t = (1/f) is determined as: (Eq 6)
Eq 6 relies on two parameters K1 and K2, for which we assume that K1 ≥ K2 and K1, K2 ≥ 0. Therefore, it is constrained to the interval [0, 1] and converges to 1 for t → ∞. The first term can be related to the force-producing reaction; i.e., larger values of K1 produce an elevated and prolonged twitch force. The second term can be related to the removal of calcium, i.e., larger values of K2 reduce twitch potentiation.
The parameters, K1 and K2 were also optimized by using either constrained Levenberg-Marquardt or trust-region-reflective algorithms, to fit the c(f) curve. It should be noted that if the number of data points (here, the number of different stimulation frequencies) is small, the fitted curve could deviate slightly more at certain frequency points (see the IMADSim optimisation example given in the supplementary toolbox [61]). Ultimately, this will affect the performance of the full six-parameter Hatze-Zakotnik model. For example, in Fig 10A and 10C, a larger deviation can be seen in FETi force prediction (where 5 frequencies were used for optimisation) compared to that of SETi (with 10 different frequencies).
We compare the Hatze-Zakotnik model with another non-linear second order model that was proposed by Wilson and colleagues [4]. The authors adapted and simplified the model of Ding [66] to obtain second order dynamics, and used it to describe the isometric response of locust skeletal muscle to SETi stimulation [4].
The Wilson model gives the muscle force F(t) and takes a pulse train u(t) as an input. The model equations are: (Eq 7)
(Eq 8)
(Eq 9) , where
(Eq 10)
Here, n defines the number of input pulses, t the time and ti the time at which the ith pulse occurs. This accounts for the assumption that the input pulses can be approximated as impulses. Therefore, the motoneuron spike is modelled as a square pulse or half-sine wave of width 1 ms, scaled to have an area of 1 under the pulse. Like the Hatze-Zakotnik model, the non-linear Wilson model has six parameters, but it consists of first-order differential equations. The variable x(t) is an intermediate stage in the model and represents a non-linear saturation. The parameters m and k define the shape of this non-linearity. A measure of [Ca2+]i is represented by the variable CN. The parameters τc, τ1 and τ2 are time constants, and A is a gain [4]. Note that Eqs 7 and 8 are very similar to a computationally efficient approximation of the original model as formulated by Hatze (see Eqs 3.27 and 3.29 to 3.31 in [44], where the normalised calcium concentration CN(t) is the equivalent to γ(t) and the nonlinear transform x(t) is equivalent to q(γ(t)) without length-dependency).
The parameter set for each animal (i.e. muscle) was estimated by minimising the least squares error between the measured force and the model output for the entire set of trials per animal, i.e., for single twitch and all stimulus frequencies. As described by Wilson et al. [4], the fitting procedure was initialised repeatedly seven times, with random parameter values drawn from a normal distribution. The optimisation was done using the MATLAB function lsqnonlin and the trust-region-reflective algorithm was used to find the best fit to the data. The system of differential equations was solved using a fixed step (0.0002 s or 5 kHz sampling rate), fourth-order Runge-Kutta method.
In a reference work on muscle modelling, Zajac [13] proposed a linear first-order model for activation dynamics. The envelopes of the rectified electromyogram (EMG) and of the low-pass filtered, rectified EMG, can be related to the neural-excitation input signal u(t) and the state variable associated with muscle activation a(t), respectively. It is a two-parameter model described by the following bilinear differential equation: (Eq 11)
(Eq 12) where 1/τact is the higher rate constant (when u(t) = 1) and β is the ratio of that over the lower rate constant, 1/ τdeact, when u(t) = 0 (relaxation). In other words, the model assumes that the build-up of activation of a fully excited muscle is faster than relaxation after termination of activation. In the case of the Zajac model, we use an iterative method for solving the differential equation numerically, as described in part II of the S1 Text.
Blümel and colleagues [2] proposed a simple model of activation dynamics for the stick insect mesothoracic extensor tibiae muscle. First-order low-pass filtering reproduces many aspects of isometric contractions in this muscle [67], so the activation dynamics are described by a single-pole, first-order low pass filter. The standard recursion equation for such a filter was used to obtain the following two-parameter model: (Eq 13) where filter sets the decay amplitude per time step and scaling is a factor that multiplies the input by a constant. u[n]and a[n] correspond to neural excitation and muscle activation, respectively. The time constant, tconst, of the filter depends on the time step duration and is related to the parameter filter according the following equation:
(Eq 14) where Δt is the time step duration. In our case, Δt was 0.0002 s in all simulations.
Wilson and colleagues [3], found that a third-order model was of optimal order for fitting isometric force responses over a range of input pulse frequencies. Their linear third-order model of activation dynamics is characterised as: (Eq 15) where a(t) is the muscle force and u(t) is the input pulse train. The model has four parameters, θ1to θ4. Here, we converted the third-order differential Eq 15 into a system of three first-order differential equations and solved it by using a fixed-step (0.0002 s), fourth-order Runge-Kutta method.
For quantitative comparison of the five models described above, with particular focus on their responses to constant frequency pulse trains, we developed a MATLAB toolbox IMADSim, or “Insect Muscle Activation Dynamics Simulation”. The toolbox consists of Matlab routines for simulating the models for arbitrary pulse trains and arbitrary parameter combinations. It was created in Matlab2009b (7.9.0) and comes with an additional installer file for Matlab versions above 2014 [61].
A graphical user interface (GUI) is provided so that a user can select a muscle activation model, adjust model or simulation parameters easily, and visualise the output (S3 Fig). The GUI permits the selection of one of two motoneuron types (SETi or FETi), spike shapes (half-sinusoidal or square) and spike generators (either constant frequency or Poisson). Arbitrary spike trains may be loaded from a file (in .mat or .txt format, comprising spike times). For constant spike frequencies, single or multiple spike frequencies can be set along with a relaxation time. There are three displays showing: (i) the time course of isometric force generation, (ii) the corresponding spike train, and (iii) the model equations. Users can save all graphs generated, as well as the corresponding data (Matlab file formats .fig or .mat).
The toolbox also includes parameter optimisation routines for the two non-linear models, i.e. the Hatze-Zakotnik and non-linear Wilson model, which are based on the Matlab LSQNONLIN solver for non-linear least squares problems. When calling the functions, several arguments can be set to customise the optimisation. If no arguments are given, default values are used. The toolbox assumes that experimental data are sampled at 5 kHz.
In addition to the toolbox, we provide experimental data from six adult female locusts (Schistocerca gregaria), comprising isometric contraction force time courses of the metathoracic extensor tibiae muscle for spike trains with different frequencies (http://doi.org/10.4119/unibi/2937068, [5]). In three animals, the slow extensor tibiae motoneuron (SETi) was stimulated. For another three animals, the fast extensor tibiae motoneuron (FETi) was stimulated.
For more details about the toolbox, functional routines, and examples, the reader is referred to the documentation file “IMADSim_Documentation.html” in the Supplementary Material [61].
Stimulus shape recorded in experiments (solid line) and used in the model (dashed line). The model stimulus is a half sine wave of length 1 ms and approximates well the depolarising phase of the signal.
https://doi.org/10.1371/journal.pcbi.1007437.s001
(TIF)
Blue lines show isometric force measurements for locust extensor muscle at different SETi stimulus frequencies are shown in the left panel (same data as in Fig 3). Forces were normalised to maximum force at a stimulation frequency of 50 Hz. Note that the tetanic force level difference between e.g. 10 and 20 Hz is larger than the difference between e.g. 40 and 50 Hz, which indicates a non-linear summation of single twitch forces. Black lines show simulated forces using the Hatze-van-Zandwijk model (Eq 1 to 3). The four parameters of Hatze’s original activation dynamics model were optimised to the single twitch of A. Van Zandwijk proposed a sigmoid scaling function (Eq 3, see inset on the right) to model the relationship between the calcium concentration and the force-producing active state. Comparison with the experimental data (blue lines) shows that the model fit is poor. For example, the tetanus fuses only at frequencies above 25 Hz and the rise time at higher stimulation frequencies is too short.
https://doi.org/10.1371/journal.pcbi.1007437.s002
(TIF)
The main panel of the GUI permits the selection of one of five published muscle activation models (two non-linear and three linear). The user can manually alter all corresponding model parameters, select one of two motor neuron types (SETi and FETi), toggle between two spike shapes (sine and square) and select one of two methods for spike generation. Experimental spike time series may be loaded from an external Matlab or text file that contains a list of individual spike times. For constant frequency stimulation, one or more spike frequencies may be set (1, 5 and 20 Hz, in the example shown). Post-stimulation relaxation time may also be set. Three displays show: (i) the time course of isometric force generation, (ii) the corresponding spike train, and (iii) the model equations. Finally, “Hold on”, “Run” and “Reset all” buttons are used to keep the multiple time courses on the display, run the simulation, or reset to default settings, respectively. Generated graphs and the corresponding data can be saved to Matlab figure (*.fig) and data (*.mat) files by selecting options in the toolbar.
https://doi.org/10.1371/journal.pcbi.1007437.s003
(TIF)
Time courses of isometric force contractions for different trains of constant frequency stimulation are shown for the two non-linear models with parameters ‘as published’ according to Hatze-Zakotnik (A) and Wilson (B). Note that model output was normalised to maximum force of the single-twitch. This was set to 0.1. Same figure details as in the top row of Fig 6, except that here the fast motor neuron (FETi) was simulated.
https://doi.org/10.1371/journal.pcbi.1007437.s004
(TIF)
Time courses of force decay after 10 s of constant frequency SETi stimulation and 1 s of constant frequency FETi stimulation were superimposed for the model (black) and experimental data (coloured). A: SETi, blue. B: FETi, red. The onset of the last stimulus spike is set at t = 0. Numbers at the start of decay indicate stimulation frequencies in Hz. Although the shape of the force signal is similar in model and experiment for SETi stimulation, the experimentally measured decay after stimulation with 20 or 40 Hz lags the onset of the modelled decay. The same is true for FETi stimulation at 50 Hz. For stimulation frequencies of 10 and 20 Hz, the experimentally measured FETi time courses show much slower decay than those computed by the model.
https://doi.org/10.1371/journal.pcbi.1007437.s006
(TIF)
Three different data sets are compared for SETi (A) and FETi (B) stimulation at different frequencies. The data sets comprise our own experimental data (black; 3 animals, per motoneuron), data published by [26] (SETi: cyan; his Fig 3F for single twitch and Fig 20 for inner and outer muscle fibre bundles; FETi: magenta; his Fig 3B for single twitch and Fig 15B for outer muscle fibre bundle), and data published by [4] (SETi: dashed blue; their Fig 2E; FETi: dashed red; their Fig 2F). Forces were normalised to the peak force at stimulation frequency 50 Hz separately for SETi and FETi.
https://doi.org/10.1371/journal.pcbi.1007437.s007
(TIF)
Model fits (black) of the Hatze-Zakotnik model (top) and non-linear Wilson model (bottom) to experimental data sets for SETi (blue) and FETi (red) stimulation. Plots for animal D (SETi) and animal 4 (FETi) show the same data in Fig 10 except that forces were not normalised to maximum force at 50 Hz stimulation frequency. For model parameter sets used see Table 2. Constant stimulating frequencies used were: 1, 7, 10, 12.5, 15, 20, 25, 30, 40 and 50 Hz for SETi, and 1, 10, 20, 30 and 50 Hz for FETi.
https://doi.org/10.1371/journal.pcbi.1007437.s009
(TIF)
We thank Greg Sutton for help with FETi data acquisition, Ansgar Büschges and colleagues for critical discussion of our results, Scott Hooper for helpful comments on an earlier version of the manuscript, and Robert Rockenfeller for constructive criticism.
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Brain-derived neurotrophic factor (BDNF) is critically involved in the pathophysiology of chronic pain. However, the mechanisms of BDNF action on specific neuronal populations in the spinal superficial dorsal horn (SDH) requires further study. We used chronic BDNF treatment (200 ng/ml, 5–6 days) of defined-medium, serum-free spinal organotypic cultures to study intracellular calcium ([Ca2+]i) fluctuations. A detailed quantitative analysis of these fluctuations using the Frequency-independent biological signal identification (FIBSI) program revealed that BDNF simultaneously depressed activity in some SDH neurons while it unmasked a particular subpopulation of ‘silent’ neurons causing them to become spontaneously active. Blockade of gap junctions disinhibited a subpopulation of SDH neurons and reduced BDNF-induced synchrony in BDNF-treated cultures. BDNF reduced neuronal excitability assessed by measuring spontaneous excitatory postsynaptic currents. This was similar to the depressive effect of BDNF on the [Ca2+]i fluctuations. This study reveals novel regulatory mechanisms of SDH neuronal excitability in response to BDNF.
Injury to, or disease of, the somatosensory system frequently generates chronic and sometimes intractable neuropathic pain1, 2. In experimental animals, peripheral nerve damage, such as that generated by chronic constriction or section of the sciatic nerve, induces pain-related behaviours that serve as a model for human neuropathic pain3, 4. Seven or more days of sciatic nerve injury promote an enduring increase in the excitability of first order primary afferent neurons5,6,7,8. These become chronically active and release a variety of mediators (cytokines, chemokines, neuropeptides, ATP and growth factors) that predispose spinal microglia to a more ‘activated’ state9,10,11,12,13,14. These in turn, release further mediators, including brain derived neurotrophic factor (BDNF) that promote a slowly developing, but persistent increase in excitability of second order neurons in the spinal dorsal horn. This ‘central sensitization’ is thought to be responsible for the allodynia, hyperalgesia, spontaneous pain and causalgia that characterize neuropathic pain3, 15. Spinal actions of BDNF involve alteration in Cl- gradients such that the normally inhibitory actions of GABA become excitatory16, 17. There is also increased excitatory synaptic drive to putative excitatory neurons9, 18. This results in an overall increase in excitability as monitored by confocal Ca2+ imaging in organotypic cultures of rat or mouse spinal cord9, 18,19,20. Despite this, the long-term effects of upregulated BDNF on neuronal plasticity in the superficial dorsal horn (SDH) are not fully understood.
In our previous work, we developed and used a model to study neuropathic pain whereby defined-medium spinal organotypic cultures are treated long-term (5–6 days) with BDNF (200 ng/ml)9, 20,21,22,23. BDNF treatment is administered at day 20 or more after organotypic cultures are established to allow neurons Xsplit Gamecaster Keygen mature to a more ‘adult-like’ state. We demonstrated that lamina II neurons in these organotypic cultures are similar to those in acute slices from young adult rats21. We also showed that the changes that occur in spinal organotypic dorsal horn neurons ‘mirror’ those observed in dorsal horn neurons from acute slices obtained from chronic constriction injury (CCI) rats at peak mechanical hypersensitivity levels20. Naïve cultures display spontaneous oscillatory changes in intracellular Ca2+ levels [Ca2+]i and we have previously shown that the amplitude and frequency of these changes in [Ca2+]i are profoundly increased following 5–6 days treatment with BDNF20.
We used this model and a new Frequency-independent biological signal identification (FIBSI)24 program to quantitatively measure the fluctuations of [Ca2+]i and examine their synchronicity. Unexpectedly, we observed two opposite effects of BDNF that appeared to occur simultaneously. First, BDNF caused an overall decrease in fluctuation size. Second, we noticed a particular population of SDH neurons in naïve cultures that did not display typical, marked fluctuations of [Ca2+]i. This population of ‘silent’ neurons was never seen in BDNF-treated cultures, suggesting that BDNF unmasks these ‘silent’ neurons and causes them to become spontaneously active. We next investigated the role of gap junctions in mediating the [Ca2+]i fluctuations; application of the gap junction blocker octanol to chronic BDNF-treated neurons revealed a subpopulation of neurons that generate low-frequency, large [Ca2+]i fluctuations. Further pharmacological experiments indicated the [Ca2+]i fluctuations in active neurons are regulated by diverse mechanisms including voltage-gated calcium and sodium channels as well as GABA and NMDA receptors. Finally, we used FIBSI to reanalyze a previous dataset of spontaneous excitatory postsynaptic current (sEPSC) recordings from BDNF-treated dorsal horn neurons in order to qualitatively compare the effects of BDNF on the sEPSCs and [Ca2+]i fluctuations in SDH neurons. The depressive effects of BDNF on the sEPSCs in delay and tonic-firing SDH neurons were consistent with the effects on the [Ca2+]i fluctuations. This study reveals novel mechanisms of BDNF regulation of dorsal horn excitability, which has implications for the study of chronic neuropathic pain physiology.
In previous work we identified that chronic BDNF treatment (200 ng/ml, 5–6 days) induces Ca2+ fluctuations compared to naïve cultures20, 25 (Supplementary Video 1A, B). In this study, we performed a detailed quantitative analysis of the fluctuations in [Ca2+]i using Fluo-4 AM Ca2+ imaging and the FIBSI analysis program24 to further establish the nature of BDNF-induced changes in lamina II neurons of the SDH. Refer to Fig. 1A1–A3 for an example of [Ca2+]i fluctuations detected by the FIBSI program. Inspection of the FIBSI-processed recordings revealed a subset of naïve neurons, which were relatively ‘silent’ (Fig. 1B, left; raw recordings in Supplemental Fig. 1) displaying only small amplitude fluctuations for the duration of the recording (‘silent’ = 1.7 ± 0.04 AU; naïve = 53.7 ± 1.6 AU; BDNF = 25.5 ± 0.7 AU). No ‘silent’ neurons were found in the BDNF treatment group (Fig. 1B, right). Refer to Table 1 for detailed statistical information. This stark contrast suggested BDNF may be unmasking, or activating, these ‘silent’ neurons. We grouped the fluctuations in the ‘silent’ neurons together for further comparison to the fluctuations in the other active naïve neurons and BDNF-treated neurons. Log-transformed fluctuation amplitudes (Fig. 1C) and area under the curve (AUC; Fig. 1D) were larger in the active naïve population and in BDNF treated neurons compared to ‘silent’ neurons. Unexpectedly, both measures were significantly decreased in the BDNF condition compared to naive. A plausible explanation for this effect was that chronic BDNF induced a concomitant enhancement of [Ca2+]i fluctuations in some neurons and depression in others. Indeed, the effects of BDNF on the relative frequencies (%) for the fluctuation amplitudes (amplitudes were normalized to the largest fluctuation in each neuron) revealed a biphasic effect; frequencies of the smaller- and larger-amplitude fluctuations were greater in the BDNF-treated neurons compared to the naïve and ‘silent’ neurons (Fig. 1E). We next compared the mean parameters between neurons in order to control for differences in sampling periods. A < 10% maximal fluctuation amplitude cut-off was applied to each neuron to reduce the effects of low-amplitude noise. As expected, nonparametric comparisons indicated that fluctuation amplitude (Fig. 1F) and AUC (Fig. 1G) were both greater in the naïve and BDNF-treated neurons compared to the ‘silent’ neurons, but were not significantly different between naïve and BDNF-treated neurons. Further analysis of the fluctuation kinetics indicated no effect on duration (Fig. 1H), a modest, but significant decrease in rise time (i.e., peak time is more negative) compared to the ‘silent’ neurons (Fig. 1I), and no effect on frequency (Fig. 1J).
BDNF-induced fluctuations of [Ca2+]I in superficial dorsal horn neurons. (A1–A3) Example recording [Ca2+]i fluctuations sampled in a SDH neuron and the processing steps used by the FIBSI event-detection program. The running median (red line in A1, window = ~ 5 s) was calculated based on the raw amplitude (AU) and time coordinate data, then the peaks below the running median were traced to form a reference line (red line in A2), and then the Ramer–Douglas–Peucker algorithm detected waveforms above the reference (red x at event peaks). (B) Left: Examples of [Ca2+]i activity detected by FIBSI in 3 different neurons. Right: The proportion of ‘silent’ neurons in the BDNF-treatment condition was significantly reduced compared to the naïve condition. Proportions were compared using a Fisher exact test. (C, D) Violin plots of the log-transformed values for fluctuation amplitude and AUC. Fluctuation amplitudes and AUC were significantly increased in naïve and BDNF-treated neurons compared to the ‘silent’ neurons, while both measures in the BDNF-treated neurons were significantly reduced compared to naïve. Neurons sampled: ‘silent’ n = 20 (849 fluctuations), naïve n = 78 (2651 fluctuations), and BDNF n = 57 (1555 fluctuations). Median with quartiles shown. Comparisons made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. (E) Relative frequency (%) of the fluctuations binned by amplitude. Amplitude values were normalized to the largest fluctuation in each neuron. (F–G) The mean fluctuation amplitude and AUC in the naïve and BDNF-treated neurons were significantly increased compared to the ‘silent’ app builder v2019 39 - Free Activators. (H–J) Further analysis revealed little to no effect of BDNF on fluctuation duration or frequency, but the fluctuations in ‘silent’ neurons exhibited longer rise times. For scatterplots F–J, the fluctuations that were < 10% the maximal fluctuation amplitude in each neuron were omitted to reduce the effects of low-amplitude noise. Scatterplots in F–J show the median with the interquartile range. All comparisons of medians in F–K were made using Kruskal–Wallis tests and Dunn’s post-hoc test. **P < 0.01, ***P < 0.001, ****P < 0.0001.
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It has been shown that gap junctions have regulatory control over network activity in the substantia gelatinosa26,27,28,29. Therefore, gap junctions may play a role in regulating the properties of [Ca2+]i across the network, but the response of individual neurons to the chronic BDNF in the culture may be masked by the complex responses of the different types of neurons to BDNF20, 30. To address this and determine the potential role of gap junctions in mediating the BDNF-induced fluctuations, we recorded [Ca2+]i from SDH neurons exposed to chronic BDNF and then applied a non-specific gap junction blocker octanol at 1 mM for ≥ 3 min. We again performed a detailed analysis using the FIBSI program and a < 10% maximal fluctuation amplitude cut-off was applied to each neuron. Inspection of the FIBSI-processed recordings revealed octanol disproportionately affected some neurons compared to others (i.e., large increases in amplitude and AUC in some neurons vs minor decreases in others). Neurons were grouped based on whether the octanol treatment caused a negative (decrease, n = 39) or positive (increase, n = 40) fold change in mean fluctuation amplitude compared to the BDNF (control) condition (Fig. 2A). Paired analyses indicated blocking gap junctions with octanol decreased the frequency of the BDNF-induced fluctuations (Fig. 2B) while increasing their duration (Fig. 2C) only in the neurons that had a positive fold change in mean fluctuation amplitude. Fluctuation rise time was also significantly decreased in the same group of neurons (Fig. 2D). These data suggest that blocking gap junctions with octanol selectively disinhibited the response to chronic BDNF in a subpopulation of SDH neurons.
Effect of gap junction blocker octanol on BDNF-induced [Ca2+]I fluctuations. (A) Scatterplot summarizing the effects of octanol on mean fluctuation amplitude and examples of FIBSI-processed recordings. The two bottom recordings show a decrease in fluctuation amplitude and the two top show an increase. (B–D) Neurons were sorted based on whether they exhibited a positive or negative fold change in mean fluctuation amplitude in response to octanol. Paired analyses showed octanol selectively and significantly decreased the mean fluctuation frequency, increased the mean fluctuation duration, and decreased the mean fluctuation rise time (i.e., more negative peak time) in the neurons with positive fold changes in mean fluctuation amplitude. Control vs octanol paired comparisons for the groups in (B–D) were made using two-way repeated measures ANOVAs and Sidak’s post-hoc test. **P < 0.01, ****P < 0.0001.
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Previous work has shown BDNF-treated SDH neurons generate synchronous [Ca2+]i fluctuations20. We verified that finding by using the FIBSI-processed recordings and comparing the degree of synchrony between neurons imaged together in naïve and BDNF-treated cultures. Adjacency matrixes were generated using Pearson r coefficients; the recording of each neuron was correlated with every other neuron imaged within the same naïve or BDNF-treated culture using a Pearson correlation matrix. The adjacency matrixes for the BDNF-treated cultures showed the neurons were more positively correlated with Pearson r coefficients closer to 1 (Fig. 3A), suggesting they exhibited more synchronous [Ca2+]i fluctuations than the naïve cultures. Indeed, we observed synchronous spikes in [Ca2+]i across multiple BDNF-treated neurons compared to the naïve neurons (Fig. 3B1–B2). Raster plots depicting the [Ca2+]i fluctuation peaks revealed the presence of multiple waves of synchronous activity in BDNF-treated neurons (Fig. 3C1–C2). To further confirm this prediction, the mean coefficient for each neuron was calculated (i.e., the mean correlation of each neuron with every other neuron in its respective culture; refer to Fig. 3A for Fisher Z transformation steps). Comparison of the mean Pearson r coefficients revealed BDNF-treated neurons were more positively correlated with other neurons in the culture than naïve neurons (Fig. 3D). Considering the role gap junctions may play in regulating network activity and synchronous activity, this result led us to anticipate that blocking gap junctions may reduce BDNF-induced synchrony within the culture. We used the same approach to measure synchrony among neurons within the same cultures treated with BDNF prior to application of octanol, and then we asked whether blocking gap junctions caused a decrease in the mean Pearson r coefficients. Indeed, octanol caused a net decrease in the mean Pearson r coefficient (Fig. 3E1,E2). Closer inspection revealed octanol decreased the mean Pearson r coefficient in 100% (25/25) of the neurons in the group with decreased fluctuation amplitudes, while only 63% (22/35) of the neurons in the group with increased fluctuation amplitudes exhibited a decreased mean Pearson r coefficient. This suggests that blocking gap junctions may actually increase [Ca2+]i fluctuation synchrony in some neurons.
Synchrony of [Ca2 +]i fluctuations in BDNF-treated cultures and the effect of octanol. (A) Example adjacency matrix constructed from Pearson r coefficients corresponding to the correlation in activity between neurons in a naïve or BDNF-treated culture, each with 8 sampled neurons. Steps to calculate the mean Pearson r for each neuron within a culture using the Fisher Z transformation are also depicted. (B1,B2) Example recordings of asynchronous [Ca2+]i fluctuations in naïve neurons and synchronous [Ca2+]i fluctuations in BDNF-treated neurons. Recordings shown were processed by FIBSI for event detection and each one was normalized (0 to 1 scale) before calculation of the adjacency matrix in A. (C1,C2) Raster plots depicting the fluctuation peak times detected by FIBSI for the recordings in B1-B2. Red stars in C2 denote the presence of synchronous waves of [Ca2+]i activity. (D) Neurons sampled from BDNF-treated cultures exhibited significantly greater mean Pearson r coefficients compared to neurons sampled from naïve cultures. Neurons were imaged from 4 naïve cultures and 3 BDNF-treated cultures, each with ≥ 8 neurons imaged. Medians shown with the 95% confidence interval. Medians were compared using a Mann–Whitney test. (E1,E2) Treatment with octanol significantly decreased the mean Pearson r in neurons sampled from cultures exposed to chronic BDNF, and this treatment effect impacted both subsets of neurons binned based on their initial response to octanol. Neurons were imaged from 4 cultures. The control vs octanol paired comparison in C1 was made using a Wilcoxon rank sum test. Control vs octanol paired comparisons for the groups in C2 were made using a measures two-way repeated measures ANOVA and Sidak’s post-hoc test. *P < 0.05, **P < 0.01, ****P < 0.0001.
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In order to further characterize the properties of the [Ca2+]i fluctuations, we used pharmacological treatments to block a variety of voltage-gated ion channels, glutamate receptors, and GABA receptors. Comparisons between the pharmacologically-treated neurons and the BDNF-treated (control) neurons revealed the [Ca2+]i fluctuation amplitudes (Fig. 4A) and frequency (Fig. 4B) are controlled by diverse mechanisms (example recordings and data available in Supplemental Fig. 2). The fluctuations were ablated when extracellular Ca2+ was removed, when all voltage-gated calcium channels (VGCCs) were blocked with Cd2+, when all TTX-sensitive voltage-gated sodium channels (VGSCs) were blocked with tetrodotoxin (TTX), when AMPA or kainite glutamate receptors were blocked with CNQX/NBQX, and when GABA was applied to the cultures. Interestingly, the fluctuations remained when we blocked T-type (with Ni2+), L-type (with nitrendipine), or N-type (with ω-conotoxin) Ca2+ channels, although the frequency was significantly reduced with no effect on amplitude. Riluzole, which blocks glutamate release and VGSCs, only affected the fluctuation frequency but not amplitude. The NMDA blocker D-AP5 was the only blocker to significantly reduce fluctuation amplitudes, but it had no effect on frequency. We also noted that applying GABA-A blockers bicuculline (10 µM) and strychnine (1 µM) to BDNF-treated slices (Supplementary Fig. 2P) abolishes higher frequency smaller amplitude fluctuations, leaving inducing the larger amplitude low frequency fluctuations that occur in naive cultures (Supplementary Fig. 2P). These data suggest the BDNF-induced [Ca2+]i fluctuations are controlled by a combination of VGCCs, TTX-sensitive VGSCs, GABA and NMDA receptors.
Pharmacology of [Ca2+]i fluctuations from BDNF-treated cultures. Summary of the effects of various pharmacological treatments on the (A) amplitude and (B) frequency of the [Ca2+]i fluctuations in BDNF-treated cultures compared to before drug treatment. Bars show the mean + standard error of the mean. *P < 0.001, paired t-test, n = 5–20 cells per condition.
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It is well established that BDNF release in the SDH is involved in neuropathic pain10, 16, 17. However, to our knowledge our findings are the first to show chronic BDNF treatment unmasks [Ca2+]i fluctuations in a subpopulation of SDH neurons while depressing activity in others. We had previously shown that sEPSCs recorded from SDH neurons are perturbed in BDNF-treated spinal cord organotypic cultures20. It is plausible that the newly discovered unmasked BDNF-induced [Ca2+]i fluctuations underlie changes in network excitability. We decided to readdress the effects of BDNF on network excitability by using the FIBSI program to quantitatively analyze a previous subset of sEPSC recordings in delay and tonic-firing SDH neurons (example recordings and low-pass filtering/matching methods are available in Supplemental Fig. 3). Inspection of the sEPSC interevent interval data indicated there might be natural clustering between single “small” sEPSCs and larger summated sEPSCs. We used partitioning around medoids (a medoid is like a centroid, but is restricted to an actual observation in the dataset) to cluster the sEPSCs in each neuron based on 3 sEPSC parameters: interevent interval (ms), amplitude (pA), and charge transfer (Q; essentially area under the curve). Optimal clustering (k medoids = 3–5 per neuron) identified 3 main clusters of sEPSCs that were present in all delay (Fig. 5A1,A2) and tonic (Fig. 5B1,B2) neurons in both control and BDNF treatment conditions (clusters named based on amplitude; interevent interval): (1) small; short, (2) small; long, and (3) large. Two other clusters were identified (small; mid and medium), but they were not present in all neurons or treatment conditions and were not analyzed for this study. We were mainly interested in the effects of BDNF on the cluster of large-amplitude sEPSCs in both neuron types (refer to Supplemental Fig. 4 for the “small” sEPSC analysis). Chronic BDNF treatment significantly decreased sEPSC amplitudes (Fig. 5C1,C2) and charge transfer (Fig. 5D1,D2) in the tonic-firing neurons. We did not observe a significant effect on sEPSC amplitudes in the delay neurons, but charge transfer was significantly decreased. The sEPSCs in the BDNF-treated delay neurons were narrower (control duration = 83.6 ± 1.2 ms; BDNF duration = 49.9 ± 0.4 ms). The sEPSC interevent intervals were significantly shorter in both neuron types (Fig. 5E1,E2), and peak time measurements indicated BDNF significantly increased activation of the sEPSCs in the tonic neuron (Fig. 5F1,F2).
Cluster analysis of spontaneous EPSCs in delay and tonic-firing superficial dorsal horn neurons. Three dimensional representations of the sEPSCs in (A1,A2) delay neurons and (B1,B2) tonic neurons plotted based on interevent interval (x-axis), amplitude (y-axis), and charge transfer (z-axis). Mapping of the clustering results and treatment shown for the total sample of sEPSCs. The sEPSCs in each neuron were clustered independently from the other neurons. Optimal partitioning around medoids identified 3 primary clusters present in all neurons in both treatment conditions (naming based on amplitude; interevent interval): small; short, small; long, and large. Two additional clusters were identified in some neurons, but were not present in all neurons in both conditions: small; mid and medium. Neurons sampled: 19 delay (5 control, 14 BDNF); 9 tonic (5 control, 4 BDNF). Total number of sEPSCs: 33,206 in delay, control; 108,261 in delay, BDNF; 40,658 in tonic, control; 36,521 in tonic, BDNF. The effects of BDNF on sEPSC amplitude, charge transfer (i.e., area under the curve), and peak time were assessed for the 3 primary clusters (large cluster shown, refer to Supplemental Fig. 3 for the two small-amplitude clusters). (C1,C2) Treatment with BDNF significantly decreased sEPSC amplitudes in tonic neurons, but a significant effect was not observed in the delay neurons. Cumulative distributions of the measured amplitudes shown (left) alongside the means ± standard error of the means (right). (D1,D2) The BDNF treatment significantly reduced sEPSC charge transfer measures in both delay and tonic neurons. (E1,E2) Interevent intervals for the sEPSCs within the large cluster were significantly decreased in the BDNF-treated neurons. (F1, F2) The sEPSC activation kinetics were significantly faster in the BDNF-treated neurons, but no significant effect was observed for the sEPSCs in the delay neurons. Large sEPSC cluster sample sizes: delay neurons control n = 753, BDNF n = 4539; tonic neurons control n = 2348, BDNF n = 2367. Comparisons between means were made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. **P < 0.01, ****P < 0.0001.
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The present study was undertaken to better understand mechanisms underlying synchronous activity among SDH neurons and the influence BDNF may have on network excitability in naïve, uninjured organotypic cultures. Our experiments indicate long-term exposure to BDNF produces a complex set of neuron-dependent changes in network excitability in the SDH. First, BDNF causes a concomitant activation of Ca2+ signaling in a subpopulation of ‘silent’ SDH neurons (~ 20% sampled) and depression of Ca2+ signaling in many others. This finding was largely unexpected as we have previously shown chronic BDNF increases the size and frequency of [Ca2+]i fluctuations20. It is plausible that the presence of the ‘silent’ neurons can greatly influence data analysis and interpretation depending on how they are grouped together, or discarded. Another possibility is that resolution differences between the two event-detection programs used (FIBSI in the current study, Mini Analysis Software (Synaptosoft, NJ)) in the previous study) could influence analysis of the [Ca2+]i fluctuations. However, the goal of the app builder v2019 39 - Free Activators study was not to compare/contrast the two programs. The second significant finding was that gap junction signaling may play a crucial role in regulating BDNF-induced changes in Ca2+ signaling in the SDH. Blockade of gap junctions with octanol unveiled a subpopulation of neurons that respond to BDNF with large-amplitude, low-frequency [Ca2+]i fluctuations. This finding suggests that gap junctions modulate global network activity, either by directly or indirectly inhibiting those particular neurons with increased Ca2+ signaling. Indeed, this is supported by the third major finding that BDNF profoundly increases synchrony of the [Ca2 +]i fluctuations, and blocking gap junctions with octanol reverses BDNF-induced synchrony.
It is known that delay neurons are mostly excitatory glutamatergic neurons expressing vesicular glutamate transporter 2 (VGLUT2) and that tonic neurons are mostly GABA- or glycinergic inhibitory neurons31. We also know from the pioneering studies of Perl and others that tonic neurons mainly possess an islet morphology and that delay neurons possess a radial morphology32. While the physiological correlates of the BDNF-related [Ca2+]i fluctuations are unclear, we do not think they are being driven by delay or tonic-firing neurons in the SDH. Unlike our analysis of the [Ca2+]i fluctuations where we describe the ‘silent’ neurons, we did not observe any unmasking effect(s) of BDNF on the sEPSCs in the delay or tonic neurons. All neurons sampled in the control (naïve) condition generated visually appreciable, large-amplitude sEPSCs. The main depressive effect of BDNF on the sEPSCs (refer also to Supplemental Fig. 4) mirrored the main effect of BDNF on comparable properties for the [Ca2+]i fluctuations. A summary of each dataset (Fig. 6, table) shows BDNF reduced the amplitude and area under the curve parameters for the [Ca2+]i fluctuations and large sEPSCs. More nuanced effects were observed for frequency (reciprocal of interevent interval) and activation kinetics: BDNF increased the frequency of the large sEPSCs but had little to no effect on the frequency of the [Ca2+]i fluctuations; BDNF increased activation of the large sEPSCs in the tonic neurons and had little to no effect on the delay neurons or [Ca2+]i fluctuations. However, these data collectively suggest PDF Shaper Full - Crack Key For U BDNF may have reduced excitability in the delay and tonic SDH neurons, and this may provide a physiological basis for the depressive effect of chronic BDNF on the [Ca2+]i fluctuations. On the contrary, the ‘silent’ neurons are predicted to have responded to chronic BDNF with [Ca2+]i fluctuations of greater amplitude, total area, frequency, and presumably activation. Our working model (Fig. 6, bottom) posits that under naïve conditions many delay and tonic SDH neurons in organotypic spinal cord cultures are generally in an active state, with some gap junction coupling with neighboring neurons. Long-term exposure to BDNF causes a homeostatic shift that pushes the SDH into a dampened state marked by synchronous, low-level activity spread across the network via increased gap junction coupling. This synchronous activity in turn mitigates the BDNF-activated ‘silent’ neurons. From our pharmacology experiments (Fig. 4 and Supplementary Fig. 1), we also suggest that activity of GABA- and glycinergic inhibitory receptor systems are necessary to support BDNF-induced Ca2+ fluctuations. Given the role of GABA and glycine in sensory processing associated with chronic pain33, it is possible that BDNF-induced fluctuations are important for pathophysiology of neuropathic pain.
Working model summarizing the effects of chronic BDNF on SDH neurons in this study.
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Several types of oscillatory and/or rhythmic bursting activity have been previously observed in spinal cord organotypic cultures34, 35 and in ex vivo slice preparations26, 36,37,38. BDNF has also been shown to induce long term potentiation (LTP) in the spinal dorsal horn39 and is critically involved in mechanisms of synaptic plasticity, which may induce synchronous activity in different pain states. Synchronous activation of groups of spinal nociceptive neurons might contribute to the ‘electric shock’-like sensations experienced by some neuropathic pain patients40. We hypothesize that [Ca2+]i fluctuations leading to synchronous network activity may be attributed to the “shooting pains” that chronic pain patients may experience. In particular, “shooting pain” is very commonly experienced by patients with radicular lower back pain (sciatica) or trigeminal neuralgia. In these patients, stimulus movement elicits so-called “traveling waves” in which neural activity sweeps across the body-part representation in somatosensory maps41. The mechanisms of this have been attributed in part to Ca2+ waves and gap junctions 41, 42. Therefore, given the properties of the [Ca2+]i fluctuations we have identified, they would make a good candidate for the neural substrate of shooting pain. In a cohort of chemotherapy-induced peripheral neuropathy (CIPN) patients, some of whom reported shooting or burning pain in the hands or feet, serum nerve growth factor (NGF) levels were much higher compared with patients with painless or absent CIPN43. Therefore, there is a precedent for the correlation of neurotrophic factors related to BDNF to the incidence of shooting pain. Indeed it has been shown that NGF regulates the expression of BDNF44. It is has also been demonstrated that BDNF overexpression induces spasticity in rodent models of spinal cord injury, which may also explain the effect of BDNF on influencing network excitability in the dorsal horn45. With regards to the involvement of the dorsal horn in shooting pain, it has been shown that dorsal root entry zone lesioning (DREZotomy) successfully reduces shooting pain caused by brachial plexus root avulsion (BPRA)46. Also, in one patient with shooting pain caused by osteoid osteoma, a partial laminectomy was able to significantly relieve pain symptoms47. Ideally however, more experiments using human ex vivo tissue are needed to determine whether these particular [Ca2+]i fluctuations correlate with shooting pain in patients.
All experimental protocols were approved by the University of Alberta Animal Care and Use Committee, which is charged with maintaining standards set forth by the Canadian Council on Animal Care.
All procedures complied with the guidelines of the Canadian Council for Animal Care and the University of Alberta Health Sciences Laboratory Animal Services Welfare Committee. Spinal cords were isolated from embryonic (E13–14) rats and transverse slices (300 ± 25 μm) were cultured using the roller-tube technique20, 22. Since tissue is obtained from embryos, sex could not be determined. Serum-free conditions were established after 5 days in vitro. Medium was exchanged with freshly prepared medium every 3–4 days. Slices were treated after 15–21 days in vitro for a period of 5–6 days with 50–200 ng/ml in serum-free medium as described previously20. Age-matched, untreated DMOTC slices served as controls.
Each organotypic slice was incubated for 1 h prior to imaging with the fluorescent Ca2+-indicator dye Fluo-4-AM (Molecular Probes, Invitrogen, Carlsbad, CA, USA). The conditions for incubating the dye were standardized across different slices to avoid uneven dye loading. After dye loading, the slice was transferred to a recording chamber and superfused with external solution containing (in mM): 131 NaCl, 2.5 KCl, 1.2 NaH2PO4, 1.3 MgSO4, 26 NaHCO3, 25 d-glucose, and 2.5 CaCl2 (20 °C, flow rate 4 ml/min). Regions of interest (ROI) corresponding to individual cell bodies of neurons were identified based on morphology and size. Changes in Ca2+-fluorescence intensity evoked by a high K+ solution (20, 35, or 50 mM, 90 s application) or other pharmacological agents, were measured in dorsal horn neurons with a confocal microscope equipped with an argon (488 nm) laser and filters (20 × XLUMPlanF1-NA-0.95 objective; Olympus FV300, Markham, Ontario, Canada). Full frame images (512 × 512 pixels) in a fixed xy plane were acquired at a scanning time of 0.8–1.08 s/frame48. In some experiments, images were cropped to accommodate faster scan rates. Selected regions of interest were drawn around distinct cell bodies depicting neurons based on morphology as described previously49 and fluorescence intensity traces were generated with FluoView v.4.3 (Olympus).
Whole cell patch-clamp recordings were obtained from neurons in organotypic slice cultures under infrared differential interference contrast optics. Neurons selected for recording were located 250–800 μm from the dorsal edge of the cultures in an area presumed to reflect the substantia gelatinosa and up to a depth of 100 μm from the surface. Neurons were categorized according to their firing pattern in response to depolarizing current commands as tonic, delay, phasic, irregular, or transient30. Recordings were obtained with an NPI SEC-05LX amplifier (ALA Scientific Ardamax Keylogger keygen, Westbury, NY, USA) in bridge balance or in discontinuous, single electrode, current or voltage-clamp mode. Neurons were sampled at 2 k Hz for 180 s when measuring sEPSCs. For recording, slices were superfused at room temperature (∼ 22 °C) with 95% O2-5% CO2-saturated aCSF that contained (in mM) 127 NaCl, 2.5 KCl, 1.2 NaH2PO4, 26 NaHCO3, 1.3 MgSO4, 2.5 CaCl2, and 25 d-glucose, pH 7.4. Patch pipettes topaz labs free download crack - Activators Patch pulled from thin-walled borosilicate glass (1.5/1.12 mm OD/ID; WPI, Sarasota, FL) to 5- to 10-MΩ resistances when filled with an internal solution containing (in mM) 130 potassium gluconate, 1 MgCl2, 2 CaCl2, 10 HEPES, 10 EGTA, 4 Mg-ATP, and 0.3 Na-GTP, pH 7.2, 290–300 mosM.
Unless otherwise stated, all chemicals were from Sigma (St. Louis, MO, USA). Fluo-4 AM dye was dissolved in a mixture of dimethyl sulfoxide (DMSO) and 20% pluronic acid (Invitrogen, Burlington, Ontario, Canada) to a 0.5 mM stock solution and kept frozen until used. The dye was thawed and sonicated thoroughly before incubating with a DMOTC slice. TTX was dissolved in distilled water as a 1 mM stock solution and stored at – 20 °C until use. TTX was diluted to a final desired concentration of 1 µM in external recording solution on the day of the experiment. Strychnine was prepared in a similar manner to TTX, and bicuculline (Tocris, Ballwin, MO, USA) was dissolved in DMSO as a 10 mM stock solution. 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX, Tocris), 2,3-dihydroxy-6-nitro-7-sulfamoyl-benzo[f]quinoxaline-2,3-dione (NBQX, Tocris) and N,N,H,-Trimethyl-5-[(tricyclo[3.3.1.13,7]dec-1-ylmethyl)amino]-1-pentanaminiumbromide hydrobromide (IEM-1460, Tocris) were prepared as 10 mM stocks dissolved in distilled water, and d(-)-2-Amino-5-phosphonopentanoic acid (d-AP5, Tocris) was prepared as 50 mM stocks dissolved in 30% 1 M NaOH. Riluzole was prepared as a 10 mM stock, kyneurinic acid and GABA as 1 M stocks, and nitrendipine as a 1 mM stock made up in distilled water. These drugs were used at a 1:1000 dilution prepared freshly with external recording solution immediately prior to the start of experiments.
Raw time-lapse calcium fluorescence intensity and whole-cell voltage-clamp recordings were analyzed using the Frequency-Independent Biological Signal Identification (FIBSI) program24 written using the Anaconda v2019.7.0.0 (Anaconda, Inc, Austin, TX) distribution of Python v3.5.2 and the NumPy and matplotlib.pyplot libraries. The FIBSI program incorporates the Ramer-Douglas-Peucker algorithm to detect significant waveforms against a time-dependent generated reference line using the least number of total points to represent said waveform, and provides quantitative measurements for each detected waveform. The calcium fluorescence traces were fit using a sliding median with a window size between 5 and 25 s, and peaks below the sliding median line were traced together to form the reference line. The voltage-clamp traces were fit using a sliding median with a window of 50 ms, and peaks above the sliding median were traced together to form the reference line. A custom Python script was used to match sEPSCs detected in filtered voltage-clamp recordings with the unfiltered recordings. The detected sEPSCs for each neuron were clustered based on their amplitude, charge transfer, and interevent interval using the partitioning around medoids function (default settings, manhattan distance used) in the cluster v2.1.0 package in R v4.0.2 (2020-06-22; R Core Team, The R Foundation for Statistical Computing, Vienna, Austria). ArcGis 10.6 Crack + Keygen Full Free Setup Download neuron was clustered independently. The average silhouette width method was used to select the optimal number of clusters for each neuron. Clusters were graphed using the plotly v4.9.2.1 package with dependency on ggplot2. The FIBSI source code, custom Python matching script, and a tutorial for using FIBSI are available on a GitHub repository (https://github.com/rmcassidy/FIBSI_program).
Statistical analysis of the calcium fluctuations and sEPSCs were performed using Prism v8.2.1 (GraphPad Software, Inc, La Jolla, CA). The comparison between the proportion of ‘silent’ neurons in the naïve and BDNF treatment conditions was made using Fisher’s exact test. The Brown–Forsythe and Welch ANOVA tests and Games–Howell post-hoc test (for n > 50) were used to compare the effects of BDNF when the calcium fluctuations from all neurons were grouped together. To control for differences in recording times, the means for each Microsoft Office Pro Plus 2020 Crack + Activator Key Free Download 2021 parameter (amplitude, AUC, etc.) were calculated for each neuron (fluctuations < 10% the maximal amplitude in each neuron were omitted to remove the effects of low-amplitude noise). The amplitude and AUC means were normalized across avast free antivirus uninstall - Crack Key For U neurons to permit direct comparisons. Normality was assessed using the D’Agostino & Pearson omnibus test, and nonparametric comparisons between group medians were made using the Kruskal–Wallis test and Dunn’s post-hoc test. Two-way repeated measures ANOVAs and Sidak’s post-hoc tests were used to compare the pre- and post-treatment effects of octanol on the calcium fluctuations. Adjacency matrixes for neurons imaged together from the same organotypic culture were constructed using Pearson correlation matrixes; the FIBSI-processed calcium fluctuation recordings were used as input (each recording was normalized on a 0 to 1 scale). Next, the mean Pearson r coefficient for each neuron within a culture was calculated using the Fisher Z transformation (steps shown in Fig. 3A). The mean Pearson r coefficients in the naïve and BDNF conditions were not normally distributed, so the comparison between the two was made using a Mann–Whitney test. Comparisons between the pre- and post-treatment effects of octanol were first made using a Wilcoxon matched-pairs signed rank test, and then using a two-way repeated measures ANOVA and Sidak’s post-hoc test. Paired t-tests were used for pharmacology experiments. Brown-Forsythe and Welch ANOVA tests and Games-Howell post-hoc test were used to compare the effects of BDNF on the sEPSC clusters. Statistical significant was set to P < 0.05 and all reported P values are two-tailed. Details for the statistical analyses can be found in Table 1.
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Download references
We thank Edgar Walters for useful discussions. We thank Klaus Ballanyi and Araya Ruangkittisakul for use of and technical assistance with confocal imaging equipment.
We acknowledge funding from the Canadian Institutes for Health Research. We acknowledge funding from the research endowment fund of the Department of Anesthesiology and Critical Care Medicine, University of New Mexico Health Sciences Center.
These authors contributed equally: Sascha R. A. Alles and Max A. Odem.
Department of Anesthesiology & Critical Care Medicine, University of New Mexico Health Sciences Center, MSC 10-6000, 2211 Lomas Blvd. NE, ACM 200, Albuquerque, NM, 87106, USA
Sascha R. A. Alles
Neuroscience and Mental Health Institute & Department of Pharmacology, University of Alberta, Edmonton, AB, Canada
Sascha R. A. Alles, Van B. Lu & Peter A. Smith
Department of Microbiology and Molecular Genetics, McGovern Medical School at UTHealth, Houston, TX, USA
Max A. Odem
Wellcome-MRC Institute of Metabolic Science, University of Cambridge, Cambridge, UK
Van B. Lu
Department of Psychiatry and Behavioral Sciences, Vanderbilt University Medical Center, Nashville, TN, USA
Ryan M. Cassidy
MicrosoftHyper-V, codenamed Viridian,[1] and briefly known before its release as Windows Server Virtualization, is a native hypervisor; it can create virtual machines on x86-64 systems running Windows.[2] Starting with Windows 8, Hyper-V superseded Windows Luminar libraries - Crack Key For U PC as the hardware virtualization component of the client editions of Windows NT. A server computer running Hyper-V can be configured to expose individual virtual machines to one or more networks. Hyper-V was first released with Windows Server 2008, and has been available without additional charge since Windows Server 2012 and Windows 8. A standalone Windows Hyper-V Server is free, but with command-line interface only.
A beta version of Hyper-V was shipped with certain x86-64 editions of Windows Server 2008. The finalized version was released on June 26, 2008 and was delivered through Windows Update.[3] Hyper-V has since been released with every version of Windows Server.[4][5][6]
Microsoft provides Hyper-V through two channels:
Hyper-V Server 2008 was released on October 1, 2008. It consists of Windows Server 2008 Server Core and Hyper-V role; other Windows Server 2008 roles are disabled, and there are limited Windows services.[8] Hyper-V Server 2008 is limited to a command-line interface used to configure the host OS, physical hardware, and software. A menu driven CLI interface and some freely downloadable script files simplify configuration. In addition, Hyper-V Server supports remote access via Remote Desktop Connection. However, administration and configuration of the host OS and the guest virtual machines is generally done over the network, using either Microsoft Management Consoles on another Windows computer or System Center Virtual Machine Manager. This allows much easier "point and click" configuration, and monitoring of the Hyper-V Server.
Hyper-V Server 2008 R2 (an edition of Windows Server 2008 R2) was made available in September 2009 and includes Windows PowerShell v2 for greater CLI control. Remote access to Hyper-V Server requires CLI configuration of network interfaces and Windows Firewall. Also using a Windows Vista PC to administer Hyper-V Server 2008 R2 is not fully supported.
Hyper-V implements isolation of virtual machines in terms of a partition. A partition is a logical unit of isolation, supported by the hypervisor, in which each guest operating system executes. There must be at least one parent partition in a hypervisor instance, running a supported version of Windows Server (2008 and later). The virtualization software runs in the parent partition and has direct access to the hardware devices. The parent partition creates child partitions which host the guest OSs. A parent partition creates child partitions using the hypercall API, which is the application programming interface exposed by Hyper-V.[9]
A child partition does not have access to the physical processor, nor does it handle its real interrupts. Instead, it has a virtual view of the processor and runs in Guest Virtual Address, which, depending on the configuration of the hypervisor, might not necessarily be the entire virtual address space. Depending on VM configuration, Hyper-V may expose only a subset of the processors to each partition. The hypervisor handles the interrupts to the processor, and redirects them to the respective partition using a logical Synthetic Interrupt Controller (SynIC). Hyper-V can hardware accelerate the address translation of Guest Virtual Address-spaces by using second level address translation provided by the CPU, referred to as EPT on Intel and RVI (formerly NPT) on AMD.
Child partitions do not have direct access to hardware resources, but instead have a virtual view of the resources, in terms of virtual devices. Any request to the virtual devices is redirected via the VMBus to the devices in the parent partition, which will manage the requests. The VMBus is a logical channel which enables inter-partition communication. The response is also redirected via the VMBus. If the devices in the parent partition are also virtual devices, it will be redirected further until it reaches the parent partition, where it will gain access to the physical devices. Parent partitions run a Virtualization Service Provider (VSP), which connects to the VMBus and handles device access requests from child partitions. Child partition virtual devices internally run a Virtualization Service Client (VSC), which redirect the request to VSPs in the parent partition via the VMBus. This entire process is transparent to the guest OS.
Virtual devices can also take advantage of a Windows Server Virtualization feature, named Enlightened I/O, for storage, networking and graphics subsystems, among others. Enlightened I/O is a specialized virtualization-aware implementation of high level communication protocols, like SCSI, that allows bypassing any device emulation layer and takes advantage of VMBus directly. This makes the communication more efficient, but requires the guest OS to support Enlightened I/O.
Currently[when?] only the following operating systems support Enlightened I/O, allowing them therefore to run faster as guest operating systems under Hyper-V than other operating systems that need to use slower emulated hardware:
The Hyper-V role is only available in the x86-64 variants airserver 5.5.9 crack Standard, Enterprise and Datacenter editions of Windows Server 2008 and later, as well as the Pro, Enterprise and Education editions of Windows 8 and later. On Windows Server, it can be installed regardless of whether the installation is a full or core installation. In addition, Hyper-V can be made available as part of the Hyper-V Server operating system, which is a freeware edition of Windows Server.[12] Either way, the host computer needs the following.[13]
The amount of memory assigned to virtual machines depends on the operating system:
The number of CPUs assigned to each virtual machine also depends on the OS:
There is also a maximum for the number of concurrently active virtual machines.
The following table lists supported guest operating systems on Windows Server 2008 R2 SP1.[20]
| Guest operating system | Virtual CPUs | ||
|---|---|---|---|
| OS | Editions | Number | Architecture |
| Windows Server 2012[a] | Hyper-V, Standard, Datacenter | 1–4 | x86-64 |
| Windows Home Server 2011 | Standard | 1–4 | x86-64 |
| Windows Server 2008 R2 SP1 | Web, Standard, Enterprise, Datacenter | 1–4 | x86-64 |
| Windows Server 2008 SP2 | Web, Standard, Enterprise, Datacenter | 1–4 | IA-32, x86-64 |
| Windows Server 2003 R2 SP2 | Web,[b] Standard, Enterprise, Datacenter | 1 or 2 | IA-32, x86-64 |
| Windows 2000 SP4 | Professional, Server, Advanced Server | 1 | IA-32 |
| Windows 7 | Professional, Enterprise, Ultimate | 1–4 | IA-32, x86-64 |
| Windows Vista | Business, Enterprise, Ultimate | 1–4 | IA-32, x86-64 |
| Windows XP SP3 | Professional | 1 or 2 | IA-32, x86-64 |
| Windows XP SP2 | Professional, Professional x64 Edition | 1 | IA-32, x86-64 |
| SUSE Linux Enterprise Server 10 SP4 or 11 SP1–SP3 | N/A | 1–4 | IA-32, x86-64 |
| Red Hat Enterprise Linux 5.5–7.0 | Red Hat Compatible Kernel | 1–4 | IA-32, x86-64 |
| CentOS 5.5–7.5 | N/A | 1–4 | IA-32, x86-64 |
| Ubuntu 12.04–20.04 | Debian Compatible Kernel | 1–4 | IA-32, x86-64 |
| Debian 7.0 | Debian Compatible Kernel | 1–4 | IA-32, x86-64 |
| Oracle Linux 6.4 | Red Hat Compatible Kernel | 1–4 | IA-32, x86-64 |
Fedora 8 or 9 are unsupported; however, they have been reported to run.[20][21][22][23]
Third-party support for FreeBSD 8.2 and later guests is provided by a partnership between NetApp and Citrix.[24] This includes both emulated and paravirtualized modes of operation, as well as several HyperV integration services.[25]
Desktop virtualization (VDI) products from third-party companies (such as Quest Software vWorkspace, Citrix XenDesktop, Systancia AppliDis Fusion[26] and Ericom PowerTerm WebConnect) provide the ability to host and centrally manage desktop virtual machines in the data center while giving end users a full PC desktop experience.
Guest operating systems with Enlightened I/O and a hypervisor-aware kernel such as Windows Server 2008 and later server versions, Windows Vista SP1 and later clients and offerings from Citrix XenServer and Novell will be able to use the host resources better since VSC drivers in these guests communicate with the VSPs directly over VMBus.[27] Non-"enlightened" operating systems will run with emulated I/O;[28] however, integration components (which include the VSC drivers) are available for Windows Server 2003 SP2, Windows Vista SP1 and Linux to achieve better performance.
On July 20, 2009, Microsoft submitted Hyper-V drivers for inclusion in the Linux kernel under the terms of the GPL.[29] Microsoft was required to submit the code when it was discovered that they had incorporated a Hyper-V network driver with GPL-licensed components statically linked to closed-source binaries.[30] Kernels Filmora 7.3.0 Serial Key with 2.6.32 may include inbuilt Hyper-V paravirtualization support which improves the performance of virtual Linux guest systems in a Windows host environment. Hyper-V provides basic virtualization support for Linux guests out of the box. Paravirtualization support requires installing the Linux Integration Components or Satori InputVSC drivers. Xen-enabled Linux guest distributions may also be paravirtualized in Hyper-V. As of 2013[update] Microsoft officially supported only SUSE Linux Enterprise Server 10 SP1/SP2 (x86 and x64) in this manner,[31] though any Xen-enabled Linux should be able to run. In February 2008, Red Hat and Microsoft signed a virtualization pact for hypervisor interoperability with their respective server operating systems, to enable Red Hat Enterprise Linux 5 to be officially supported on Hyper-V.[32]
Hyper-V in Windows Server 2012 and Windows Server 2012 R2 changes the support list above as follows:[33]
Hyper-V on Windows Server 2012 R2 added the Generation 2 VM.[34]
Hyper-V, like Microsoft Virtual Server and Windows Virtual PC, saves each guest OS to a single virtual hard disk file. It supports the older .vhd format, as well as the newer .vhdx. Older .vhd files from Virtual Server 2005, Virtual PC 2004 and Virtual PC 2007 can be copied and used in Hyper-V, but any old virtual machine integration software (equivalents of Hyper-V Integration Services) must be removed from the virtual machine. After the migrated guest OS is configured and started using Hyper-V, the guest OS will detect changes to the (virtual) hardware. Installing "Hyper-V Integration Services" installs five services to improve performance, at the same time adding the new guest video and network card drivers.
Hyper-V does not virtualize audio hardware. Before Windows 8.1 and Windows Server 2012 R2, it was possible to work around this issue by connecting to the virtual machine with Remote Desktop Connection over a network connection and use its audio redirection feature.[35][36] Windows 8.1 and Windows Server 2012 R2 add the enhanced session mode which provides redirection without a network connection.[37]
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Open Access
Peer-reviewed
Roles Formal analysis, Investigation, Software, Validation, Visualization, Writing – original draft, Writing – review & editing
Current address: Department of Electrical and Electronic Engineering, Faculty of Engineering, University of Peradeniya, Peradeniya, Sri Lanka
Affiliations Biological Cybernetics, Faculty of Biology, Bielefeld University, Bielefeld, Germany, Cognitive Interaction Technology—Center of Excellence (CITEC), Bielefeld University, Bielefeld, Germany
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In computational modelling of sensory-motor control, the dynamics of muscle contraction is an important determinant of movement timing and joint stiffness. This is particularly so in animals with many slow muscles, as is the case in insects—many of which are important models for sensory-motor control. A muscle model is generally used to transform motoneuronal input into muscle force. Although standard models exist for vertebrate muscle innervated by many motoneurons, there is no agreement on a parametric model for single motoneuron stimulation of invertebrate muscle. Although several different models have been proposed, they have never been evaluated using a common experimental data set. We evaluate five models for isometric force production of a well-studied model system: the locust hind leg tibial extensor muscle. The response of this muscle to motoneuron spikes is best modelled as a non-linear low-pass system. Linear first-order models can approximate isometric force time courses well at high spike rates, but they cannot account for appropriate force time courses at low spike rates. A linear third-order model performs better, but only non-linear models can account for frequency-dependent change of decay time and force potentiation at intermediate stimulus frequencies. Some of the differences among published models are due to differences among experimental data sets. We developed a comprehensive toolbox for modelling muscle activation dynamics, and optimised model parameters using one data set. The “Hatze-Zakotnik model” that emphasizes an accurate single-twitch time course and uses frequency-dependent modulation of the twitch for force potentiation performs best for the slow motoneuron. Frequency-dependent modulation of a single twitch works less well for the fast motoneuron. The non-linear “Wilson” model that optimises parameters to all data set parts simultaneously performs better here. Our open-access toolbox provides powerful tools for researchers to fit appropriate models to a range of insect muscles.
Insects are important study organisms in the neuroscience of sensory-motor systems. Since the dynamics of muscle contraction and associated changes in force, torque or stiffness are central to our understanding of sensory-motor systems in general, the choice of the most appropriate model for insect muscle matters. Computational modelling of muscle properties typically separates activation dynamics from contraction dynamics. The former models the development of muscle force in response to motoneuron activity, whereas the latter describes how this force is affected by the current physical state of the muscle: its length and contraction velocity. We evaluate five published activation dynamics models for insect muscle. We explain differences between them, suggest how to decide which one to use, and provide an open-source toolbox for activation dynamics modelling. We further show that non-linear models are the best choice if: (i) the time course of a single muscle twitch is slow, (ii) the spike frequency ranges between one and thirty spikes per second, or (iii) the sensory-motor system tends to execute movements in a similar manner even if the demand on joint torque or stiffness changes.
Citation: Harischandra N, Clare AJ, Zakotnik J, Blackburn LML, Matheson T, Dürr V (2019) Evaluation of linear and non-linear activation dynamics models for insect muscle. PLoS Comput Biol 15(10): e1007437. https://doi.org/10.1371/journal.pcbi.1007437
Editor: Joseph Ayers, Northeastern University, UNITED STATES
Received: May 3, 2019; Accepted: September 25, 2019; Published: October 14, 2019
Copyright: © 2019 Harischandra et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The Matlab Toolbox IMADSim vs 1.0.4 is available on Matlab Central (https://de.mathworks.com/matlabcentral/fileexchange/72487-insect-muscle-activation-dynamics-simulator-imadsim). The experimental data are available from Bielefeld University under http://doi.org/10.4119/unibi/2937068
Funding: Supported by a grant from the Studienstiftung des Deutschen Volkes to JZ, project grants of the cluster of excellence EXC277 of the German Research Council (DFG) to NH and VD, a Biotechnology and Biological Sciences Research Council (UK) studentship to LMLB, a BBSRC research grant (BB/H014047/1) and a BBSRC Research Development Fellowship (BB/I019065/1) to TM. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The small number of motoneurons per muscle in insects has permitted detailed analyses of context-dependent adaptation of motor patterns and behaviour in several model species. Counter-intuitively, however, this small number of motoneurons poses a problem for the modelling of sensory-motor control, and the computational neuroscience of behaviour in general. This is because it necessitates accurate modelling of the spike-by-spike activation dynamics of the muscle, rather than simpler modelling based on average motor spike rate. Several muscle activation models have been proposed for insect muscle [1–4], but these have very different properties. Furthermore, although their published parameters were obtained from fits to experimental data on the same leg muscle (extensor tibiae) and, in three of four cases, the same leg and species (the metathoracic jumping leg of the locust Schistocerca gregaria [1, 3, 4]), the models yield very different force time courses when driven with identical inputs. This calls for a comparative evaluation. To identify the most reliable and computationally sound model of insect muscle activation dynamics, we provide a comprehensive muscle activation model toolbox for Matlab, comprising three linear and two non-linear models. We use this toolbox to identify differences caused by the models themselves, as opposed to effects caused by differences in the sample data used to tune the published versions of the models. To this end, we use a data set [5] comprising isometric force contraction time courses in response to metathoracic fast extensor tibiae (FETi) and slow extensor tibiae (SETi) motoneuron stimulation in the locust. In particular, we compare the model predictions of frequency-dependencies of maximum force, rise- and decay times using our own and published experimental data. Understanding arthropod muscle and being able to model its actions is important, because insects, crustaceans and spiders are widely and increasingly being used as models for biomimetic and robotic applications (e.g., flight: [6]; walking: [7, 8]; jumping: [9]). The active and passive properties of arthropod muscle provide remarkable solutions to the conflicting demands of flexibility and stability of movement, driven by relatively simple nervous control systems. Engineers struggle to manufacture equally versatile actuators, in part because the properties of muscles and their patterns of activation and modulation are still not fully understood.
Muscle models come in two types: ‘physiological’ and ‘conceptual’. Physiological models describe the bitdefender total security review - Crack Key For U mechanisms underlying muscle contraction, e.g., by formal description of calcium release dynamics, binding-unbinding kinetics and diffusion processes. Such models can predict muscle forces accurately and reproduce effects such as potentiation and depression [10,11], but often contain many parameters that are unknown or difficult to determine (indirectly) from force measurements or behavioural data. Conceptual models, on the other hand, estimate the transfer function from motoneuron spike activity (input) to muscle contraction (output). Generally, they have fewer parameters because they do not model details of the underlying physiological mechanisms. In some cases, they may not model any physiological mechanism at all but instead implement an abstract transfer function [12].
The generation of muscle force is typically modelled as a two-stage process that separates the activation dynamics of muscle from its contraction dynamics [13]. Contraction dynamics describe how much force is produced given the active state and the current muscle length and any length change, taking into account the force-length-velocity characteristics of the muscle, and the elastic properties of muscle and tendon. In essence, the equations describing contraction dynamics scale the isometric force predicted by the activation dynamics model. Activation dynamics describe the transform of the neural signal to the active state of the muscle, which corresponds to the number of actin-myosin cross-bridges. This determines the muscular tension produced without muscle shortening, i.e., the isometric force. Given the finding that the force-length dependency of contraction dynamics shifts with calcium concentration [14] which, in turn, governs activation, the computational separation of the activation and contraction dynamics may be challenged. For example, Rockenfeller and Günther recently proposed a model where muscle activation depends on the volumetric density of calcium binding sites, which in turn is a function of muscle fibre length [15,16].
The present study focuses on conceptual models of muscle activation dynamics for typical isometric force measurements (varying motoneuron firing frequency at constant muscle length). A simple and very widely used model was proposed by Zajac [13]. It applies a first-order low-pass filter to transform a rectified electromyogram (EMG) signal into a force output. The “Zajac model” is sufficient for many studies in vertebrate muscle, where muscle activation is largely determined by the number of motoneurons recruited. Other first-order models have been applied to the smooth muscle of Aplysia [17,18], or to the stomatogastric system of crustaceans [19]. Variants of these first-order models have also been applied to the mesothoracic extensor tibiae muscle of the stick insect Carausius morosus [2].
In contrast to these first-order models, models that use higher-order differential equations can create a delayed response to a pulsed input such as an action potential [20,21]. The “Hatze model” uses a pair of non-linear second-order differential equations, based on theoretical assumptions regarding the transformation of the neural signal to force [20]. Zakotnik [1] introduced a variant of this model for locust extensor tibiae muscle that includes a frequency-dependent modulation of twitch shape (Fig 1). More recently, Wilson and co-workers applied a set of different models to experimental data from the same insect muscle, finding that a linear third-order model captured well the activation dynamics following slow motoneuron stimulation [3]. Subsequently, they found that a non-linear fourth-order model was still better-suited to modelling the responses to different—slow and fast—motoneuron types [4].
Fig 1. Twitch shape parameters of the Hatze-Zakotnik model.
Black to grey curves show changes in single-twitch shape with variation of parameters θ3 (A) and θ4 (B) in Eq 2. Parameter θ3 modulates the twitch peak force and decay rate at a constant area under the twitch, whereas θ4 jointly modulates peak force and single-twitch duration. Zakotnik’s extension to Hatze’s original model includes a spike-frequency-dependent modulation of parameter θ4 (see Eq 5).
https://doi.org/10.1371/journal.pcbi.1007437.g001
Three of the five models compared in the present study, including our own former work [1] and that of Wilson [3,4], have focussed on the extensor tibiae muscle of the locust hind leg, so we have used exemplar data from that muscle for this comparative analysis. The physiology and the underlying neuronal control of this muscle have been well studied [22–25]. Time constants for force production, and descriptions of history-dependent effects such as potentiation are available [26]. The muscle consists of slow, intermediate and fast fibre types and is innervated by two excitatory motoneurons, one inhibitory motoneuron, and a neuromodulatory dorsal unpaired median (DUM) neuron [26–28]. The fast extensor tibiae motoneuron (FETi) usually fires 12 to 17 spikes at up to 70 s-1 as part of a stereotyped motor sequence during jumping and kicking [29,30], but is generally not active in walking [31]. It fires up to 9 spikes during each cycle of a scratching movement [32]. In contrast, the slow extensor tibiae motoneuron (SETi) is active during both fast (jumping and kicking) and slower behaviours: for example only 2 to 3 SETi spikes per step allow the muscle to produce forces sufficient for propulsion [31]. During rhythmical scratching, SETi fires bursts of up to 30 spikes at a median instantaneous rate of 96 sp s-1, with transient peak frequencies up to approx. 400 s-1 [33]. The locust metathoracic extensor tibiae muscle is thus used in a range of natural behaviours, including walking, scratching, kicking and jumping. While the hind leg is adapted for jumping, this adaptation has more to do with muscle volume, biomechanical specialisation of the joint and motor patterns than with muscle fibre physiology. Its innervation by fast, slow and common inhibitory motor neurons is the same as that of the stick insect mesothoracic extensor tibiae muscle modelled by Blümel et al. [2].
The activation dynamics model of a muscle becomes critical when the time course of force generation is not a simple function of average motoneuron spike rate. This is the case not only for the hind leg extensor muscle of locusts: but also for insect muscles in general, which are typically innervated by only a small number of app builder v2019 39 - Free Activators, and where single muscle potentials can generate considerable force. We present a detailed comparative evaluation of three linear and two non-linear activation dynamics models that is important for the use of insect model systems in neuroscience. We explain app builder v2019 39 - Free Activators non-linear models are clearly superior whenever low and intermediate spike rates up to 30 Hz are of interest; and we demonstrate how suitable the two available non-linear models are in accounting for the properties of different motoneuron and muscle types. Moreover, we present an open-source muscle activation dynamics toolbox for research and education.
Modelling realistic time courses of muscle contraction in insects requires appropriate consideration of the characteristic properties of insect muscle, including the importance of single twitches, the long-lasting twitch force, and non-linear force potentiation. As these properties may differ considerably among particular muscle activation dynamics models, we present a comparative evaluation of five models that have been proposed in the literature. We do so in four steps. First, we focus on the time course of the single twitch because it can be considered the elementary muscle contraction app builder v2019 39 - Free Activators. Second, we relate the properties of the single twitch to those of isometric muscle contractions in response to arbitrary motoneuron spike rates. Third, we compare and evaluate the properties of five published models. To illustrate the differences of the models in exactly the way that they were proposed originally, we initially adhere to the published parameter sets. Fourth, we evaluate the properties of the two most accurate models after optimising their parameters to one exemplar experimental data set.
Single twitch contractions of insect muscle are generally fairly slow: for both slow and fast motoneurons, a single action potential may lead to a change in isometric muscle force for 200 ms or longer (Fig 2). In response to an SETi action potential (Fig 2A and 2B), the time course of force development in the extensor tibiae muscle was of sigmoid shape with maximum rising velocity at approximately 25 ms. The time to peak force was on average 67 ms (SD: 9 ms) so force development was always faster than force decay, which could take up to 1 s. In response to an FETi action potential (Fig 2C and 2D) the overall shape of the isometric force time course was similar to that following an SETi action potential, but peak force was considerably higher. The time to peak force was on average 61 ms (SD: 6 ms). In some experiments on FETi stimulation, a single stimulus caused additional weak contractions of unknown origin that followed the initial strong twitch (Fig 2C and 2D).
Fig 2. Modelling a single twitch.
Top row shows an isometric twitch response to a single SETi spike (blue) and corresponding model simulation results for that single twitch (solid black) using the Hatze-Zakotnik model (A) and non-linear Wilson model (B). Force is normalised to the maximum SETi-induced force of the extensor tibiae muscle of this particular animal. The spike onset is at time t = 0. The grey lines in A and C show the force responses predicted by Zajac’s first-order model, where the rise time is much shorter than in a real twitch. C, D show an isometric twitch response to a single FETi spike (red) and corresponding simulation results (black) for the same two models. Force is normalised to the maximum FETi-induced force of the extensor tibiae muscle of this particular animal. Note that for this figure, model parameters were optimised to fit the single twitch response only. In the Hatze-Zakotnik model, there were four parameters only (parameters K1 and K2 were kept constant). In the Wilson non-linear model, there were six.
https://doi.org/10.1371/journal.pcbi.1007437.g002
In Fig 2, both the Hatze-Zakotnik and the non-linear Wilson models were fitted to experimental recordings of single twitches. In the case of the Hatze-Zakotnik model, this involved only four parameters (θ1-θ4) because no parameters were needed to model force potentiation (see Eqs 1, 4 and 6: c(f) = 1 for f = 1, irrespective of K1 and K2). In contrast, Wilson's model requires optimisation of six parameters. Both models achieved very good fits. For example, the Hatze-Zakotnik model accurately captured single twitch force time courses with an average root mean square error (RMSE) of 1.1% of peak twitch force for SETi stimulation, and 6.6% for FETi stimulation (solid black lines in Fig 2). For comparison, modelling the twitch response with a first-order model (e.g., [13]) leads to an exceedingly short rise time and lacks the rounded peak observed in experiments (dashed lines in Fig 2). The third-order, linear Wilson model can reproduce the delayed, sigmoid increase and rounded peak [3].
Modelling force potentiation (i.e., the supra-linear increase of peak force with increasing stimulation frequency) requires a non-linearity. One way to incorporate such a non-linearity is to alter the shape of a single twitch in a frequency-dependent manner. In the case of Hatze's model (Eqs 1 and 2), this can be achieved by varying parameters that affect not only the peak twitch force, but also the rates of increase and decay (θ3) or the duration of the twitch (θ4, see Fig 1). In the Hatze-Zakotnik model, θ4 is modified by the frequency-dependent factor c(f)). As a result, summation of twitches is adjusted so as to capture frequency-dependent force potentiation.
The effect of non-linear force potentiation is best observed in the Hatze-Zakotnik model when increasing the motoneuron spike rate from 10 Hz to 50 Hz, for both slow (SETi) and fast (FETi) motoneuron stimulation (Fig 3). The increase in peak force was much larger when the stimulus frequency was changed from 10 Hz to 20 Hz than when it was changed from 20 Hz to 50 Hz. At stimulus frequencies above 20 Hz, single twitches fused into a smooth tetanic contraction for both SETi (Fig 3A) and FETi (Fig 3B).
Fig 3. Frequency-dependent modulation of single twitch shape can explain non-linear force potentiation.
Measured isometric force traces in response to SETi (blue, A) and FETi stimulation (red, B) and corresponding simulation using Hatze-Zakotnik model (black). Stimulation frequencies (Hz) are indicated to the right. Parameters θ1 - θ4 were optimised for the single twitch (as in Fig 2A and 2C) and then θ4 was optimized separately for each spike frequency. Each SETi time course could be fitted extremely well except for the first three twitches at 10 and 12.5 Hz. For FETi, the model slightly overestimates the rise time at high frequencies and underestimates the rise time at low frequencies.
https://doi.org/10.1371/journal.pcbi.1007437.g003
Two extensions of Hatze’s model have been proposed to account for non-linear force potentiation. For vertebrate muscle, van Zandwijk and colleagues [34] suggested scaling the peak twitch force by Adobe Acrobat Reader 19.5 Crack - Crack Key For U sigmoid function. However, application of this Hatze-van-Zandwijk model to the insect data shown in Fig 3 proves insufficient. S2 Fig shows the result after optimising the sigmoid function parameters A and γ0 of Eq 3 by fitting force traces of all stimulation frequencies with constant coefficients θ1 to θ4. While the Hatze-van-Zandwijk model performs reasonably well for low stimulation frequencies (S2 Fig), it fails to replicate the experimental data for higher stimulation frequencies. The time taken to reach maximum tetanic force is shorter in the model than in experiments. For example, a full tetanus for SETi stimulation at 50 Hz is generated after approximately 300 ms in malwarebytes license key model, but only after 600 ms in the experiment. In addition, force values in the model are too low at SETi frequencies above 10 Hz. Furthermore, the tetanus fuses only at frequencies above 25 Hz.
A much better fit to experimental data was obtained when the sigmoid relationship was replaced with a non-linear, frequency-dependent force potentiation. Fig 3 shows a direct comparison of measured and simulated force profiles for a representative SETi stimulation experiment, with a constant set of parameters θ1 to θ3 (optimised to single twitch) and frequency-specific optimisation of parameter θ4 (see Eq 4). The resulting model predictions match the experimental data extremely well across all stimulation frequencies. Fig 3 thus demonstrates the concept of the Hatze-Zakotnik model that scales parameter θ4 by the frequency-dependent factor c(f) (Eq 5). Table 1 lists the improvement achieved by optimisation of θ4. The fit quality improved in all cases tested. Although additional optimisation of θ3 improved fit quality substantially in one test data set, it led to much less improvement or even lead to worse results in other cases. To show how factor θ4 needs to be modulated with spike frequency, Fig 4 plots the optimal values of c(f) relative to the inter-spike interval 1/f, superimposed on the time course of a single twitch. Potentiation factor c(f) is 1 for a single twitch (i.e., no potentiation) and less than 1 for higher stimulus frequencies. For both SETi and FETi stimulation, c(f) was lowest at an inter-spike interval of 0.05 s, equivalent to f = 20 Hz (dotted line in Fig 4). The overall shape of the function graph c(1/f) resembles the shape of an inverted twitch, with its minimum approximately 15 ms before peak twitch force. During microsoft office 365 crack kickass - Activators Patch spike train, force potentiation is thus largest if a spike occurs just prior to the maximum twitch force caused by the preceding spike. In the experimental data, this effect was consistent across all preparations.
Fig 4. Frequency-dependent scaling of single twitch force.
Values of function c(t) of Eq 6 for different SETi stimulation frequencies. Black circles and solid vertical lines indicate medians and inter-quartile ranges. Smaller values indicate a stronger potentiation of twitches, with a minimum at 20 Hz (dashed line). The smaller this value, the stronger is force potentiation (see Fig 1). As a reference, the values are superimposed on a single twitch (black curve and shaded area indicate mean and inter-quartile range of experimental data, n = 5). Maximum potentiation was achieved when a spike occurred approximately 15 ms before peak twitch force generated by the preceding spike (displacement of dashed and dotted lines). The lower scale relates frequency to inter-spike interval because in Eq 5 factor c is a function of frequency, whereas in Eq 6 it is a function of time.
https://doi.org/10.1371/journal.pcbi.1007437.g004
Table 1. RMS error values for three kinds of optimization runs for the Hatze-Zakotnik model prediction for SETi data (11 frequencies of 1 s duration from each animal).
Optimising θ4 (middle column) always improves fit quality compared to the reference parameters of Zakotnik (2006). Additional optimization of θ3 (right column) may or may not further improve fit quality.
https://doi.org/10.1371/journal.pcbi.1007437.t001
The frequency-dependent potentiation factor c(f) in the Hatze-Zakotnik model is implemented as a Michaelis-Menten function (Eq 6). The two parameters of this function were obtained by least-square fits to the experimental data from four preparations (two SETi, and two FETi stimulation experiments). As shown in Fig 5, Eq 6 approximated well the determined values for c: it was 1 for a single twitch (equivalent to t = 1 s) and reached a minimum at about t = 0.04s (24.9 Hz). Mean parameter values for SETi were K1 = 1.46·10−2 (SD: 2.3·10−3) and K2 = 3.9·10−4 (SD: 2.6·10−4). For FETi, mean parameter values were K1 = 7.8·10−2 (SD: 2.6·10−2) and K2 = 8.0·10−4 (SD: 1.8·10−4). For an overview of all model parameters see Table 2.
Fig 5. Values of force potentiation factor c as a function of inter-spike interval.
Both SETi (A) and FETi (B) panels show data from two animals (different symbols for different animals). For each symbol, factor c was calculated after frequency-specific optimisation of θ4 to measured force traces, as shown in Fig 3. Fits are Michaelis-Menten-type functions according to Eq 6 (solid and dashed lines). Each function fit was weighted by the stimulus frequency, improving fit quality at small inter-spike intervals.
https://doi.org/10.1371/journal.pcbi.1007437.g005
Table 2. Estimated model parameters for six locusts when fitting to SETi data (animals A, B, and D) or FETi data (animals 2, 3, and 4), using both the Hatze-Zakotnik and Wilson non-linear models.
Published model parameters are shown for comparison.
https://doi.org/10.1371/journal.pcbi.1007437.t002
To quantitatively compare the properties of available muscle activation models “as published”, we examined their responses to constant-frequency stimulation of the slow motoneuron SETi (Fig 6, normalised to single twitch force), the corresponding responses to more natural random spike trains (Fig 7), the dependence of maximum isometric force on stimulation frequency (Fig 8A), and the rise and decay rates in response to a step onset or offset of stimulation (Fig 9A and 9C). Additionally, we compared the properties of the two non-linear models with parameter sets for a fast motoneuron (constant-frequency stimulation: S4 Fig; random stimulation: S5 Fig; peak force: Fig 8B; rise and decay rates: Fig 9B). Note that these comparisons were based on the parameter sets that were published in the original model descriptions. Accordingly, differences may, to some extent, depend on the particular properties of the experimental data sets that had been used to obtain these parameters sets in the first place.
Fig 6. Constant frequency responses of five muscle activation models “as published”.
A, B show the time course of isometric SETi contractions at different stimulation frequencies (1–50 Hz) for two kinds of second-order, non-linear models: the Hatze-Zakotnik model [65] and the non-linear Wilson model ([4]. Note that, for immediate comparison, model output was normalised to maximum force of the single-twitch. This was set to 0.1. C-E show corresponding time courses of three published linear models: (C) [13], (D) [2] (both first order), and (E) [3] (third order). Constant frequency stimulation started at t = 0 s and persisted for 2 s. For comparison with FETi contractions see S4 Fig.
https://doi.org/10.1371/journal.pcbi.1007437.g006
Fig 7. Response to random activation.
Comparison of simulated isometric SETi contraction forces in response to two Poisson spike trains with mean frequencies of 20 Hz (A) and 5 Hz (B). The same models and model parameters are used as in Fig 6. Differences between models are most prominent where force potentiation is strongest, i.e., when inter-spike intervals are approximately 50 ms. The time courses predicted by the Hatze-Zakotnik and linear Wilson models are relatively similar, as are those of the two first-order models. The non-linear Wilson model deviates most strongly from the others. For comparison with FETi contractions see S5 Fig.
https://doi.org/10.1371/journal.pcbi.1007437.g007
Fig 8. Frequency dependence of peak isometric force.
As a summary of Fig 6 (A: SETi) and S4 Fig (B: FETi), normalised peak isometric force was plotted as a function of spike frequency for constant stimulation. Data were normalised to the peak force for a stimulation frequency of 50 Hz. For linear models, peak force linearly depends on stimulation frequency. With the published parameter sets, the Hatze-Zakotnik model has a saturating, supra-linear non-linearity for SETi stimulation, whereas the non-linear Wilson model has a sub-linear non-linearity. For FETi stimulation, both models are supra-linear, with stronger saturation for the Hatze-Zakotnik model.
https://doi.org/10.1371/journal.pcbi.1007437.g008
Fig 9. Half-maximal rise and decay times.
Rise time to 50% of peak force at a given constant stimulation frequency for SETi (A) and FETi (B) is shown in the top row. C, D: Decay time from peak to 50% of peak force for SETi and FETi, respectively. The results were derived for the same models as used in Figs 6–8. Linear models show no frequency-dependence of decay time, and only the third-order Wilson linear model has frequency-dependent rise time. The two non-linear models differ most strongly with regard to their decay time, particularly for FETi stimulation.
https://doi.org/10.1371/journal.pcbi.1007437.g009
The two first-order linear models (Zajac and Blümel) stand out in three ways: first, they cannot replicate the typical rounded peaks of muscle twitches (Fig 6C and 6D; Fig 7); second, they imply perfectly linear force potentiation with increasing spike frequency (Fig 8A); third, they show very little frequency-dependence in rise time to half-maximal force (Fig 9A)—and none at all for decay time (Fig 9C). With regard to all of these properties, they clearly differ from the properties of muscle. Examining the properties of the third-order linear Wilson model shows that some of these short-comings are a consequence of the first-order dynamics of the Blümel and Zajac models, whereas others are a consequence of linearity. For example, the linear Wilson model produces natural looking rounded twitch peaks (Fig 6E; Fig 7) and shows a strong change of rise time for stimulation frequencies between 10 and 20 Hz (Fig 9A). Both of these properties are related to the third-order dynamics of the model. In contrast, the linear Wilson model behaves the same as the other linear models in that it has perfectly linear force potentiation (Fig 8A) and lacks frequency-dependence of decay time (Fig 9C). Only the non-linear muscle activation models can capture the latter two properties of insect muscle.
With their parameter settings “as published”, the two non-linear models have considerably different properties. For example, the tetanic force for 50 Hz stimulation of a slow motoneuron reached more than thirty times the peak force of a single twitch in the non-linear Wilson model (Fig 6B), whereas it was less than ten times peak twitch force in the Hatze-Zakotnik model (Fig 6A). Differences between models are most prominent where force potentiation is strongest, i.e., when the spike frequency is around 20 Hz (Fig 7A). This is very similar for both the SETi stimulation (Fig 6) and the FETi stimulation (S4 Fig). The difference is reflected in a supra-linear (saturating) frequency dependence of maximum isometric force for the Hatze-Zakotnik model, and a sub-linear dependence for the non-linear Wilson model (Fig 8). Note that in the linear force-potentiation range (i.e., at low stimulation frequencies), the linear Wilson model behaves in a similar way to the non-linear Hatze-Zakotnik model (Fig 7A). For random FETi stimulation, the overall time courses of the two non-linear models were similar, though with considerably higher peak force for the Hatze-Zakotnik model (S5 Fig).
Although all of the higher-order models have a very similar frequency-dependence of half-maximal rise time (Fig 9A), they differ strongly in their frequency-dependence of decay time. At the end of a spike train of either SETi or FETi, measured extensor tibiae muscle force decayed in an exponential manner (S6 Fig). Similarly, all muscle activation models show an exponential decay after stimulus offset (Fig 6). The half-maximal decay time of the Hatze-Zakotnik model is strongly frequency-dependent (Fig 9C and 9D), whereas in the non-linear Wilson model it is hardly frequency-dependent in the case of SETi stimulation (Fig 9C), and shows nearly opposite behaviour to the Hatze-Zakotnik model for FETi stimulation (Fig 9D). In the latter case, decay time peaked at 10 Hz and then decreased with increasing frequency in the Hatze-Zakotnik model, whereas it reached a minimum at 5 Hz and then increased with increasing frequency in the non-linear Wilson model.
Finally, the tested muscle activation models differ strongly with regard to computational efficiency (Table 3). Whereas the two linear-first order models clearly outperform all other models, the third most efficient model is the iterative solution of the Hatze-Zakotnik model. It is only fifteen times slower than the linear first-order models, more than ten times faster than the linear Wilson model and almost 50 times faster than the non-linear Wilson model.
Table 3. Comparison of maximum CPU-time for simulating a 40 Hz spike train of 2 s followed by a relaxation time of 1 s (n = 10).
The iterative version of the Hatze-Zakotnik model is only fifteen times slower than the linear first-order models, and nearly 50 times faster than the non-linear Wilson model.
https://doi.org/10.1371/journal.pcbi.1007437.t003
Given the considerable differences among the published models, it was important to determine the extent to which these differences are due to the computational properties of the models, or rather due to differences in the experimental data for which the published parameter sets had been optimised. To resolve this issue, we fitted the two non-linear models to the same experimental data set, obtaining three particular solutions per model for both SETi and FETi stimulation (see Table 2 for model parameters and performance measures). Owing to the limitations of the linear models, as mentioned in the previous section, we did not consider them further.
The overall fit quality of the two models is shown in Fig 10. Both models can simulate isometric force time courses similarly well for SETi stimulation (Fig 10A and 10B). The core of the Hatze-Zakotnik model is an accurate model of a single twitch response that is then modulated in a frequency-dependent manner, so the model’s fit quality is best for low stimulation frequencies. With increasing frequency it tends to overestimate the half-maximal rise time, such that force build-up is slightly slower than in the experimental data (Fig 10A). In contrast, parameters of the non-linear Wilson model are optimised across all spike frequencies simultaneously, such that fit quality is similar for all stimulation frequencies. As a consequence, the fits to the single twitch and to responses to low-frequency stimulation are less good than those of the Hatze-Zakotnik model. For intermediate SETi stimulation frequencies, the non-linear Wilson model tends to underestimate rise time (Fig 10B). With regard to FETi stimulation, the Hatze-Zakotnik model performs less well than for SETi stimulation. It underestimates both the rise time and the maximum force for intermediate stimulation frequencies (Fig 10C). In comparison, the non-linear Wilson model achieves a better fit of the maximum force, while slightly overestimating rise time (Fig 10D). When comparing fit quality across experimental data sets, both non-linear models prove to capture the frequency-dependent force potentiation equally well for SETi stimulation (Fig 11A). Regarding FETi stimulation, the non-linear Wilson model achieves better fits (Fig 11B, compare model results with shaded area of experimental data). Despite the fact that the Hatze-Zakotnik model could replicate the saturating force potentiation curves well for SETi, optimisation did not lead to a similarly good match for FETi stimulation.
Fig 10. Comparison of non-linear models optimised to the same experimental data.
Model fits (black) to experimental data sets for SETi (A, B, blue) and FETi (C, D, red) stimulation. Both models have six free parameters. Since the non-linear Wilson model optimises the complete parameter set for all force traces simultaneously, its single-twitch fit is worse than that of the Hatze-Zakotnik model. In the latter, four parameters are optimised for the single twitch, and the remaining two describe the frequency-dependent modulation of the single-twitch time course. Constant stimulation frequencies used were: 1, 7, 10, 12.5, 15, 20, 25, 30, 40 and 50 Hz for SETi, and 1, 10, 20, 30 and 50 Hz for FETi.
https://doi.org/10.1371/journal.pcbi.1007437.g010
Fig 11. Frequency-dependent force potentiation.
Both non-linear models were fit to data from six preparations, three for SETi (A) and three for FETi (B). The experimental data range (N = 3) is shown in grey. Values were normalised to peak force obtained for stimulation at 50 Hz.
https://doi.org/10.1371/journal.pcbi.1007437.g011
As described above for the models ‘as published’ (Fig 9), frequency-dependent change of half-maximal rise time is similar in both optimised models (Fig 12A: SETi, 12B: FETi). For low frequency stimulation of two SETi data sets, the non-linear Wilson model overestimated rise time by more than 100 ms (Fig 12A). The Hatze-Zakotnik model overestimated rise time to a similar degree for one FETi data set (Fig 12B). Concerning the half-maximal decay time, the two models differ in much the same way as described for Fig 9. Whereas the Hatze-Zakotnik model shows the longest decay times for low to medium stimulation frequencies, with faster decay after high-frequency stimulation (Fig 12C and 12D), five parameter sets of the non-linear Wilson model lead to increasingly slower decay with increasing stimulation frequency. When comparing the models with experimental data, the Hatze-Zakotnik model underestimated decay time at high frequencies, whereas the non-linear Wilson model underestimated decay at intermediate frequencies of SETi stimulation (Fig 12C). S6 Fig compares the decay time courses of the Hatze-Zakotnik model with experimental data. For SETi stimulation, the model fit is very good for low frequencies (5 and 10 Hz). At higher stimulation frequencies (20 and 40 Hz) the main difference between model and experimental data was the delayed onset of decay in the experiment. As a consequence, the decay time course of the model looks very similar to the experimental time course, but the former leads the latter by approximately 70 ms. For FETi stimulation, the model differed from the experimental data in two ways: for stimulation at 50 Hz, the experimental decay again lagged the model decay; the decay of the experimental data was also considerably slower than in the model for stimulation at 10 and 20 Hz (S6 Fig).
Fig 12. Half-maximal rise and decay times.
A, B: Rise time to 50% peak force at a given constant stimulation frequency for SETi (A) and FETi (B). C, D: Decay time from peak to 50% peak force for SETi (C) and FETi (D). Both models were fit to the same experimental data sets as used for Fig 11. The experimental data range is shown in grey (N = 3). No experimental data are available for frequency dependence of FETi decay. Continuous and dashed colour lines depict best-fit results for the Hatze-Zakotnik and non-linear Wilson models, respectively.
https://doi.org/10.1371/journal.pcbi.1007437.g012
In summary, the non-linear models are similarly capable of replicating the properties of slow motoneuron (SETi) induced isometric contraction, with the Hatze-Zakotnik model being slightly superior with regard to the half-maximal decay time. For stimulation of the fast motoneuron (FETi), the non-linear Wilson model achieves better fits, particularly with regard to frequency-dependent force potentiation.
The choice of an appropriate muscle activation model is important whenever dynamic properties of movement sequences are of interest. In the simplest case, muscle activation is a function of the number of motoneurons recruited and can be related to the overall instantaneous firing rate. This is the rule in vertebrates, where Zajac’s linear first-order model of muscle activation [13] is usually appropriate, and isometric muscle force can be considered a low-pass transform of the rectified EMG. Although first-order muscle activation models have been applied also to muscles of molluscs, crustaceans and insects [2,17–19], they can capture neither the delayed slow rise and rounded peak of a single twitch, nor the non-linear force potentiation observed in the frequency range between 10 and 30 Hz. Our model comparison (Figs 6–9) illustrates that rounded twitch peaks and the frequency-dependent increase in half-maximal rise time typical of invertebrate muscle require the activation model to have higher-order dynamics. On the other hand, frequency-dependent potentiation and change in decay time require a non-linearity. The two models that fulfil these criteria require three times more parameters than first-order linear models, and may require considerably more computation time (Table 3). For this reason, it is worth considering when the properties of higher-order non-linear models are of particular relevance, which one of the models is preferable, and what other aspects require consideration when modelling insect muscle. The following sections address these three questions when, which and what:
Given the strong, sustained force of a single twitch (Fig 2), and the pronounced non-linear force potentiation for stimulation frequencies between 10 and 30 Hz (Fig 3), one may argue that a muscle activation model should account for these properties whenever the system to be modelled commonly experiences motoneuron frequencies below 30 Hz. On the other hand, even under these conditions, model-dependent differences in isometric force may be small compared to the strong attenuating effects of changes in muscle length and contraction velocity [13]. Thus, in movement sequences where limb kinematics require the use of a wide range iskysoft toolbox for android free download - Activators Patch muscle lengths and contraction velocities, variation in muscle force is likely to be governed more by muscle contraction dynamics than by activation dynamics. However, animals often execute the same movement sequence with very similar kinematics, despite marked changes of the mechanical demand. For example, insects can compensate for changes in load without significant changes in kinematics. In this case, length- and velocity-dependent changes in force as described by a muscle contraction dynamics model cannot counter the altered mechanical demand. Similarly, passive muscle properties, which have a strong effect on the dynamics of limb movements in insects [1,35–38] cannot account for the compensation of altered load without an associated change in kinematics. In other words, if load compensation occurs without a corresponding change in kinematics, an appropriate change of muscle activation is strictly necessary. For example, scratching locusts can compensate for substantial additional loads to the hind leg with essentially no change in joint angle time courses [39]. They control the target-specific movement by appropriate activation of antagonist muscles [32]. In response to a change in load, the associated change in motoneuronal spike number per burst is very small, but this causes a substantial change in co-contraction of antagonist muscles and, as a consequence, a change in net joint torque [1].
In walking or climbing insects, changes in load occur naturally whenever the animal changes its body attitude or encounters a change in inclination of the substrate. For example, during steady-state walking on upward and downward slopes (±45°), stick insects neither change their average speed nor do they show strong changes in leg kinematics. Nevertheless, at the same time, joint torques change drastically with the change in body weight distribution among legs [40]. As for scratching locusts, walking stick insects alter the relative activation of antagonist muscles during the early stance phase, suggesting that they maintain similar kinematics by regulating joint stiffness rather than joint angle or velocity. In both cases, antagonist muscles tend to be activated by alternating bursts of motoneuron activity; but the onset and offset of bursts is not always well defined, and a burst of motoneuron activity in one muscle may be opposed by only a few, or sometimes even a single spike of an antagonist motoneuron. Moreover, owing to the strong and long-lasting time course of a single twitch force, antagonist muscle co-activation can arise even without co-activity of antagonist motoneurons. Thus, even if intermittent spikes between “typical bursts” are rare, the effect of non-linear force potentiation may be substantial because the decay of force after a motoneuron burst may last well into the force build-up cause by an antagonist motoneuron burst. As a consequence, the choice of an appropriate muscle activation model is particularly relevant if the motor behaviour involves co-activation of antagonist muscles, for example for context-dependent regulation of joint stiffness or net torque.
The first decision to make when selecting an activation dynamics model should be whether or not a linear model is sufficient for the purpose of modelling. As outlined in the previous section, this decision should depend on whether or not low to medium motoneuron spike rates (1–30 Hz) commonly occur in the movement being modelled. If so, this would call for appropriate consideration of non-linear force potentiation that is most pronounced in this frequency range. On the other hand, if antagonistic motoneurons are consistently firing bursts in alternation and at high frequency (>50 Hz), and temporal separation of antagonistic bursts suggests little or no overlap in the activation time courses of antagonistic muscles, neither non-linear force potentiation nor non-linear effects on force decay would be expected and a linear muscle activation model is likely to be sufficient. Whenever context-dependent modulation of antagonist co-activation occurs (see examples on load compensation above), non-linear force potentiation will be an issue.
Comparing the properties of the two non-linear models, Table 3 shows that the Hatze-Zakotnik model is much more efficient computationally than the non-linear Wilson model. ProPresenter 7.2.0 (117571592) Crack + License Key Free Download 2020 10 and 11 suggest that modulating a single parameter of the single-twitch model works very well for activation by the slow motoneuron SETi, while revealing considerable mismatch for activation by the fast motoneuron FETi (Table 2; S9 Fig). Apparently, the transition from single twitch to tetanus does not follow the same physiological principles for SETi and FETi activation. Since the non-linear Wilson model does not depend on the modulation of a single twitch, it is possible to find suitable parameter sets for both SETi and FETi activation.
For the slow motoneuron, the Hatze-Zakotnik model has the advantage of being based on a physiologically plausible underlying concept. Twitches are maximally potentiated at a stimulus frequency of 20 Hz (Fig 4), which corresponds approximately to the time from twitch onset to twitch maximum; i.e. maximal potentiation occurs when the following twitch is triggered at the maximum of the preceding twitch. This can also be observed in cat gastrocnemius muscle [40, 41]. Here, potentiation occurs if an additional spike is triggered during the twitch-falling phase, due to reaction kinetics of calcium dynamics, and this leads to increased twitch amplitude and slower force decay. Studies of calcium dynamics in single barnacle muscle fibres show that enhanced release of calcium is the main reason for twitch potentiation [41].
In the Hatze-Zakotnik model, calcium release is included in the second differential equation (Eq 2), in which a single parameter is modulated through a bi-sigmoid Michaelis-Menten type equation (Eq 6). As a formal description of calcium release and removal, it is an intuitive conceptual model of the process underlying the potentiation of twitch force. Frequency-dependent modulation of twitch shape introduces only two additional parameters, and the low complexity makes optimisation of the model parameters to experimental data feasible. This leads to non-redundant solutions, which can be directly compared for different muscles or motoneurons. An extension to the model of muscle force which includes more detail and additional observations can be implemented by including twitch potentiation state as a more complex model parameter. For example, a framework for decomposition of tetanic forces into a series of individual twitches of different sizes was proposed by [42].
As the present study focuses on properties of isometric force time courses under the assumption of no change in muscle fibre length, it is important to note that muscle activation dynamics gets more complicated as soon as muscle is allowed to shorten, or if isometric force time courses are to be compared for different muscle lengths. This is because the force-length dependency of contraction dynamics depends on the level of activation (e.g., [14]). In a recent study, Rockenfeller and Günther discussed a range of activation dynamics models that include length-dependency [16] that transform the normalised calcium concentration γ(t) (Eq 4) into a length-dependent active state q(γ, l), where l denotes the relative fibre length. In their own model and in all of Hatze’s own variants [20,43,44], the nonlinear transform q contains the product of γ and a lever function ϖ(l) (see Table 2 and Eqs 2 and 3 of [16]). Thus, under the assumption of constant fibre length, all of these length-dependent activation dynamics models scale the normalised calcium concentration γ with a constant factor ϖ fix. Since none of the nonlinear transforms used in the three model variants discussed here includes length-dependency (Hatze-Zandwijk, Eq 3; Hatze-Zakonik, Eq 5; Wilson non-linear, Eq 8) they can be related to other models by setting ϖ (l) = ϖ fix = 1. Given the non-linear, frequency-dependent force facilitation shown in Fig 11 or S7 Sketchup pro 2019 license key and authorization number, a non-linear transform q(γ) is necessary even without any change in muscle length. Future experiments will need to elucidate the potential interaction of motoneuron firing frequency on the one hand and muscle fibre length on the other.
Both non-linear muscle activation models can account for frequency-dependent, non-linear force potentiation (Fig 10), however, they do not include long-term potentiation of twitches. For example, Brown and colleagues [46] modelled the sag effect, i.e., a slow decay in muscle force with long stimulation times, by including increased calcium removal at different stimulus frequencies. The effect of sag on twitches (cf. Fig 4 in [46]) resembles twitch modulation in the Hatze-Zakotnik model, so future work could lead to an incorporation of long-term potentiation into this model.
Another effect not included in current models is long term potentiation of force after a break [26] and the catch-effect, i.e., a prolonged increase in force production resulting from as few as one spike in an otherwise constant frequency stimulus sequence (p. 238 in [47]). Catch-like tension is thought to be based on complex calcium dynamics in the muscle [10].
A critical aspect of muscle models in general is that even an optimal set of model parameters can only account for the most typical behaviour in the face of very strong inter-individual variation of muscle properties [48]. Blümel and colleagues showed that for stick insect extensor tibiae muscle contraction dynamics, the use of individually fitted model parameters can halve the error of model estimates [2].
Our own experimental data also suggest that inter-individual variation can result in a considerable range of parameters of activation dynamics (see parameter ranges of the three per-animal fits in Table 2). As a consequence, individually optimised parameter sets lead to different half-maximal rise and decay times (Fig 11), whereas force potentiation appears to be affected less (at least for a given type of model, Fig 10). The fact that the Hatze-Zakotnik and non-linear Wilson models differ substantially more when using the parameters “as published” (Figs 6 to 9; S4 and S5 Figs) than after parameter optimisation to the same data set (Figs 10–12) app builder v2019 39 - Free Activators indicates that variation among experimental data is substantial. To illustrate this further, we compared the peak forces and half-maximal rise and decay times among three experimental data sets obtained from the same muscle in the same leg of the same insect species, i.e., the hind leg extensor tibiae muscle of the locust Schistocerca gregaria ([4,26] and our own data). Frequency-dependent potentiation of peak force was different in our experimental data compared to that of [4,26] for both SETi and FETi (S7 Fig). Our data showed stronger force potentiation at low frequencies. The situation for half-maximal rise time was quite different however: here, the data set of [26] stood out, with up to ten times slower rise and decay of force for low and medium frequencies compared to the data of [4] and our own (S8A Fig). Variation among data sets was smaller in the case of half-maximal rise and decay times during FETi stimulation (S8B Fig).
Physiologically meaningful differences in experimental data both within and between studies may be attributed to a number of factors including inter-individual or genetic strain differences in neuromuscular function, diurnal changes in physiology, the effects of circulating neuromodulators, or direct muscle inhibition. Differences in methodological procedures in different studies could markedly affect modulatory effects in particular. For example, levels of the insect neuromodulator octopamine are elevated following stress [49,50] which may vary among animals and between studies. Octopamine is also released peripherally from Dorsal Unpaired Median (DUM) neurons during particular types of behaviour such as flight or kicking [51,52]. Octopamine modulates both neuromuscular transmission and muscle contractile properties [53]. Moreover, Common Inhibitor motoneurons (for review see [54]) release γ-aminobutyric acid (GABA) onto insect skeletal muscles, which influences muscle contraction and relaxation dynamics [55–57]. In the locust, Common Inhibitor activation reduces extensor muscle relaxation times [58] and may reduce the force generated by excitatory motor spikes, thus facilitating fast cyclical leg movements [56,57,59]. For pharmacological induction of muscle relaxation related to inhibitory innervation see [60].
In summary, much of the difference between the two published non-linear activation dynamics models is due to considerable differences between the experimental data sets, but this inter-individual variation is an important aspect of muscle physiology. In the face of this inter-individual variation, a test for generality of any computational model involving muscle properties will require systematic variation of model parameters within the documented parameter ranges. Our muscle activation toolbox for Matlab [61] and the corresponding experimental data set [5] will facilitate the comprehensive and computationally efficient use of different activation dynamics models, and will also help us to learn more about variation of muscle properties in different preparations, types of muscle, and species.
Experiments were carried out on adult male and female Schistocerca gregaria, from crowded colonies at the Department of Zoology, University of Cambridge, UK or Department of Biology, University of Leicester, UK. Locusts were fixed in modelling clay, ventral side uppermost. The right hind leg was immobilised with dental cement (Protemp, ESPE) with the femur at right angles to the body, and the femoro-tibial angle initially set at 140° or 90°, for SETi and FETi experiments respectively to facilitate the subsequent attachment of the extensor muscle to the force transducer. For SETi experiments, a window was cut in the distal end of femur, and the accessory flexor muscle, overlying trachea and air sacs were removed. The end of the apodeme of the extensor muscle was grasped with a pair of forceps attached to a force transducer (see below) and the apodeme was cut distal to the forceps. Stimulation of FETi causes high extensor muscle forces that can fracture the extensor muscle apodeme at the point where it is grasped by forceps. To avoid this in FETi experiments, the proximal tibia was first braced in an aluminium sleeve which was then tightly attached to the force transducer with suture thread, such that the attachment point was aligned with the axis of pull of the extensor muscle. The distal femoral cuticle of the femoro-tibial joint was dissected away, VMWare Workstation 15 Full Download - Free Activators that the extensor muscle remained attached solely to the transducer through its natural point of attachment to the dorsal proximal tibia. For both SETi and FETi experiments, the app builder v2019 39 - Free Activators muscle length was adjusted to correspond to 90° femur-tibia angle before stimulation.
A window was cut into the ventral thoracic cuticle, and overlying air sacs were removed. Contractions in the extensor tibiae muscle were elicited by stimulation of Nerve 3b (N3b) or Nerve 5 (N5) in the thorax, for SETi and FETi respectively, through a pair of 50 μm silver hook electrodes placed under the nerve and insulated with petroleum jelly. Stimulation strength was set just above the threshold for eliciting a single twitch reliably. The axon of common inhibitor motoneuron CI1 runs in the same nerve as that of SETi, but it was not activated in these experiments. This was confirmed by making intracellular recordings from slow extensor tibiae muscle fibres during N3b stimulation. Such recordings revealed excitatory junctional potentials (from SETi) but no inhibitory potentials [62].
Stimuli were generated using a Master 8 stimulator (AMPI, Jerusalem, Israel). Series of pulses at various frequencies were delivered for 10 s or 1 s, for SETi and FETi respectively, with a 60 s gap between each stimulus train to allow the muscle to recover. FETi stimulus trains were restricted to 1 s to prevent damage to the apodeme insertion point. Muscle forces were measured using an isometric force transducer (Model 305C, Aurora Scientific, Canada), digitised at 5 kHz using a micro1401 interface and Spike 2 software (both Cambridge Electronic Design, Cambridge, UK).
To focus on the time course of muscle activation, measured forces were normalised to the interval [0,1], where 1 corresponds to maximum measured force per animal. They were not filtered. In the model, the stimulus protocol was shifted by a fixed delay (e.g. 10 ms) to account for the time taken by neural signals to be conducted to the muscle from the point of nerve stimulation.
Either constrained Levenberg-Marquardt or trust-region-reflective optimisation algorithms, implemented in Matlab (Mathworks Inc, Natick, USA), were used to fit the model parameters. These algorithms minimised a least squares error function that captured the distance between the measured and stimulated forces. The fitting procedure was repeatedly initialised with randomly distributed values to avoid local minima. Both algorithms were applied to the same experimental data set, and the one with the better performance (lower error) used for further analysis. This is mentioned in the data structures of the optimised data provided with the supplementary MATLAB IMADSim toolbox [61]. In the following sections, we introduce the five muscle activation models compared in this study.
Hatze [20, 44] proposed a second-order model comprising a pair of coupled second-order inhomogeneous differential equations, to capture two stages of signal processing (Eqs 16 and 17 in [20]; Eqs 3.21 and 3.22 with ρ*(ξ) = 1 in [44]). The first stage describes the transformation of neural activity α(t) at the motor endplate to membrane potential β(t) in the T-tubular system of a muscle. α(t) is zero except for 1 ms periods in which each motoneuron spike is modelled as a half sine wave, idealising the depolarised portion of an action potential (S1 Fig). When entering the T-tubular system, the action potential α(t) is transformed to signal β(t) according to the differential Eq 1: (Eq 1)
The second stage of the model transforms β(t) into the intracellular free ionic calcium concentration [Ca2+]i and therefore models calcium release and re-uptake by the sarcoplasmic reticulum. Differential Eq 2 determines the calcium concentration γ(t): (Eq 2)
Both processes can be viewed as over-critically damped, second-order systems, each with a different parameter set [20]. See Part I of the Supplementary Appendix S10 for a numerical solution for γ(t).
The relationship between the concentration of released calcium and the active state of the muscle is typically described by a sigmoid function as measured in [63] and [64]. Hatze [20] proposed that the active state also depended on muscle fibre length (his Eq 14), and varied this dependency in different versions (for a detailed treatise of this length dependence and its significance, see [16]). Given the lack of experimental data with systematic co-variation of both motoneuron frequency and muscle fibre length, the present study focuses on variants of the Hatze model that assume muscle fibre length to remain constant. For example, van Zandwijk and colleagues [45] expanded the Hatze model by including a sigmoid relationship between the calcium concentration, i.e., γ(t), and the force-producing active state: (Eq 3) where A and γ0 describe the slope and offset of the sigmoid, respectively. Owing to an unsatisfactory fit of the Hatze-van-Zandwijk model to isometric force measurements in insect muscle (S2 Fig), Zakotnik [65] replaced van Zandwijk’s sigmoid (Eq 3) with a frequency-dependent potentiation factor c(f) (hence the name Hatze-Zakotnik model). Reordering of Eq 2 shows that the calcium concentration γ is divided by the parameter value θ4:
(Eq 4)
Therefore, if value θ4 is decreased, the twitch force γ increases and has a slower decay. In contrast, parameter θ3 controls the twitch shape such that tetanic force does not change (Fig 1). To model force potentiation in the Hatze-Zakotnik model, θ4 is multiplied by a factor c(f) that depends on the instantaneous motor spike frequency f: (Eq 5)
Assuming that there is no potentiation for a single twitch (f = 1, c(1) = 1), values for c(f) can be determined by optimising parameter θ4 for each stimulation frequency independently and dividing this value by the measured θ4 at single twitch stimulation. Note that, for this procedure, the other three parameters, θ1 to θ3, remain fixed.
Because calcium kinetics are thought to be the main reason for force potentiation, a Michaelis-Menten-type equation is used for c(f). It is related to a [Ca2+]i reaction process that potentiates a twitch, and a calcium pump that reduces both [Ca2+]i and twitch size. The value for c(t) depending on the time since the previous stimulus t = (1/f) is determined as: (Eq 6)
Eq 6 relies on two parameters K1 and K2, for which we assume that K1 ≥ K2 and K1, K2 ≥ 0. Therefore, it is constrained to the interval [0, 1] and converges to 1 for Splice Sounds of KSHMR Vol.1,2,3 Free Download → ∞. The first term can be related to the force-producing reaction; i.e., larger values of K1 produce an elevated and prolonged twitch force. The second term can be related to the removal of calcium, i.e., larger values of K2 reduce twitch potentiation.
The parameters, K1 and K2 were also optimized by using either constrained Levenberg-Marquardt or trust-region-reflective algorithms, to fit the c(f) curve. It should be noted that if the number of data points (here, the number of different stimulation frequencies) is small, the fitted curve could deviate slightly more at certain frequency points (see the IMADSim optimisation example given in the supplementary toolbox [61]). Ultimately, this will affect the performance of the full six-parameter Hatze-Zakotnik model. For example, in Fig 10A and 10C, a larger deviation can be seen in FETi force prediction (where 5 frequencies were used for optimisation) compared to that of SETi (with 10 different frequencies).
We compare the Hatze-Zakotnik model with another non-linear second order model that was proposed by Wilson and colleagues [4]. The authors adapted and simplified the model of Ding [66] to obtain second order dynamics, and used it to describe the isometric response of locust skeletal Home Designer Pro 2017 Crack + Keygen Full Version Free to SETi stimulation [4].
The Wilson model gives the muscle force F(t) and takes a pulse train u(t) as an input. The model equations are: (Eq 7)
(Eq 8)
(Eq 9)where
(Eq 10)
Here, n defines the number of input pulses, t the time and ti the time at which the ith pulse occurs. This accounts for the assumption that the input pulses can be approximated as impulses. Therefore, the motoneuron spike is modelled as a square pulse or half-sine wave of width 1 ms, scaled to have an area of 1 under the pulse. Like the Hatze-Zakotnik model, the non-linear Wilson model has six parameters, but it consists of first-order differential equations. The variable x(t) is an intermediate stage in the model and represents a non-linear saturation. The parameters m and k define the shape of this non-linearity. A measure of [Ca2+]i is represented by the variable CN. The parameters τc, τ1 and τ2 are time constants, and A is a gain [4]. Note that Eqs 7 and 8 are very similar to a computationally efficient approximation of the original model as formulated by Hatze (see Eqs 3.27 and 3.29 to 3.31 in [44], where the normalised calcium concentration CN(t) is the equivalent to γ(t) and the nonlinear transform x(t) is equivalent to q(γ(t)) without length-dependency).
The parameter set for each animal (i.e. muscle) was estimated by minimising the least squares error between the measured force and the model output for the entire set of trials per animal, i.e., for single twitch and all stimulus frequencies. As described by Wilson et al. [4], the fitting procedure was initialised repeatedly seven times, with random parameter values drawn from a normal distribution. The optimisation was done using the MATLAB function lsqnonlin and the trust-region-reflective algorithm was used to find the best fit to the data. The system of differential equations was solved using a fixed step (0.0002 s or 5 kHz sampling rate), fourth-order Runge-Kutta method.
In a reference work on muscle modelling, Zajac [13] proposed a linear first-order model for activation dynamics. The envelopes of the rectified electromyogram (EMG) and of the low-pass filtered, rectified EMG, can be related to the neural-excitation input signal u(t) and the state variable associated with muscle activation a(t), respectively. It is a two-parameter model described by the following bilinear differential equation: (Eq 11)
(Eq 12) where 1/τact is the higher rate constant (when u(t) = 1) and β is the ratio of that over the lower rate constant, 1/ τdeact, when u(t) = 0 (relaxation). In other words, the model assumes that the build-up of activation of a fully excited muscle is faster than relaxation after termination of activation. In the case of the Zajac model, we use an iterative method for solving the differential equation numerically, as described in part II of the S1 Text.
Blümel and colleagues [2] proposed a simple model of activation dynamics for the stick insect mesothoracic extensor tibiae muscle. First-order low-pass filtering reproduces many aspects of isometric contractions in this muscle [67], so the activation dynamics are described by a single-pole, first-order low pass filter. The standard recursion equation for such a filter was used to obtain the following two-parameter model: (Eq 13) where filter sets the decay amplitude per time step and scaling is a factor that multiplies the input by a constant. u[n]and a[n] correspond to neural excitation and muscle activation, respectively. The time constant, tconst, of the filter depends on the time step duration and is related to the parameter filter according the following equation:
(Eq 14) where Δt is the time step duration. In our case, Δt was 0.0002 s in all simulations.
Wilson and colleagues [3], found that a third-order model was of optimal order for fitting isometric force responses over a range of input pulse frequencies. Their linear third-order model of activation dynamics is characterised as: (Eq 15) where a(t) is the muscle force and u(t) is the input pulse train. The model has four parameters, θ1to θ4. Here, we converted the third-order differential Eq 15 into a system of three first-order differential equations and solved it by using a fixed-step (0.0002 s), fourth-order Runge-Kutta method.
For quantitative comparison of the five models described above, with particular focus on their responses to constant frequency pulse trains, we developed a MATLAB toolbox IMADSim, or “Insect Muscle Activation Dynamics Simulation”. The toolbox consists of Matlab routines for simulating the models for arbitrary pulse trains and arbitrary parameter combinations. It was created in Matlab2009b (7.9.0) and comes with an additional installer file for Matlab versions above 2014 [61].
A graphical user interface (GUI) is provided so that a user can select a muscle activation model, adjust model or simulation parameters easily, and visualise the output (S3 Fig). The GUI permits the selection of one of two motoneuron types (SETi or FETi), spike shapes (half-sinusoidal utorrent pro download square) and spike generators (either constant frequency or Poisson). Arbitrary spike trains may be loaded from a file (in .mat or .txt format, comprising spike times). For constant spike frequencies, single or multiple spike frequencies can be set along with a relaxation time. There are three displays showing: (i) the time course of isometric force generation, (ii) the corresponding spike train, and (iii) the model equations. Users can save all graphs generated, as well as the corresponding data (Matlab file formats .fig or .mat).
The toolbox also includes parameter optimisation routines for the two non-linear models, i.e. the Hatze-Zakotnik and non-linear Wilson model, which are based on the Matlab LSQNONLIN solver for non-linear least squares problems. When calling the functions, several arguments can be set to customise the optimisation. If no arguments are given, default values are used. The toolbox assumes that experimental data are sampled at 5 kHz.
In addition to the toolbox, we provide experimental data from six adult female locusts (Schistocerca gregaria), comprising isometric contraction force time courses of the metathoracic extensor tibiae muscle for spike trains with different frequencies (http://doi.org/10.4119/unibi/2937068, [5]). In three animals, the slow extensor tibiae motoneuron (SETi) was stimulated. For another three animals, the fast extensor tibiae motoneuron (FETi) was stimulated.
For more details about the toolbox, functional routines, and examples, the reader is referred to the documentation file “IMADSim_Documentation.html” in the Supplementary Material [61].
Stimulus shape recorded in experiments (solid line) and used in the model (dashed line). The model stimulus is a half sine wave of length 1 ms and approximates well the depolarising phase of the signal.
https://doi.org/10.1371/journal.pcbi.1007437.s001
(TIF)
Blue lines show isometric force measurements for locust extensor muscle at different SETi stimulus frequencies are shown in the left panel (same data as in Fig 3). Forces were normalised to maximum force at a stimulation frequency of 50 Hz. Note that the tetanic force level difference between e.g. 10 and 20 Hz is larger than the difference between e.g. 40 and 50 Hz, which indicates a non-linear summation of single twitch forces. Black lines show simulated forces using the Hatze-van-Zandwijk model (Eq 1 to 3). The four parameters of Hatze’s original activation dynamics model were optimised to the single twitch of A. Van Zandwijk proposed a sigmoid scaling function (Eq 3, see inset on the right) to model the relationship between the calcium concentration and the force-producing active state. Comparison with the experimental data (blue lines) shows that the model fit is poor. For example, the tetanus fuses only at frequencies above 25 Hz and the rise time at higher stimulation frequencies is too short.
https://doi.org/10.1371/journal.pcbi.1007437.s002
(TIF)
The main panel of the GUI permits the selection of one of five published muscle activation models (two non-linear and three linear). The user can manually alter all corresponding model parameters, select one of two motor neuron types (SETi and FETi), toggle between two spike shapes (sine and square) and select one of two methods for spike generation. Experimental spike time series may be loaded from an external Matlab or text file that contains a list of individual spike times. For constant frequency stimulation, one or more spike frequencies may be set (1, 5 and 20 Hz, in the example shown). Post-stimulation relaxation time may also be set. Three displays show: (i) the time course of isometric force generation, (ii) the corresponding spike train, and (iii) the model equations. Finally, “Hold on”, “Run” and “Reset all” buttons are used to keep the multiple time courses on the display, run the simulation, or reset to default settings, respectively. Generated graphs and the corresponding data can be saved to Matlab figure (*.fig) and data (*.mat) files by selecting options in the toolbar.
https://doi.org/10.1371/journal.pcbi.1007437.s003
(TIF)
Time courses of isometric force contractions for different trains of constant frequency stimulation are shown for the two non-linear models with parameters ‘as published’ according to Hatze-Zakotnik (A) and Wilson (B). Note that model output was normalised to maximum force of the single-twitch. This was set to 0.1. Same figure details as in the top row of Fig 6, except that here the fast motor neuron (FETi) was simulated.
https://doi.org/10.1371/journal.pcbi.1007437.s004
(TIF)
Time courses of force decay after 10 s of constant frequency SETi stimulation and 1 s of constant frequency FETi stimulation were superimposed for the model (black) and experimental data (coloured). A: SETi, blue. B: FETi, red. The onset of the last stimulus spike is set at t = 0. Numbers at the start of decay indicate stimulation frequencies in Hz. Although the shape of the force signal is similar in model and experiment for SETi stimulation, the experimentally measured decay after stimulation with 20 or 40 Hz lags the onset of the modelled decay. The same is true for FETi stimulation at 50 Hz. For stimulation frequencies of 10 and 20 Hz, the experimentally measured FETi time courses show much slower decay than those computed by the model.
https://doi.org/10.1371/journal.pcbi.1007437.s006
(TIF)
Three different data sets are compared for SETi (A) and FETi (B) stimulation at different frequencies. The data sets comprise our own experimental data (black; 3 animals, per motoneuron), data published by [26] (SETi: cyan; his Fig 3F for single twitch and Fig 20 for inner and outer muscle fibre bundles; FETi: magenta; his Fig 3B for single twitch and Fig 15B for outer muscle fibre bundle), and data published by [4] (SETi: dashed blue; their Fig 2E; FETi: dashed red; their Fig 2F). Forces were normalised to the peak force at stimulation frequency 50 Hz separately for SETi and FETi.
https://doi.org/10.1371/journal.pcbi.1007437.s007
(TIF)
Model fits (black) of the Hatze-Zakotnik model (top) and non-linear Wilson model (bottom) to experimental data sets for SETi (blue) and FETi (red) stimulation. Plots for animal D (SETi) and animal 4 (FETi) show the same data in Fig 10 except that forces were not normalised to maximum force at 50 Hz stimulation frequency. For model parameter sets used see Table 2. Constant stimulating frequencies used were: 1, 7, 10, 12.5, 15, 20, 25, 30, 40 SUMo Pro License key 50 Hz for SETi, and 1, 10, 20, 30 and 50 Hz for FETi.
https://doi.org/10.1371/journal.pcbi.1007437.s009
(TIF)
We thank Greg Sutton for help with FETi data acquisition, Ansgar Büschges and colleagues for critical discussion of our results, Scott Hooper for helpful comments on an earlier version of the manuscript, and Robert Rockenfeller for constructive criticism.
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The global prevalence of migraine as a primary headache has been estimated as 14.4% in both sexes. Migraine headache has been ranked as the highest contributor to disability in under 50 years old population in the world. Extensive research has been conducted in order to clarify the pathological mechanisms of migraine. Although uncertainties remains, it has been indicated that vascular dysfunction, cortical spreading depression (CSD), activation of the trigeminovascular pathway, pro-inflammatory and oxidative state may play a putative role in migraine pain generation. Knowledge about pathophysiological mechanisms of migraine should be integrated into a multimodal treatment approach to increase quality of life in patients. With respect to this, within the integrative health studies growing interest pertains to dietary interventions. Although the number of studies concerning effects of diet on headache/migraine is not yet very large, the current article will review the available evidence in this area. All publications on headache/migraine and dietary interventions up to May 2019 were included in the present review through a PubMed/MEDLINE and ScienceDirect database search. According to the current findings, Ketogenic diet and modified Atkins diet are thought to play a role in neuroprotection, improving mitochondrial function and energy metabolism, compensating serotoninergic dysfunction, decreasing calcitonin gene-related peptide (CGRP) level and suppressing neuro-inflammation. It can also be speculated that prescription of low glycemic diet may be promising in headache/migraine control through attenuating the inflammatory state. Moreover, obesity and headaches including migraine could be attributed to each other through mechanisms like inflammation, and irregular hypothalamic function. Thereby, applying dietary strategies for weight loss may also ameliorate headache/migraine. Another important dietary intervention that might be effective in headache/migraine improvement is related to balance between the intake of essential fatty acids, omega-6 and omega-3 which also affect inflammatory responses, platelet function and regulation of vascular tone. Regarding elimination diets, it appears that targeted these diets in migraine patients with food sensitivities could be effective in headache/migraine prevention. Taken together, dietary approaches that could be considered as effective strategies in headache/migraine prophylaxis include weight loss diets in obese headache patients, ketogenic and low-calorie diets, reducing omega-6 and increasing omega-3 fatty acid intakes.
According to the reports of global burden of headache, 2016 [1], The global prevalence of migraine as a primary headache has been estimated as 14.4% in both sexes [1]. Migraine headache has been ranked as the highest contributor to disability in under 50 years old population in the world [2]. Furthermore, it has been evident that women are affected by migraine 2 or 3 times more than men and also experience more disabling, more severe attacks with longer duration and increased risk of recurrent headaches [3]. Based on the number of headache days in a month, migraine is classified into episodic migraine ((EM): having < 15 headache days /month) or chronic migraine ((CM): having ≥15 headache days /month with experiencing migraine features in at least 8 days/month) [4]. Suffering from concurrent disorders such as other neurologic and psychiatric disorders, chronic pain, cardiovascular diseases, gastrointestinal (GI) complaints, allergy or /asthma, and obesity would also make the treatment more complicated. These comorbidities may additionally be involved in the transformation from EM to CM [5, 6]. Irrespective of treatment modalities applied, trigger control, and lifestyle modification are indispensable to the successful management of migraine [7].
Therefore, knowledge about pathophysiological mechanisms of migraine should be integrated into a multimodal treatment approach to increase quality of life in patients. With respect to this, within the integrative health studies growing interest pertains to dietary interventions. Although the number of studies concerning effects of diet on headache/migraine is not yet very large, the current article will review the available evidence in this area. The dietary approaches that will be discussed throughout this manuscript include fasting and carbohydrate restricted diets (ketogenic diet (KD), low-calorie diet, modified Atkins diet (MAD), low glycemic diet (LGD),), weight loss diets, low-fat diet, elimination diet and low sodium diet. Afterwards, the possible mechanisms underlying each diet in protecting against primary headache with a focus on migraine pathogenies will be explored at the end of each section.
All publications on headache/migraine and dietary interventions up to May 2019 were included in the present narrative review through a PubMed/MEDLINE, Science Direct and Google Scholar database search. The following keywords were used: “diet”, OR “nutrition”, OR “dietary intervention”, OR “ketosis”, OR “fasting”, OR “glycemic index”, OR “carbohydrate”, OR “fat”, OR “protein”, OR “weigh reduction”, OR “obesity”, OR “food elimination”, OR “sodium”, AND “chronic migraine”, OR “episodic migraine”, OR “tension type headache”, OR “headache”, OR “treatment” AND “inflammation”, “endothelial function”, OR “platelet aggregation”, OR “pain”, OR “nociception”. Reference of included articles was evaluated, and relevant studies were also included in the current review. All eligible studies were written in the English language and were performed on adults. A description of the studies on dietary interventions in adults with headache is summarized in Additional file 1 and the studies on pediatric patients and adolescents are summarized in Additional file 2. The majority of included articles were case studies, case-series, case-control and clinical trials.
Among lifestyle modalities, nutraceuticals and diet play a notable role in headache/migraine and therefore adjusting one’s diet could be useful in preventing and treating headaches [8,9,10]. The main components of a comprehensive diet include carbohydrates, proteins, fats, vitamins, and ions. It is not clear whether these dietary factors prevent or provoke headache attacks [11]. The initiation of a headache/migraine attack may occur following consumption of specific food items. These food items should be identified and eliminated [12, 13]. Moreover, making specific dietary recommendations based on patients’ needs, and types of comorbidities could be effective in reducing the frequency of headache or even preventing initiation of an attack [12]. The underlying comorbidities of the patients that have gained a special attention when making dietary advices consist of obesity, seizure, GI disorders, depression and anxiety, and food intolerance [12].
So far, the effects of different types of diets have been studied in relation to migraine and headache [14,15,16,17,18]. It is speculated that dietary interventions could affect headache/ migraine characteristics through a variety of mechanisms. These mechanisms may include affecting serotoninergic dysfunction, neuronal excitability, levels of factors with a role in migraine pathogenesis (such as Calctonin-Gene-Related-Peptide (CGRP), nitric oxide (NO), adiponectin, and leptin), brain mitochondrial function, neuro-inflammation, hypothalamic function, and platelet aggregation [17, 19,20,21,22,23,24,25,26,27,28,29,30]. For example, obesity, which is highly related to western dietary patterns [31], is also thought to be prevalent among headache patients [32]. It has been proposed that headaches could be improved following reducing excessive weight [32,33,34].
Before long the great philosophers applied fasting as a means of therapy (https://www.allaboutfasting.com/history-of-fasting.html). KD was initially designed in order to stimulate the ketotic effects of fasting. Using KD for treating refractory epilepsy dates back to the time of Hippocrates [35]. Then after, in recent years it received growing attention as a potential treatment for other neurological disorders [36, 37]. In the following section, the effects of different types of fasting, low caloric, ketogenic, MAD and LGD on headache/migraine will be explored.
In one case report and one case series [38, 39], modified fasting was the mean of producing ketosis in headache patients. In the specific case report, a woman with chronic headache used modified fasting diet with 3–4 high protein/low carbohydrate (200 kcal) shake a day. After ketosis establishment, the headache attacks disappeared. This effect remained for 7 months after stopping fasting [38]. In the mentioned case series study, 51 adults with chronic migraine followed a low-calorie diet (1200–1500 kcal/d) for several months. Significant reduction in headaches days and abortive medication consumption was observed. Twenty eight percent of the sample reached complete remission from migraine headache. Also, continuous improvement was noted for 3 months after stopping low calorie diet [39].
Both KD and the MAD have been widely prescribed in the treatment of patients with intractable epilepsy [15]. Carbohydrate content in KD is highly restricted which results in inducing fasting in addition to rapid weight loss and elevating metabolism of fat and thus producing ketone bodies (KB) [15, 40, 41]. While carbohydrate restriction in the MAD is lesser than KD, MAD does not require a prior induced fasting state, restriction of energy, protein or fluid and is thereby applied on out-patients [40, 42]. Likewise to anticonvulsant medications, the beneficial effects of KD and MAD on the conditions such as neurodegenerative disorders, brain tumors, autism, amyotrophic lateral sclerosis, and migraine have also been of interest as a therapeutic strategy [15, 41, 43]. These type of diets are thought to play a role in neuroprotection, mitochondrial function and enhancement of production of ATP [15, 43].
The application of KD for treating headache dates back to 1928 [44]. In the first case-series, a group of 18 migraineurs treated with KD, half of the studied population reported some relief [44]. Since then up to now, only a few case reports have addressed specifically the encouraging effect of KD on migraine/headache [38, 40, 45, 46]. In a recent study [40], 18 adults suffering from migraine without aura were investigated during interictal state. They were prescribed with KD for 1 month. Significant improvements were reported in frequency and duration of their migraine attacks. This could be explained by the fact that KD regulates the balance between inhibition and excitation at the cortical level as shown by normalizing the neurophysiologic tests findings including interictally decreased visual (VEPs) and median nerve somatosensory (SSEPs) evoked potentials habituation [40].
Di Lorenzo and colleagues compared ketogenic and low-calorie (1200–1500 kcal/d) diets, in a group of 108 migraineurs [14]. KD was superior to low-calorie diet with a 90% responder rate, while low calorie diet was not effective [14]. Di Lorenzo et al. [41] have additionally reported migraine remission following ketosis in an open-label study on 96 migraine suffers. During the ketogenic phase (the first month of intervention) in ketogenic very-low-calorie diet or KD group (n = 45) a significant improvement in headache related features including frequency of attacks, number of headache days and medications use was demonstrated independent of weight reduction. Continuous improvement was also observed for a couple of months after stopping the diet. It is of note that although the frequency of headache days rapidly decreased in patients who followed KD, it worsened when they stopped the diet throughout the transition period (from first vs. second month of study). Moreover, there was a significant reduction in frequency of headache days and medications use and headache attacks frequency following prescribing an standard low-calorie diet in the other intervention group (n = 51) after 3 and 6 months, respectively [41].
Consuming more than half of the total energy (50–55%) from carbohydrate has been generally accepted as a healthy diet for several decades [11]. In some circumstances such as epilepsy, weight management, diabetes, and hyperlipidemia, LGD has been proved as an effective alternative [47]. In LGD, daily carbohydrate intake is restricted to 40–60 g with a glycemic index (GI) of less than 50 relative to glucose [48]. So consumption of white bread, sugar, chocolate, sweets, pastries, rice, potato, corn, jams, honey, molasses, ready-made fruit juices, sugary carbohydrate drinks, watermelon, and melon would be limited [49]. LGD is a therapeutic dietary option with remarkable advantages including its increased tolerability and a low incidence of side effects [50]. In LGD carbohydrate is mainly come from legumes, vegetables, fruits, and high fiber cereals [47].
In a RCT by Evcili et al. in 2018 [49], 350 migraineurs were allocated (n = 1:1) either to a low GI diet group or to a prophylactic medications group (who received propranolol, flunarazine, amitriptyline). One month after dietary restriction, frequency of attacks were reduced significantly in both groups. After 3 months, headache intensity was also significantly reduced following the low glycemic index diet [49]. According to the results of different studies, LGD, at least in part, imposes its effect by modifying inflammatory responses. In a clinical study designed to evaluate the impacts of a low glycemic (GI 38) legume-enriched (250 g/d) diet compared to a healthy American diet (GI 69), it was shown that soluble TNF-a receptors II and CRP levels were significantly attenuated by adherence to LGD [51]. To summarize, it can be speculated that prescription of LGD may be promising in migraine control, However, more RCTs are required in order to fully elucidate the effects of LGD on migraine characteristics.
Despite several animal studies have been conducted on the effects of ketosis on different aspects of metabolism [52, 53], the exact pathway by which it may affect on CSD, and trigeminal activation has not yet been clarified. However, several mechanisms have been proposed in the literature [14, 52, 53]. According to in vitro research, it is thought that ketosis may attenuate the severity of migraine headache through compensating serotoninergic dysfunction, inhibition of neuronal excitability, decreasing CGRP synthesis and release, and cortical spreading depression (CSD) and by brain mitochondrial function improvement [14, 52, 53]. Moreover, studies on mouse models indicated that ketosis may prevent neurogenic inflammation [19, 54] which is believed to play an important role in migraine pathogenesis [28]. Also, animal studies revealed that ketosis might increase neuropeptide Y (NPY) and agouti-related protein (AgRP) levels through stimulating ghrelin secretion from the stomach during fasting state. It has been suggested that hypothalamic AgRP and NPY may reduce CGRP level subsequently [55].
The relationship between primary headaches and obesity was first suggested by Scher and colleagues in 2003 [56]. In a prospective population-based 11 months follow-up, 3% of controls developed chronic daily headache (CDH). Obese subjects (body mass index (BMI) ≥ 30), had a fivefold increase in the relative risk of developing CDH compared to normal weight individuals. Odds of CDH were three times higher in overweight patients (BMI: 25–29) [56]. In this regard, weight reduction is among the suggested interventions for headaches due to idiopathic intracranial hypertension [33]. Although data about the effects of weight loss on primary headache control is limited, the association between migraine and obesity has been a growing field of interest in the recent years. According to an observational study, subjects with obesity would experience more frequent and severe headaches compared to normal-weight individuals [34]. Besides, both abdominal and general obesity have been reported to be independent risk factors for headache development [32]. Studies concerning the effects of weight reduction on migraine applied two approaches including non-surgical modalities, especially dietary intervention, or surgical approaches, particularly bariatric surgery [57,58,59]. An open-label trial compared a low-calorie diet with KD found that achieving a significant weight loss through each of these interventions could result in a decrease of headache frequency [41]. In a recent trial, bariatric surgery was compared with a multi-intervention approach including low-calorie diet and aerobic exercise program. With the comparable amount of weight loss, bariatric surgery offered a better improvement in headache days and attack duration than non-surgical interventions [58]. Also, 2 observational studies [57, 59] proposed a decrease in migraine intensity, frequency and disability in obese women suffered from migraine after bariatric surgery [57, 59]. Indeed, these studies showed promising results [57,58,59]. However, a recent small single blind trial examined the effect of low-calorie diet on migraine and found no significant effect [14].
To conclude, although considerable efforts have been made in this field, the results about the direct effects of weight loss dietary intervention on migraine/headache are not conclusive yet. Differences in assessing and modifying life style parameters might influence the results of the studies on the effect of weight loss in controlling headache attacks. In the following part, the proposed mechanisms of the association between excess body weight and migraine will be discussed. Notwithstanding, when it comes to shared pathophysiological pathways, inflammation received most attention.
In addition to obesity, hypertension, dyslipidemia, insulin resistance, and augmented inflammation, that all are believed to be components of metabolic syndrome, tend to be highly prevalent diseases in migraineurs [49, 60]. Recent studies have reported that insulin level may also be higher among migraineurs. About 11.1% of these patients may suffer from IR [61]. It was also noted that IR might correlate with attacks duration in migraine suffers [62]. Migraine and metabolic syndrome are usually comorbid, though no causal relationship has yet been established [60]. Moreover, the association between metabolic syndrome components and migraine characteristics including frequency of headache attacks, severity and duration needs further studies [60]. Although no specific treatment for migraine and concurrent metabolic syndrome has been proposed to date, general recommendations are given on following a weight reduction plan including diet and physical activity, proper sleep duration, and reducing stress levels [60].
The hypothetical relationship between obesity and migraine has been linked to an elevated release of pro-inflammatory markers and neuroinflammation that might be principally involved in migraine pain genesis [28]. Among the studied proinflammatory agents, elevated level of C-reactive protein (CRP), which is known as a marker of systemic inflammation, has been reported both in obese individuals and patients with migraine. It seems there could be an epidemiological association between CRP and migraine headache onset [63, 64]. Furthermore, a rise in pro-inflammatory factors, such as interleukin (IL)-1β, IL-6, tumor necrosis factor (TNF)-α and leptin were reported in obese individuals, while anti-inflammatory agents including adiponectin seem to be increased in this population. These events finally lead to a persistent low-grade inflammatory status [20,21,22]. On the other aspect, IL-1β, IL-6, TNF-α levels have also been shown to be elevated in migraineurs especially during their attack phases [63, 65,66,67,68,69].
In addition, due to the important role of CGRP in migraine pathogenesis, the neuropeptide and its receptors are predominantly important targets for migraine treatment [29]. On the other hand, evidence proposed an increase in plasma CGRP level in adults with obesity, which is also observed in patients with migraine [29, 30, 70, 71]. Moreover, it has been proposed that the administration of CGRP induces the accumulation of fat in obese animal models. Also in murine model, an elevation of CGRP level was reported before obesity onset [70,71,72,73]. Substance P (SP) in another factor which is likely to play a role in migraine attack pathogenesis that was also detected in adipose tissue and shares a role in fat accumulation and the start of the inflammatory cascade related to obesity [70].
Further, in recent years, the relationship between adipocytes released factors, known as adipokines (e.g. adiponectin and leptin) and migraine headache has given more insight into the contribution of adipose tissue in the pathophysiology of migraine [23, 24]. Although more studies are required to make a definite conclusion, the current evidence propose that adiponectin concentration might be increased between attack phases whereas it may be decreased during migraine attacks [23, 24]. It has also been mentioned that the level of this factor might be mediated following prophylactic treatment of migraine with topiramate [23]. Thus, it could be hypothesized that chronic rise in adiponectin level might be beneficial in migraine improvement [23]. This issue might be related to the reported correlation of low level of adiponectin with proinflammatory cytokine secretion and platelet aggregation [22, 74]. Adiponectin may block TNF-α and IL-6 production and on the other hand, it could induce IL-10 and IL-1 receptor antagonist (IL-1 RA) formation [74]. Thus, at lower than normal levels, adiponectin might be nociceptive [58]. On the other hand, reduced adiponectin level appears be involved in increasing the risk of developing obesity, atherosclerosis and diabetes [22, 75]. Additionally, increased leptin level is thought to induce secretion of proinflammatory factors that play a role in migraine (IL-6 and TNF- α) and NO, through NFκβ signaling pathway [24, 74]. The current findings reported leptin administration in Wistar rats could diminish the threshold of pain [76]. On the flip side, enhancement of leptin levels following weight reduction has been noted [23]. However, the results of the researches concerning the association between leptin levels and migraine have not been conclusive yet [23]. Nonetheless, it is likely that migraineurs might have lower leptin levels in ictal phases and higher levels of leptin during inter attack periods. Besides, there may be a negative relationship between leptin and pain intensity [23].
Some neurotransmitters, such as serotonin (5HT), are responsible for food consumption and body weight regulation which are controlled by hypothalamus and seems to be involved in sense of fullness [25]. On the other hand, the increment in serotonin concentrations in ictal periods in migraine can possibly be attributed to the secretion of serotonin from platelets that induce vasoconstriction of arteries and affects CSD development [26, 27].
Another appetite regulator, which also might be contributed to migraine, is orexin A. An increase in CSF level of orexin has been observed in migraineurs [63]. Orexin A could have antinociceptive characteristic and may probably play a role in compensatory reaction to pain and also contribute to hunger perception [63]. Additionally, orexinergic system dysfunction may be associated with homeostatic pathways which are involved in risk of attack generation, migraine nociception and characteristics, as well as its premonitory stage including appetite alteration [77]. Evidence showed that orexin A administration in murine model stimulates hunger and postpones the sense of fullness [63]. Therefore, applying the medications that can affect the orexinergic system may ameliorate migraine associated gastrointestinal features [77]. However, more studies are needed in order to address the association between orexin A and migraine headache in obese and non-obese individuals and explore the effects of drugs that target this system in migraineurs.
Besides, available data suggest that hypothalamic NPY may contribute to the etiology of weight gain among migraineurs who received prophylactic treatment [77]. For instance, NPY concentrations in plasma of migraine suffers may be elevated following treatment with flunarizine or amitriptyline. It has been also proposed that the weight gain following drug therapy may probably be related to changes in leptin transport system or sensitivity to leptin [77].
Five studies addressed the effect of low-fat diets as means of migraine/headache prophylaxis [78,79,80,81,82]. In 1999, a trial conducted to assess the role of fat-reduced diet for migraine control in 54 adults. Patients were instructed to restrict their fat intake to less than 20 g/d for 12 weeks, after 28 days of run-in period. They reported a notable reduction in headache frequency, intensity, and need for abortive medication [78]. An open-label, randomized cross-over trial investigated the effect of a diet change in comparison with a placebo supplement on migraine patients. For the first 16-week duration, 42 individuals were randomly allocated in intervention group (who were prescribed a low-fat vegan diet for 4 weeks, followed by elimination diet for 4 weeks, followed by reintroduction diet for the last 8 weeks group, n = 21) or placebo group (who were supplemented with 10 mcg alpha-linolenic acid + 10 mcg vitamin E as placebo, n = 21). Then after a 4-week washout period was considered and the studied subjects in either group crossed over to the other group. A decrease in headache intensity, frequency, and use of abortive medication were observed following the intervention; However, in the mentioned trial, details regarding dietary fat composition were not noted [79]. In another cross-over study on 63 adults with episodic or chronic migraine, low lipid diet (< 20% of total daily energy intake) for 3 months significantly reduced frequency and severity of headache attacks. In this study, participants did not reduce total fat intake to less than 45 g/d and used olive oil as the main source of fat intake [80]. Additionally, based on the theory of the probable effects of different fat types on headache characteristics, a randomized study assessed the effect of omega-3 and omega-6 intake. Fifty-five adults with CM were either reduced omega-6 fats intake or reduced omega-6 fats along with increased omega-three consumption. After 12 weeks, individuals on high omega-3 combined with low omega-6 diet showed a higher headache improvement compared to headache patients on the reduced omega-6 diet [81]. In another randomized double-blind controlled trial on 80 patients with EM, effect of omega-3 supplementation (2500 mg/d) was compared with either nano-curcumin (80 mg/d) or placebo. After 2 months of supplementation, combination of nano-curcumin and ω-3 fatty acids lowered the expression of TNF-α mRNA and serum level of TNF-α. Headache frequency was also reduced in all treatment groups (incl. Nano-curcumin, omega-3, and combination of omega-3/nano-curcumin), with two-fold higher effect in combination group [82].
Amount and type of fat intake affect inflammatory responses [16]. The balance between the omega-6 and omega-3, two main fatty acids that compete with arachidonic acid as eicosanoid biosynthesis precursor, contribute to inflammatory control in response to the environmental metabolic changes. Prostaglandins (PG), which are made from essential fatty acids, take part in platelet function and regulation of vascular tone. PGs also play the principal role in controlling acute and chronic inflammation [16]. PGE1, downstream metabolite of linoleic acid (omega-6), is one of the most potent vasodilators. PGE1 has been shown to cause headache [83]. On the other hand, omega-3 fatty acids (i.e. eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA)) might probably attenuate platelet aggregation [84] and affect serotonin biosynthesis pathway or the function of 5HT receptors [85].
It is generally believed that high-fat diet elevates plasma LDL-cholesterol and consequently increases platelet agreeability [86]. Studies reported hypercoagulation in serum samples obtained from healthy subjects after a high-fat meal [78, 87]. On the flip side, it has been suggested that migraine attack could be initiated following and condition that causes platelet aggregation, through serotonin secretion and its consequent effects on blood vessels, and NO and PGs production. The secretion of these factors simultaneously may contribute to head pain initiation in migraine [88].
In particular, it is proposed that vulnerability to migraine is likely to be related to constant low concentration of serum serotonin and increased sensitivity to serotonin agonists during attacks, probably due to a defect in serotonin metabolism [26, 27]. In this regard, suppressing platelet aggregation might have therapeutic value in migraine prevention [88]. Therefore, any modalities in dietary fat intake that results in modulating plasma free fatty acids and plasma lipid profile and consequently reducing platelet aggregation, seems to decrease the frequency and duration of migraine headache [78, 88].
Each headache patients may have a specific trigger or a unique set of triggers. It is known that certain types of foods and beverages can act as headache triggers [13]. Cheese, chocolate, citrus fruit, alcohol, coffee, tomatoes, carbohydrates, leavened products and red wine are among the proposed foods that may trigger migraine attacks [13, 89, 90]. However, there is not any consensus between previous studies on identifying food triggers in headache. For example, as mentioned, chocolate have been introduced as one of triggering foods of headache; while a double-blind trial by Marcus et al. [91] performed in order to assess the effects of chocolate compared to carob on 63 female subjects suffering from with chronic headache, yielded different results. The trial was conducted following prescribing a diet in which vasoactive amine-rich foods were restricted for 2 weeks. However, after administration of chocolate and carob (both two samples), there was not any differences in provocative effects of these agents on headache [91].
Moreover, there has been speculation about the way that food triggers may act in migraine attack initiation and some probable mechanisms are proposed: the ‘amine hypothesis’, “allergic” mechanism or histamine/NO caused vasodilatation; though, none of these suggested mechanisms has yet been established by adequate evidence [13].
In the following paragraphs, the studies regarding elimination diets in headache patients have been described. A few studies assessed the effect of elimination diets in controlling headache among adults. Two randomized controlled trials (RCT), applied the personalized method for eliminating trigger food in migraine suffers, using IgG antibodies to food antigens [92, 93]. Although, the 12-week parallel-group trial on patients with migraine like headaches that examined the impact of the elimination diet compared to a sham diet failed to show any differences between the 2 studied arms [93], the other study demonstrated beneficial effects in reducing migraine headaches [92]. In this research, the effect of the eliminating diet in migraineurs, who also suffered from irritable bowel syndrome was explored. It was reported that a diet excluding provocative foods in comparison with provocation diet could effectively attenuate the symptoms of both disorders with positive effects on patients quality of life [92]. Similarly, a small cross-over RCT showed that individualized elimination diet could reduce migraine frequency and abortive medication need, compared to a standard diet after six-week [18].
Some headache suffers report that specific foods only provoke headache in combination with stress or extended physical exercise. Both conditions are identified as headache triggers and lead to the production of proinflammatory cytokines [18]. Therefore, it seems that applying elimination diets in headache patients with food sensitivities could be effective in preventing migraine attacks, though more studies are needed [94].
Some patients may also be sensitive to other triggers like histamine, which is related to impairment of detoxification caused by low activity of di-amino-oxidase. Histamine thus, may play an additional role which should be considered in the elimination diets [18]. A sample of 28 patients who suffered from chronic headache attacks were assigned to complete a histamine-free diet for 4 weeks. Nineteen patients had a 50% or greater decline in their headache attacks. Also, the number of headache attacks and analgesic medication consumption significantly decreased following the histamine-free diet [95].
With respect to the probable link between migraine and allergy [96, 97], and due to the beneficial effects of histamine-reduced diet on histamine intolerance symptoms and allergic disorders [98, 99], it can be hypothesized that this type of diet might be promising in migraine control particularly among allergic patients.
The mechanisms of IgG-mediated food allergy have not been entirely clarified, but it has been suggested that an increase in production of pro-inflammatory mediators and IgG antibodies through food allergy reaction can induce an inflammatory state that may play a crucial role in the migraine pathophysiology [92]. In both migraine and food sensitivities, inflammation induced by food could make the pro-inflammatory environment which is needed for the induction of headache by other triggers [18]. In this regard, when we focus on inflammation caused by food, a specific indicator is required. Except for IgG4, all IgG subclasses can cause an inflammatory response in exposure to the respective antigen [18]. Accordingly, specific IgG can thus be considered as an ideal tool for a vast number of foods to identify individually suspected food items and enables adjusting eating habits in order to prevent chronic inflammation and occurrence of migraine in sensitized patients [18].
According to the findings of a large sample population-based cohort study there might be a negative relationship between blood pressure and headache occurrence [100].
Therefore, it may be logical to anticipate that dietary interventions that reduce blood pressure, could also lower headache occurrence. In this regard, certain nutritional strategies for lowering blood pressure including dietary approach to stop hypertension (DASH) diet and controlling the amount of sodium intake [101, 102], could be considered as a matter of interest in studies on headache prophylaxis. Available evidence on sodium intake in relation to headache has mainly focused on monosodium glutamate (MSG) intake on headache initiation [103]. Otherwise, in a descriptive study on 266 women with migraine headache, severe headache (measured by visual analogue scale (VAS): 8–10) was 46% less prevalent in subjects with the greatest adherence to the DASH diet. Also, the frequency of moderate headaches (VAS: 4–7) was 36% lower in this group compared to the individuals with lowest adherence [104]. However, the result of the only multicenter, randomized clinical trial on the effect of DASH and low-sodium diet on headache is to somehow different. The study was performed on 390 participants in three 30-days phases (i.e. 1: high sodium diet, 2: intermediate sodium diet and 3: low sodium diet in a random allocation). The occurrence of headaches was not different in DASH group compared to controls, following either phases of low, intermediate and high sodium diets. However, headache risk was lower in low versus high sodium intake, both in DASH diet and control groups [105]. In sum, according to these findings, the current data on the effects of dietary sodium intakes on headache characteristic is not conclusive yet. Thereby, except for those migraineurs suffering from concurrent hypertension [106], more researches are needed to be able to make certain advice for optimal sodium intake in migraine patients.
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Brain-derived neurotrophic factor (BDNF) is critically involved in the pathophysiology of chronic pain. However, the mechanisms of BDNF action on specific neuronal populations in the spinal superficial dorsal horn (SDH) requires further study. We used chronic BDNF treatment (200 ng/ml, 5–6 days) of defined-medium, serum-free spinal organotypic cultures to study intracellular calcium ([Ca2+]i) fluctuations. A detailed quantitative analysis of these fluctuations using the Frequency-independent biological signal identification (FIBSI) program revealed that BDNF simultaneously depressed activity in some SDH neurons while it unmasked a particular subpopulation of ‘silent’ neurons causing them to become spontaneously active. Blockade of gap junctions disinhibited a subpopulation of SDH neurons and reduced BDNF-induced synchrony in BDNF-treated cultures. BDNF reduced neuronal excitability assessed by measuring spontaneous excitatory postsynaptic currents. This was similar to the depressive effect of BDNF on the [Ca2+]i fluctuations. This study reveals novel regulatory mechanisms of SDH neuronal excitability in response to BDNF.
Injury to, or disease of, the somatosensory system frequently generates chronic and sometimes intractable neuropathic pain1, 2. In experimental animals, peripheral nerve damage, such as that generated by chronic constriction or section of the sciatic nerve, induces pain-related behaviours that serve as a model for human neuropathic pain3, 4. Seven or more days of sciatic nerve injury promote an enduring increase in the excitability of first order primary afferent neurons5,6,7,8. These become chronically active and release a variety of mediators (cytokines, chemokines, neuropeptides, ATP and growth factors) that predispose spinal microglia to a more ‘activated’ state9,10,11,12,13,14. These in turn, release further mediators, including brain derived neurotrophic factor (BDNF) that promote a slowly developing, but persistent increase in excitability of second order neurons in the spinal dorsal horn. This ‘central sensitization’ is thought to be responsible for the allodynia, hyperalgesia, spontaneous pain and causalgia that characterize neuropathic pain3, 15. Spinal actions of BDNF involve alteration in Cl- gradients such that the normally inhibitory actions of GABA become excitatory16, 17. There is also increased excitatory synaptic drive to putative excitatory neurons9, 18. This results in an overall increase in excitability as monitored by confocal Ca2+ imaging in organotypic cultures of rat or mouse spinal cord9, 18,19,20. Despite this, the long-term effects of upregulated BDNF on neuronal plasticity in the superficial dorsal horn (SDH) are not fully understood.
In our previous work, we developed and used a model to study neuropathic pain whereby defined-medium spinal organotypic cultures are treated long-term (5–6 days) with BDNF (200 ng/ml)9, 20,21,22,23. BDNF treatment is administered at day 20 or more after organotypic cultures are established to allow neurons to mature to a more ‘adult-like’ state. We demonstrated that lamina II neurons in these organotypic cultures are similar to those in acute slices from young adult rats21. We also showed that the changes that occur in spinal organotypic dorsal horn neurons ‘mirror’ those observed in dorsal horn neurons from acute slices obtained from chronic constriction injury (CCI) rats at peak mechanical hypersensitivity levels20. Naïve cultures display spontaneous oscillatory changes in intracellular Ca2+ levels [Ca2+]i and we have previously shown that the amplitude and frequency of these changes in [Ca2+]i are profoundly increased following 5–6 days treatment with BDNF20.
We used this model and a new Frequency-independent biological signal identification (FIBSI)24 program to quantitatively measure the fluctuations of [Ca2+]i and examine their synchronicity. Unexpectedly, we observed two opposite effects of BDNF that appeared to occur simultaneously. First, BDNF caused an overall decrease in fluctuation size. Second, we noticed a particular population of SDH neurons in naïve cultures that did not display typical, marked fluctuations of [Ca2+]i. This population of ‘silent’ neurons was never seen in BDNF-treated cultures, suggesting that BDNF unmasks these ‘silent’ neurons and causes them to become spontaneously active. We next investigated the role of gap junctions in mediating the [Ca2+]i fluctuations; application of the gap junction blocker octanol to chronic BDNF-treated neurons revealed a subpopulation of neurons that generate low-frequency, large [Ca2+]i fluctuations. Further pharmacological experiments indicated the [Ca2+]i fluctuations in active neurons are regulated by diverse mechanisms including voltage-gated calcium and sodium channels as well as GABA and NMDA receptors. Finally, we used FIBSI to reanalyze a previous dataset of spontaneous excitatory postsynaptic current (sEPSC) recordings from BDNF-treated dorsal horn neurons in order to qualitatively compare the effects of BDNF on the sEPSCs and [Ca2+]i fluctuations in SDH neurons. The depressive effects of BDNF on the sEPSCs in delay and tonic-firing SDH neurons were consistent with the effects on the [Ca2+]i fluctuations. This study reveals novel mechanisms of BDNF regulation of dorsal horn excitability, which has implications for the study of chronic neuropathic pain physiology.
In previous work we identified that chronic BDNF treatment (200 ng/ml, 5–6 days) induces Ca2+ fluctuations compared to naïve cultures20, 25 (Supplementary Video 1A, B). In this study, we performed a detailed quantitative analysis of the fluctuations in [Ca2+]i using Fluo-4 AM Ca2+ imaging and the FIBSI analysis program24 to further establish the nature of BDNF-induced changes in lamina II neurons of the SDH. Refer to Fig. 1A1–A3 for an example of [Ca2+]i fluctuations detected by the FIBSI program. Inspection of the FIBSI-processed recordings revealed a subset of naïve neurons, which were relatively ‘silent’ (Fig. 1B, left; raw recordings in Supplemental Fig. 1) displaying only small amplitude fluctuations for the duration of the recording (‘silent’ = 1.7 ± 0.04 AU; naïve = 53.7 ± 1.6 AU; BDNF = 25.5 ± 0.7 AU). No ‘silent’ neurons were found in the BDNF treatment group (Fig. 1B, right). Refer to Table 1 for detailed statistical information. This stark contrast suggested BDNF may be unmasking, or activating, these ‘silent’ neurons. We grouped the fluctuations in the ‘silent’ neurons together for further comparison to the fluctuations in the other active naïve neurons and BDNF-treated neurons. Log-transformed fluctuation amplitudes (Fig. 1C) and area under the curve (AUC; Fig. 1D) were larger in the active naïve population and in BDNF treated neurons compared to ‘silent’ neurons. Unexpectedly, both measures were significantly decreased in the BDNF condition compared to naive. A plausible explanation for this effect was that chronic BDNF induced a concomitant enhancement of [Ca2+]i fluctuations in some neurons and depression in others. Indeed, the effects of BDNF on the relative frequencies (%) for the fluctuation amplitudes (amplitudes were normalized to the largest fluctuation in each neuron) revealed a biphasic effect; frequencies of the smaller- and larger-amplitude fluctuations were greater in the BDNF-treated neurons compared to the naïve and ‘silent’ neurons (Fig. 1E). We next compared the mean parameters between neurons in order to control for differences in sampling periods. A < 10% maximal fluctuation amplitude cut-off was applied to each neuron to reduce the effects of low-amplitude noise. As expected, nonparametric comparisons indicated that fluctuation amplitude (Fig. 1F) and AUC (Fig. 1G) were both greater in the naïve and BDNF-treated neurons compared to the ‘silent’ neurons, but were not significantly different between naïve and BDNF-treated neurons. Further analysis of the fluctuation kinetics indicated no effect on duration (Fig. 1H), a modest, but significant decrease in rise time (i.e., peak time is more negative) compared to the ‘silent’ neurons (Fig. 1I), and no effect on frequency (Fig. 1J).
BDNF-induced fluctuations of [Ca2+]I in superficial dorsal horn neurons. (A1–A3) Example recording [Ca2+]i fluctuations sampled in a SDH neuron and the processing steps used by the FIBSI event-detection program. The running median (red line in A1, window = ~ 5 s) was calculated based on the raw amplitude (AU) and time coordinate data, then the peaks below the running median were traced to form a reference line (red line in A2), and then the Ramer–Douglas–Peucker algorithm detected waveforms above the reference (red x at event peaks). (B) Left: Examples of [Ca2+]i activity detected by FIBSI in 3 different neurons. Right: The proportion of ‘silent’ neurons in the BDNF-treatment condition was significantly reduced compared to the naïve condition. Proportions were compared using a Fisher exact test. (C, D) Violin plots of the log-transformed values for fluctuation amplitude and AUC. Fluctuation amplitudes and AUC were significantly increased in naïve and BDNF-treated neurons compared to the ‘silent’ neurons, while both measures in the BDNF-treated neurons were significantly reduced compared to naïve. Neurons sampled: ‘silent’ n = 20 (849 fluctuations), naïve n = 78 (2651 fluctuations), and BDNF n = 57 (1555 fluctuations). Median with quartiles shown. Comparisons made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. (E) Relative frequency (%) of the fluctuations binned by amplitude. Amplitude values were normalized to the largest fluctuation in each neuron. (F–G) The mean fluctuation amplitude and AUC in the naïve and BDNF-treated neurons were significantly increased compared to the ‘silent’ neurons. (H–J) Further analysis revealed little to no effect of BDNF on fluctuation duration or frequency, but the fluctuations in ‘silent’ neurons exhibited longer rise times. For scatterplots F–J, the fluctuations that were < 10% the maximal fluctuation amplitude in each neuron were omitted to reduce the effects of low-amplitude noise. Scatterplots in F–J show the median with the interquartile range. All comparisons of medians in F–K were made using Kruskal–Wallis tests and Dunn’s post-hoc test. **P < 0.01, ***P < 0.001, ****P < 0.0001.
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It has been shown that gap junctions have regulatory control over network activity in the substantia gelatinosa26,27,28,29. Therefore, gap junctions may play a role in regulating the properties of [Ca2+]i across the network, but the response of individual neurons to the chronic BDNF in the culture may be masked by the complex responses of the different types of neurons to BDNF20, 30. To address this and determine the potential role of gap junctions in mediating the BDNF-induced fluctuations, we recorded [Ca2+]i from SDH neurons exposed to chronic BDNF and then applied a non-specific gap junction blocker octanol at 1 mM for ≥ 3 min. We again performed a detailed analysis using the FIBSI program and a < 10% maximal fluctuation amplitude cut-off was applied to each neuron. Inspection of the FIBSI-processed recordings revealed octanol disproportionately affected some neurons compared to others (i.e., large increases in amplitude and AUC in some neurons vs minor decreases in others). Neurons were grouped based on whether the octanol treatment caused a negative (decrease, n = 39) or positive (increase, n = 40) fold change in mean fluctuation amplitude compared to the BDNF (control) condition (Fig. 2A). Paired analyses indicated blocking gap junctions with octanol decreased the frequency of the BDNF-induced fluctuations (Fig. 2B) while increasing their duration (Fig. 2C) only in the neurons that had a positive fold change in mean fluctuation amplitude. Fluctuation rise time was also significantly decreased in the same group of neurons (Fig. 2D). These data suggest that blocking gap junctions with octanol selectively disinhibited the response to chronic BDNF in a subpopulation of SDH neurons.
Effect of gap junction blocker octanol on BDNF-induced [Ca2+]I fluctuations. (A) Scatterplot summarizing the effects of octanol on mean fluctuation amplitude and examples of FIBSI-processed recordings. The two bottom recordings show a decrease in fluctuation amplitude and the two top show an increase. (B–D) Neurons were sorted based on whether they exhibited a positive or negative fold change in mean fluctuation amplitude in response to octanol. Paired analyses showed octanol selectively and significantly decreased the mean fluctuation frequency, increased the mean fluctuation duration, and decreased the mean fluctuation rise time (i.e., more negative peak time) in the neurons with positive fold changes in mean fluctuation amplitude. Control vs octanol paired comparisons for the groups in (B–D) were made using two-way repeated measures ANOVAs and Sidak’s post-hoc test. **P < 0.01, ****P < 0.0001.
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Previous work has shown BDNF-treated SDH neurons generate synchronous [Ca2+]i fluctuations20. We verified that finding by using the FIBSI-processed recordings and comparing the degree of synchrony between neurons imaged together in naïve and BDNF-treated cultures. Adjacency matrixes were generated using Pearson r coefficients; the recording of each neuron was correlated with every other neuron imaged within the same naïve or BDNF-treated culture using a Pearson correlation matrix. The adjacency matrixes for the BDNF-treated cultures showed the neurons were more positively correlated with Pearson r coefficients closer to 1 (Fig. 3A), suggesting they exhibited more synchronous [Ca2+]i fluctuations than the naïve cultures. Indeed, we observed synchronous spikes in [Ca2+]i across multiple BDNF-treated neurons compared to the naïve neurons (Fig. 3B1–B2). Raster plots depicting the [Ca2+]i fluctuation peaks revealed the presence of multiple waves of synchronous activity in BDNF-treated neurons (Fig. 3C1–C2). To further confirm this prediction, the mean coefficient for each neuron was calculated (i.e., the mean correlation of each neuron with every other neuron in its respective culture; refer to Fig. 3A for Fisher Z transformation steps). Comparison of the mean Pearson r coefficients revealed BDNF-treated neurons were more positively correlated with other neurons in the culture than naïve neurons (Fig. 3D). Considering the role gap junctions may play in regulating network activity and synchronous activity, this result led us to anticipate that blocking gap junctions may reduce BDNF-induced synchrony within the culture. We used the same approach to measure synchrony among neurons within the same cultures treated with BDNF prior to application of octanol, and then we asked whether blocking gap junctions caused a decrease in the mean Pearson r coefficients. Indeed, octanol caused a net decrease in the mean Pearson r coefficient (Fig. 3E1,E2). Closer inspection revealed octanol decreased the mean Pearson r coefficient in 100% (25/25) of the neurons in the group with decreased fluctuation amplitudes, while only 63% (22/35) of the neurons in the group with increased fluctuation amplitudes exhibited a decreased mean Pearson r coefficient. This suggests that blocking gap junctions may actually increase [Ca2+]i fluctuation synchrony in some neurons.
Synchrony of [Ca2 +]i fluctuations in BDNF-treated cultures and the effect of octanol. (A) Example adjacency matrix constructed from Pearson r coefficients corresponding to the correlation in activity between neurons in a naïve or BDNF-treated culture, each with 8 sampled neurons. Steps to calculate the mean Pearson r for each neuron within a culture using the Fisher Z transformation are also depicted. (B1,B2) Example recordings of asynchronous [Ca2+]i fluctuations in naïve neurons and synchronous [Ca2+]i fluctuations in BDNF-treated neurons. Recordings shown were processed by FIBSI for event detection and each one was normalized (0 to 1 scale) before calculation of the adjacency matrix in A. (C1,C2) Raster plots depicting the fluctuation peak times detected by FIBSI for the recordings in B1-B2. Red stars in C2 denote the presence of synchronous waves of [Ca2+]i activity. (D) Neurons sampled from BDNF-treated cultures exhibited significantly greater mean Pearson r coefficients compared to neurons sampled from naïve cultures. Neurons were imaged from 4 naïve cultures and 3 BDNF-treated cultures, each with ≥ 8 neurons imaged. Medians shown with the 95% confidence interval. Medians were compared using a Mann–Whitney test. (E1,E2) Treatment with octanol significantly decreased the mean Pearson r in neurons sampled from cultures exposed to chronic BDNF, and this treatment effect impacted both subsets of neurons binned based on their initial response to octanol. Neurons were imaged from 4 cultures. The control vs octanol paired comparison in C1 was made using a Wilcoxon rank sum test. Control vs octanol paired comparisons for the groups in C2 were made using a measures two-way repeated measures ANOVA and Sidak’s post-hoc test. *P < 0.05, **P < 0.01, ****P < 0.0001.
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In order to further characterize the properties of the [Ca2+]i fluctuations, we used pharmacological treatments to block a variety of voltage-gated ion channels, glutamate receptors, and GABA receptors. Comparisons between the pharmacologically-treated neurons and the BDNF-treated (control) neurons revealed the [Ca2+]i fluctuation amplitudes (Fig. 4A) and frequency (Fig. 4B) are controlled by diverse mechanisms (example recordings and data available in Supplemental Fig. 2). The fluctuations were ablated when extracellular Ca2+ was removed, when all voltage-gated calcium channels (VGCCs) were blocked with Cd2+, when all TTX-sensitive voltage-gated sodium channels (VGSCs) were blocked with tetrodotoxin (TTX), when AMPA or kainite glutamate receptors were blocked with CNQX/NBQX, and when GABA was applied to the cultures. Interestingly, the fluctuations remained when we blocked T-type (with Ni2+), L-type (with nitrendipine), or N-type (with ω-conotoxin) Ca2+ channels, although the frequency was significantly reduced with no effect on amplitude. Riluzole, which blocks glutamate release and VGSCs, only affected the fluctuation frequency but not amplitude. The NMDA blocker D-AP5 was the only blocker to significantly reduce fluctuation amplitudes, but it had no effect on frequency. We also noted that applying GABA-A blockers bicuculline (10 µM) and strychnine (1 µM) to BDNF-treated slices (Supplementary Fig. 2P) abolishes higher frequency smaller amplitude fluctuations, leaving inducing the larger amplitude low frequency fluctuations that occur in naive cultures (Supplementary Fig. 2P). These data suggest the BDNF-induced [Ca2+]i fluctuations are controlled by a combination of VGCCs, TTX-sensitive VGSCs, GABA and NMDA receptors.
Pharmacology of [Ca2+]i fluctuations from BDNF-treated cultures. Summary of the effects of various pharmacological treatments on the (A) amplitude and (B) frequency of the [Ca2+]i fluctuations in BDNF-treated cultures compared to before drug treatment. Bars show the mean + standard error of the mean. *P < 0.001, paired t-test, n = 5–20 cells per condition.
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It is well established that BDNF release in the SDH is involved in neuropathic pain10, 16, 17. However, to our knowledge our findings are the first to show chronic BDNF treatment unmasks [Ca2+]i fluctuations in a subpopulation of SDH neurons while depressing activity in others. We had previously shown that sEPSCs recorded from SDH neurons are perturbed in BDNF-treated spinal cord organotypic cultures20. It is plausible that the newly discovered unmasked BDNF-induced [Ca2+]i fluctuations underlie changes in network excitability. We decided to readdress the effects of BDNF on network excitability by using the FIBSI program to quantitatively analyze a previous subset of sEPSC recordings in delay and tonic-firing SDH neurons (example recordings and low-pass filtering/matching methods are available in Supplemental Fig. 3). Inspection of the sEPSC interevent interval data indicated there might be natural clustering between single “small” sEPSCs and larger summated sEPSCs. We used partitioning around medoids (a medoid is like a centroid, but is restricted to an actual observation in the dataset) to cluster the sEPSCs in each neuron based on 3 sEPSC parameters: interevent interval (ms), amplitude (pA), and charge transfer (Q; essentially area under the curve). Optimal clustering (k medoids = 3–5 per neuron) identified 3 main clusters of sEPSCs that were present in all delay (Fig. 5A1,A2) and tonic (Fig. 5B1,B2) neurons in both control and BDNF treatment conditions (clusters named based on amplitude; interevent interval): (1) small; short, (2) small; long, and (3) large. Two other clusters were identified (small; mid and medium), but they were not present in all neurons or treatment conditions and were not analyzed for this study. We were mainly interested in the effects of BDNF on the cluster of large-amplitude sEPSCs in both neuron types (refer to Supplemental Fig. 4 for the “small” sEPSC analysis). Chronic BDNF treatment significantly decreased sEPSC amplitudes (Fig. 5C1,C2) and charge transfer (Fig. 5D1,D2) in the tonic-firing neurons. We did not observe a significant effect on sEPSC amplitudes in the delay neurons, but charge transfer was significantly decreased. The sEPSCs in the BDNF-treated delay neurons were narrower (control duration = 83.6 ± 1.2 ms; BDNF duration = 49.9 ± 0.4 ms). The sEPSC interevent intervals were significantly shorter in both neuron types (Fig. 5E1,E2), and peak time measurements indicated BDNF significantly increased activation of the sEPSCs in the tonic neuron (Fig. 5F1,F2).
Cluster analysis of spontaneous EPSCs in delay and tonic-firing superficial dorsal horn neurons. Three dimensional representations of the sEPSCs in (A1,A2) delay neurons and (B1,B2) tonic neurons plotted based on interevent interval (x-axis), amplitude (y-axis), and charge transfer (z-axis). Mapping of the clustering results and treatment shown for the total sample of sEPSCs. The sEPSCs in each neuron were clustered independently from the other neurons. Optimal partitioning around medoids identified 3 primary clusters present in all neurons in both treatment conditions (naming based on amplitude; interevent interval): small; short, small; long, and large. Two additional clusters were identified in some neurons, but were not present in all neurons in both conditions: small; mid and medium. Neurons sampled: 19 delay (5 control, 14 BDNF); 9 tonic (5 control, 4 BDNF). Total number of sEPSCs: 33,206 in delay, control; 108,261 in delay, BDNF; 40,658 in tonic, control; 36,521 in tonic, BDNF. The effects of BDNF on sEPSC amplitude, charge transfer (i.e., area under the curve), and peak time were assessed for the 3 primary clusters (large cluster shown, refer to Supplemental Fig. 3 for the two small-amplitude clusters). (C1,C2) Treatment with BDNF significantly decreased sEPSC amplitudes in tonic neurons, but a significant effect was not observed in the delay neurons. Cumulative distributions of the measured amplitudes shown (left) alongside the means ± standard error of the means (right). (D1,D2) The BDNF treatment significantly reduced sEPSC charge transfer measures in both delay and tonic neurons. (E1,E2) Interevent intervals for the sEPSCs within the large cluster were significantly decreased in the BDNF-treated neurons. (F1, F2) The sEPSC activation kinetics were significantly faster in the BDNF-treated neurons, but no significant effect was observed for the sEPSCs in the delay neurons. Large sEPSC cluster sample sizes: delay neurons control n = 753, BDNF n = 4539; tonic neurons control n = 2348, BDNF n = 2367. Comparisons between means were made using Brown-Forsythe and Welch’s ANOVA tests and Games-Howell post-hoc test. **P < 0.01, ****P < 0.0001.
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The present study was undertaken to better understand mechanisms underlying synchronous activity among SDH neurons and the influence BDNF may have on network excitability in naïve, uninjured organotypic cultures. Our experiments indicate long-term exposure to BDNF produces a complex set of neuron-dependent changes in network excitability in the SDH. First, BDNF causes a concomitant activation of Ca2+ signaling in a subpopulation of ‘silent’ SDH neurons (~ 20% sampled) and depression of Ca2+ signaling in many others. This finding was largely unexpected as we have previously shown chronic BDNF increases the size and frequency of [Ca2+]i fluctuations20. It is plausible that the presence of the ‘silent’ neurons can greatly influence data analysis and interpretation depending on how they are grouped together, or discarded. Another possibility is that resolution differences between the two event-detection programs used (FIBSI in the current study, Mini Analysis Software (Synaptosoft, NJ)) in the previous study) could influence analysis of the [Ca2+]i fluctuations. However, the goal of the present study was not to compare/contrast the two programs. The second significant finding was that gap junction signaling may play a crucial role in regulating BDNF-induced changes in Ca2+ signaling in the SDH. Blockade of gap junctions with octanol unveiled a subpopulation of neurons that respond to BDNF with large-amplitude, low-frequency [Ca2+]i fluctuations. This finding suggests that gap junctions modulate global network activity, either by directly or indirectly inhibiting those particular neurons with increased Ca2+ signaling. Indeed, this is supported by the third major finding that BDNF profoundly increases synchrony of the [Ca2 +]i fluctuations, and blocking gap junctions with octanol reverses BDNF-induced synchrony.
It is known that delay neurons are mostly excitatory glutamatergic neurons expressing vesicular glutamate transporter 2 (VGLUT2) and that tonic neurons are mostly GABA- or glycinergic inhibitory neurons31. We also know from the pioneering studies of Perl and others that tonic neurons mainly possess an islet morphology and that delay neurons possess a radial morphology32. While the physiological correlates of the BDNF-related [Ca2+]i fluctuations are unclear, we do not think they are being driven by delay or tonic-firing neurons in the SDH. Unlike our analysis of the [Ca2+]i fluctuations where we describe the ‘silent’ neurons, we did not observe any unmasking effect(s) of BDNF on the sEPSCs in the delay or tonic neurons. All neurons sampled in the control (naïve) condition generated visually appreciable, large-amplitude sEPSCs. The main depressive effect of BDNF on the sEPSCs (refer also to Supplemental Fig. 4) mirrored the main effect of BDNF on comparable properties for the [Ca2+]i fluctuations. A summary of each dataset (Fig. 6, table) shows BDNF reduced the amplitude and area under the curve parameters for the [Ca2+]i fluctuations and large sEPSCs. More nuanced effects were observed for frequency (reciprocal of interevent interval) and activation kinetics: BDNF increased the frequency of the large sEPSCs but had little to no effect on the frequency of the [Ca2+]i fluctuations; BDNF increased activation of the large sEPSCs in the tonic neurons and had little to no effect on the delay neurons or [Ca2+]i fluctuations. However, these data collectively suggest chronic BDNF may have reduced excitability in the delay and tonic SDH neurons, and this may provide a physiological basis for the depressive effect of chronic BDNF on the [Ca2+]i fluctuations. On the contrary, the ‘silent’ neurons are predicted to have responded to chronic BDNF with [Ca2+]i fluctuations of greater amplitude, total area, frequency, and presumably activation. Our working model (Fig. 6, bottom) posits that under naïve conditions many delay and tonic SDH neurons in organotypic spinal cord cultures are generally in an active state, with some gap junction coupling with neighboring neurons. Long-term exposure to BDNF causes a homeostatic shift that pushes the SDH into a dampened state marked by synchronous, low-level activity spread across the network via increased gap junction coupling. This synchronous activity in turn mitigates the BDNF-activated ‘silent’ neurons. From our pharmacology experiments (Fig. 4 and Supplementary Fig. 1), we also suggest that activity of GABA- and glycinergic inhibitory receptor systems are necessary to support BDNF-induced Ca2+ fluctuations. Given the role of GABA and glycine in sensory processing associated with chronic pain33, it is possible that BDNF-induced fluctuations are important for pathophysiology of neuropathic pain.
Working model summarizing the effects of chronic BDNF on SDH neurons in this study.
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Several types of oscillatory and/or rhythmic bursting activity have been previously observed in spinal cord organotypic cultures34, 35 and in ex vivo slice preparations26, 36,37,38. BDNF has also been shown to induce long term potentiation (LTP) in the spinal dorsal horn39 and is critically involved in mechanisms of synaptic plasticity, which may induce synchronous activity in different pain states. Synchronous activation of groups of spinal nociceptive neurons might contribute to the ‘electric shock’-like sensations experienced by some neuropathic pain patients40. We hypothesize that [Ca2+]i fluctuations leading to synchronous network activity may be attributed to the “shooting pains” that chronic pain patients may experience. In particular, “shooting pain” is very commonly experienced by patients with radicular lower back pain (sciatica) or trigeminal neuralgia. In these patients, stimulus movement elicits so-called “traveling waves” in which neural activity sweeps across the body-part representation in somatosensory maps41. The mechanisms of this have been attributed in part to Ca2+ waves and gap junctions 41, 42. Therefore, given the properties of the [Ca2+]i fluctuations we have identified, they would make a good candidate for the neural substrate of shooting pain. In a cohort of chemotherapy-induced peripheral neuropathy (CIPN) patients, some of whom reported shooting or burning pain in the hands or feet, serum nerve growth factor (NGF) levels were much higher compared with patients with painless or absent CIPN43. Therefore, there is a precedent for the correlation of neurotrophic factors related to BDNF to the incidence of shooting pain. Indeed it has been shown that NGF regulates the expression of BDNF44. It is has also been demonstrated that BDNF overexpression induces spasticity in rodent models of spinal cord injury, which may also explain the effect of BDNF on influencing network excitability in the dorsal horn45. With regards to the involvement of the dorsal horn in shooting pain, it has been shown that dorsal root entry zone lesioning (DREZotomy) successfully reduces shooting pain caused by brachial plexus root avulsion (BPRA)46. Also, in one patient with shooting pain caused by osteoid osteoma, a partial laminectomy was able to significantly relieve pain symptoms47. Ideally however, more experiments using human ex vivo tissue are needed to determine whether these particular [Ca2+]i fluctuations correlate with shooting pain in patients.
All experimental protocols were approved by the University of Alberta Animal Care and Use Committee, which is charged with maintaining standards set forth by the Canadian Council on Animal Care.
All procedures complied with the guidelines of the Canadian Council for Animal Care and the University of Alberta Health Sciences Laboratory Animal Services Welfare Committee. Spinal cords were isolated from embryonic (E13–14) rats and transverse slices (300 ± 25 μm) were cultured using the roller-tube technique20, 22. Since tissue is obtained from embryos, sex could not be determined. Serum-free conditions were established after 5 days in vitro. Medium was exchanged with freshly prepared medium every 3–4 days. Slices were treated after 15–21 days in vitro for a period of 5–6 days with 50–200 ng/ml in serum-free medium as described previously20. Age-matched, untreated DMOTC slices served as controls.
Each organotypic slice was incubated for 1 h prior to imaging with the fluorescent Ca2+-indicator dye Fluo-4-AM (Molecular Probes, Invitrogen, Carlsbad, CA, USA). The conditions for incubating the dye were standardized across different slices to avoid uneven dye loading. After dye loading, the slice was transferred to a recording chamber and superfused with external solution containing (in mM): 131 NaCl, 2.5 KCl, 1.2 NaH2PO4, 1.3 MgSO4, 26 NaHCO3, 25 d-glucose, and 2.5 CaCl2 (20 °C, flow rate 4 ml/min). Regions of interest (ROI) corresponding to individual cell bodies of neurons were identified based on morphology and size. Changes in Ca2+-fluorescence intensity evoked by a high K+ solution (20, 35, or 50 mM, 90 s application) or other pharmacological agents, were measured in dorsal horn neurons with a confocal microscope equipped with an argon (488 nm) laser and filters (20 × XLUMPlanF1-NA-0.95 objective; Olympus FV300, Markham, Ontario, Canada). Full frame images (512 × 512 pixels) in a fixed xy plane were acquired at a scanning time of 0.8–1.08 s/frame48. In some experiments, images were cropped to accommodate faster scan rates. Selected regions of interest were drawn around distinct cell bodies depicting neurons based on morphology as described previously49 and fluorescence intensity traces were generated with FluoView v.4.3 (Olympus).
Whole cell patch-clamp recordings were obtained from neurons in organotypic slice cultures under infrared differential interference contrast optics. Neurons selected for recording were located 250–800 μm from the dorsal edge of the cultures in an area presumed to reflect the substantia gelatinosa and up to a depth of 100 μm from the surface. Neurons were categorized according to their firing pattern in response to depolarizing current commands as tonic, delay, phasic, irregular, or transient30. Recordings were obtained with an NPI SEC-05LX amplifier (ALA Scientific Instruments, Westbury, NY, USA) in bridge balance or in discontinuous, single electrode, current or voltage-clamp mode. Neurons were sampled at 2 k Hz for 180 s when measuring sEPSCs. For recording, slices were superfused at room temperature (∼ 22 °C) with 95% O2-5% CO2-saturated aCSF that contained (in mM) 127 NaCl, 2.5 KCl, 1.2 NaH2PO4, 26 NaHCO3, 1.3 MgSO4, 2.5 CaCl2, and 25 d-glucose, pH 7.4. Patch pipettes were pulled from thin-walled borosilicate glass (1.5/1.12 mm OD/ID; WPI, Sarasota, FL) to 5- to 10-MΩ resistances when filled with an internal solution containing (in mM) 130 potassium gluconate, 1 MgCl2, 2 CaCl2, 10 HEPES, 10 EGTA, 4 Mg-ATP, and 0.3 Na-GTP, pH 7.2, 290–300 mosM.
Unless otherwise stated, all chemicals were from Sigma (St. Louis, MO, USA). Fluo-4 AM dye was dissolved in a mixture of dimethyl sulfoxide (DMSO) and 20% pluronic acid (Invitrogen, Burlington, Ontario, Canada) to a 0.5 mM stock solution and kept frozen until used. The dye was thawed and sonicated thoroughly before incubating with a DMOTC slice. TTX was dissolved in distilled water as a 1 mM stock solution and stored at – 20 °C until use. TTX was diluted to a final desired concentration of 1 µM in external recording solution on the day of the experiment. Strychnine was prepared in a similar manner to TTX, and bicuculline (Tocris, Ballwin, MO, USA) was dissolved in DMSO as a 10 mM stock solution. 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX, Tocris), 2,3-dihydroxy-6-nitro-7-sulfamoyl-benzo[f]quinoxaline-2,3-dione (NBQX, Tocris) and N,N,H,-Trimethyl-5-[(tricyclo[3.3.1.13,7]dec-1-ylmethyl)amino]-1-pentanaminiumbromide hydrobromide (IEM-1460, Tocris) were prepared as 10 mM stocks dissolved in distilled water, and d(-)-2-Amino-5-phosphonopentanoic acid (d-AP5, Tocris) was prepared as 50 mM stocks dissolved in 30% 1 M NaOH. Riluzole was prepared as a 10 mM stock, kyneurinic acid and GABA as 1 M stocks, and nitrendipine as a 1 mM stock made up in distilled water. These drugs were used at a 1:1000 dilution prepared freshly with external recording solution immediately prior to the start of experiments.
Raw time-lapse calcium fluorescence intensity and whole-cell voltage-clamp recordings were analyzed using the Frequency-Independent Biological Signal Identification (FIBSI) program24 written using the Anaconda v2019.7.0.0 (Anaconda, Inc, Austin, TX) distribution of Python v3.5.2 and the NumPy and matplotlib.pyplot libraries. The FIBSI program incorporates the Ramer-Douglas-Peucker algorithm to detect significant waveforms against a time-dependent generated reference line using the least number of total points to represent said waveform, and provides quantitative measurements for each detected waveform. The calcium fluorescence traces were fit using a sliding median with a window size between 5 and 25 s, and peaks below the sliding median line were traced together to form the reference line. The voltage-clamp traces were fit using a sliding median with a window of 50 ms, and peaks above the sliding median were traced together to form the reference line. A custom Python script was used to match sEPSCs detected in filtered voltage-clamp recordings with the unfiltered recordings. The detected sEPSCs for each neuron were clustered based on their amplitude, charge transfer, and interevent interval using the partitioning around medoids function (default settings, manhattan distance used) in the cluster v2.1.0 package in R v4.0.2 (2020-06-22; R Core Team, The R Foundation for Statistical Computing, Vienna, Austria). Each neuron was clustered independently. The average silhouette width method was used to select the optimal number of clusters for each neuron. Clusters were graphed using the plotly v4.9.2.1 package with dependency on ggplot2. The FIBSI source code, custom Python matching script, and a tutorial for using FIBSI are available on a GitHub repository (https://github.com/rmcassidy/FIBSI_program).
Statistical analysis of the calcium fluctuations and sEPSCs were performed using Prism v8.2.1 (GraphPad Software, Inc, La Jolla, CA). The comparison between the proportion of ‘silent’ neurons in the naïve and BDNF treatment conditions was made using Fisher’s exact test. The Brown–Forsythe and Welch ANOVA tests and Games–Howell post-hoc test (for n > 50) were used to compare the effects of BDNF when the calcium fluctuations from all neurons were grouped together. To control for differences in recording times, the means for each fluctuation parameter (amplitude, AUC, etc.) were calculated for each neuron (fluctuations < 10% the maximal amplitude in each neuron were omitted to remove the effects of low-amplitude noise). The amplitude and AUC means were normalized across all neurons to permit direct comparisons. Normality was assessed using the D’Agostino & Pearson omnibus test, and nonparametric comparisons between group medians were made using the Kruskal–Wallis test and Dunn’s post-hoc test. Two-way repeated measures ANOVAs and Sidak’s post-hoc tests were used to compare the pre- and post-treatment effects of octanol on the calcium fluctuations. Adjacency matrixes for neurons imaged together from the same organotypic culture were constructed using Pearson correlation matrixes; the FIBSI-processed calcium fluctuation recordings were used as input (each recording was normalized on a 0 to 1 scale). Next, the mean Pearson r coefficient for each neuron within a culture was calculated using the Fisher Z transformation (steps shown in Fig. 3A). The mean Pearson r coefficients in the naïve and BDNF conditions were not normally distributed, so the comparison between the two was made using a Mann–Whitney test. Comparisons between the pre- and post-treatment effects of octanol were first made using a Wilcoxon matched-pairs signed rank test, and then using a two-way repeated measures ANOVA and Sidak’s post-hoc test. Paired t-tests were used for pharmacology experiments. Brown-Forsythe and Welch ANOVA tests and Games-Howell post-hoc test were used to compare the effects of BDNF on the sEPSC clusters. Statistical significant was set to P < 0.05 and all reported P values are two-tailed. Details for the statistical analyses can be found in Table 1.
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We thank Edgar Walters for useful discussions. We thank Klaus Ballanyi and Araya Ruangkittisakul for use of and technical assistance with confocal imaging equipment.
We acknowledge funding from the Canadian Institutes for Health Research. We acknowledge funding from the research endowment fund of the Department of Anesthesiology and Critical Care Medicine, University of New Mexico Health Sciences Center.
These authors contributed equally: Sascha R. A. Alles and Max A. Odem.
Department of Anesthesiology & Critical Care Medicine, University of New Mexico Health Sciences Center, MSC 10-6000, 2211 Lomas Blvd. NE, ACM 200, Albuquerque, NM, 87106, USA
Sascha R. A. Alles
Neuroscience and Mental Health Institute & Department of Pharmacology, University of Alberta, Edmonton, AB, Canada
Sascha R. A. Alles, Van B. Lu & Peter A. Smith
Department of Microbiology and Molecular Genetics, McGovern Medical School at UTHealth, Houston, TX, USA
Max A. Odem
Wellcome-MRC Institute of Metabolic Science, University of Cambridge, Cambridge, UK
Van B. Lu
Department of Psychiatry and Behavioral Sciences, Vanderbilt University Medical Center, Nashville, TN, USA
Ryan M. Cassidy
MicrosoftHyper-V, codenamed Viridian,[1] and briefly known before its release as Windows Server Virtualization, is a native hypervisor; it can create virtual machines on x86-64 systems running Windows.[2] Starting with Windows 8, Hyper-V superseded Windows Virtual PC as the hardware virtualization component of the client editions of Windows NT. A server computer running Hyper-V can be configured to expose individual virtual machines to one or more networks. Hyper-V was first released with Windows Server 2008, and has been available without additional charge since Windows Server 2012 and Windows 8. A standalone Windows Hyper-V Server is free, but with command-line interface only.
A beta version of Hyper-V was shipped with certain x86-64 editions of Windows Server 2008. The finalized version was released on June 26, 2008 and was delivered through Windows Update.[3] Hyper-V has since been released with every version of Windows Server.[4][5][6]
Microsoft provides Hyper-V through two channels:
Hyper-V Server 2008 was released on October 1, 2008. It consists of Windows Server 2008 Server Core and Hyper-V role; other Windows Server 2008 roles are disabled, and there are limited Windows services.[8] Hyper-V Server 2008 is limited to a command-line interface used to configure the host OS, physical hardware, and software. A menu driven CLI interface and some freely downloadable script files simplify configuration. In addition, Hyper-V Server supports remote access via Remote Desktop Connection. However, administration and configuration of the host OS and the guest virtual machines is generally done over the network, using either Microsoft Management Consoles on another Windows computer or System Center Virtual Machine Manager. This allows much easier "point and click" configuration, and monitoring of the Hyper-V Server.
Hyper-V Server 2008 R2 (an edition of Windows Server 2008 R2) was made available in September 2009 and includes Windows PowerShell v2 for greater CLI control. Remote access to Hyper-V Server requires CLI configuration of network interfaces and Windows Firewall. Also using a Windows Vista PC to administer Hyper-V Server 2008 R2 is not fully supported.
Hyper-V implements isolation of virtual machines in terms of a partition. A partition is a logical unit of isolation, supported by the hypervisor, in which each guest operating system executes. There must be at least one parent partition in a hypervisor instance, running a supported version of Windows Server (2008 and later). The virtualization software runs in the parent partition and has direct access to the hardware devices. The parent partition creates child partitions which host the guest OSs. A parent partition creates child partitions using the hypercall API, which is the application programming interface exposed by Hyper-V.[9]
A child partition does not have access to the physical processor, nor does it handle its real interrupts. Instead, it has a virtual view of the processor and runs in Guest Virtual Address, which, depending on the configuration of the hypervisor, might not necessarily be the entire virtual address space. Depending on VM configuration, Hyper-V may expose only a subset of the processors to each partition. The hypervisor handles the interrupts to the processor, and redirects them to the respective partition using a logical Synthetic Interrupt Controller (SynIC). Hyper-V can hardware accelerate the address translation of Guest Virtual Address-spaces by using second level address translation provided by the CPU, referred to as EPT on Intel and RVI (formerly NPT) on AMD.
Child partitions do not have direct access to hardware resources, but instead have a virtual view of the resources, in terms of virtual devices. Any request to the virtual devices is redirected via the VMBus to the devices in the parent partition, which will manage the requests. The VMBus is a logical channel which enables inter-partition communication. The response is also redirected via the VMBus. If the devices in the parent partition are also virtual devices, it will be redirected further until it reaches the parent partition, where it will gain access to the physical devices. Parent partitions run a Virtualization Service Provider (VSP), which connects to the VMBus and handles device access requests from child partitions. Child partition virtual devices internally run a Virtualization Service Client (VSC), which redirect the request to VSPs in the parent partition via the VMBus. This entire process is transparent to the guest OS.
Virtual devices can also take advantage of a Windows Server Virtualization feature, named Enlightened I/O, for storage, networking and graphics subsystems, among others. Enlightened I/O is a specialized virtualization-aware implementation of high level communication protocols, like SCSI, that allows bypassing any device emulation layer and takes advantage of VMBus directly. This makes the communication more efficient, but requires the guest OS to support Enlightened I/O.
Currently[when?] only the following operating systems support Enlightened I/O, allowing them therefore to run faster as guest operating systems under Hyper-V than other operating systems that need to use slower emulated hardware:
The Hyper-V role is only available in the x86-64 variants of Standard, Enterprise and Datacenter editions of Windows Server 2008 and later, as well as the Pro, Enterprise and Education editions of Windows 8 and later. On Windows Server, it can be installed regardless of whether the installation is a full or core installation. In addition, Hyper-V can be made available as part of the Hyper-V Server operating system, which is a freeware edition of Windows Server.[12] Either way, the host computer needs the following.[13]
The amount of memory assigned to virtual machines depends on the operating system:
The number of CPUs assigned to each virtual machine also depends on the OS:
There is also a maximum for the number of concurrently active virtual machines.
The following table lists supported guest operating systems on Windows Server 2008 R2 SP1.[20]
| Guest operating system | Virtual CPUs | ||
|---|---|---|---|
| OS | Editions | Number | Architecture |
| Windows Server 2012[a] | Hyper-V, Standard, Datacenter | 1–4 | x86-64 |
| Windows Home Server 2011 | Standard | 1–4 | x86-64 |
| Windows Server 2008 R2 SP1 | Web, Standard, Enterprise, Datacenter | 1–4 | x86-64 |
| Windows Server 2008 SP2 | Web, Standard, Enterprise, Datacenter | 1–4 | IA-32, x86-64 |
| Windows Server 2003 R2 SP2 | Web,[b] Standard, Enterprise, Datacenter | 1 or 2 | IA-32, x86-64 |
| Windows 2000 SP4 | Professional, Server, Advanced Server | 1 | IA-32 |
| Windows 7 | Professional, Enterprise, Ultimate | 1–4 | IA-32, x86-64 |
| Windows Vista | Business, Enterprise, Ultimate | 1–4 | IA-32, x86-64 |
| Windows XP SP3 | Professional | 1 or 2 | IA-32, x86-64 |
| Windows XP SP2 | Professional, Professional x64 Edition | 1 | IA-32, x86-64 |
| SUSE Linux Enterprise Server 10 SP4 or 11 SP1–SP3 | N/A | 1–4 | IA-32, x86-64 |
| Red Hat Enterprise Linux 5.5–7.0 | Red Hat Compatible Kernel | 1–4 | IA-32, x86-64 |
| CentOS 5.5–7.5 | N/A | 1–4 | IA-32, x86-64 |
| Ubuntu 12.04–20.04 | Debian Compatible Kernel | 1–4 | IA-32, x86-64 |
| Debian 7.0 | Debian Compatible Kernel | 1–4 | IA-32, x86-64 |
| Oracle Linux 6.4 | Red Hat Compatible Kernel | 1–4 | IA-32, x86-64 |
Fedora 8 or 9 are unsupported; however, they have been reported to run.[20][21][22][23]
Third-party support for FreeBSD 8.2 and later guests is provided by a partnership between NetApp and Citrix.[24] This includes both emulated and paravirtualized modes of operation, as well as several HyperV integration services.[25]
Desktop virtualization (VDI) products from third-party companies (such as Quest Software vWorkspace, Citrix XenDesktop, Systancia AppliDis Fusion[26] and Ericom PowerTerm WebConnect) provide the ability to host and centrally manage desktop virtual machines in the data center while giving end users a full PC desktop experience.
Guest operating systems with Enlightened I/O and a hypervisor-aware kernel such as Windows Server 2008 and later server versions, Windows Vista SP1 and later clients and offerings from Citrix XenServer and Novell will be able to use the host resources better since VSC drivers in these guests communicate with the VSPs directly over VMBus.[27] Non-"enlightened" operating systems will run with emulated I/O;[28] however, integration components (which include the VSC drivers) are available for Windows Server 2003 SP2, Windows Vista SP1 and Linux to achieve better performance.
On July 20, 2009, Microsoft submitted Hyper-V drivers for inclusion in the Linux kernel under the terms of the GPL.[29] Microsoft was required to submit the code when it was discovered that they had incorporated a Hyper-V network driver with GPL-licensed components statically linked to closed-source binaries.[30] Kernels beginning with 2.6.32 may include inbuilt Hyper-V paravirtualization support which improves the performance of virtual Linux guest systems in a Windows host environment. Hyper-V provides basic virtualization support for Linux guests out of the box. Paravirtualization support requires installing the Linux Integration Components or Satori InputVSC drivers. Xen-enabled Linux guest distributions may also be paravirtualized in Hyper-V. As of 2013[update] Microsoft officially supported only SUSE Linux Enterprise Server 10 SP1/SP2 (x86 and x64) in this manner,[31] though any Xen-enabled Linux should be able to run. In February 2008, Red Hat and Microsoft signed a virtualization pact for hypervisor interoperability with their respective server operating systems, to enable Red Hat Enterprise Linux 5 to be officially supported on Hyper-V.[32]
Hyper-V in Windows Server 2012 and Windows Server 2012 R2 changes the support list above as follows:[33]
Hyper-V on Windows Server 2012 R2 added the Generation 2 VM.[34]
Hyper-V, like Microsoft Virtual Server and Windows Virtual PC, saves each guest OS to a single virtual hard disk file. It supports the older .vhd format, as well as the newer .vhdx. Older .vhd files from Virtual Server 2005, Virtual PC 2004 and Virtual PC 2007 can be copied and used in Hyper-V, but any old virtual machine integration software (equivalents of Hyper-V Integration Services) must be removed from the virtual machine. After the migrated guest OS is configured and started using Hyper-V, the guest OS will detect changes to the (virtual) hardware. Installing "Hyper-V Integration Services" installs five services to improve performance, at the same time adding the new guest video and network card drivers.
Hyper-V does not virtualize audio hardware. Before Windows 8.1 and Windows Server 2012 R2, it was possible to work around this issue by connecting to the virtual machine with Remote Desktop Connection over a network connection and use its audio redirection feature.[35][36] Windows 8.1 and Windows Server 2012 R2 add the enhanced session mode which provides redirection without a network connection.[37]
Optical drives virtualized in the guest VM are read-only.[38] Officially Hyper-V does not support the host/root operating system's optical drives to pass-through in guest VMs. As a result, burning to discs, audio CDs, video CD/DVD-Video playback are not supported; however, a workaround exists using the iSCSI protocol. Setting up an iSCSI target on the host machine with the optical drive can then be talked to by the standard Microsoft iSCSI initiator. Microsoft produces their own iSCSI Target software or alternative third party products can be used.[39]
Hyper-V uses the VT-x on Intel or AMD-V on AMD x86 virtualization. Since Hyper-V is a native hypervisor, as long as it is installed, third-party software cannot use VT-x or AMD-V. For instance, the Intel HAXM Android device emulator (used by Android Studio or Microsoft Visual Studio) cannot run while Hyper-V is installed.[40]
x64 SKUs of Windows 8, 8.1, 10 Pro, Enterprise, Education, come with a special version Hyper-V called Client Hyper-V.[41]
Windows Server 2012 introduced many new features in Hyper-V.[6]
With Windows Server 2012 R2 Microsoft introduced another set of new features.[46]
Hyper-V in Windows Server 2016 and Windows 10 1607 adds[52]
Hyper-V in Windows Server 2019 and Windows 10 1809 adds[55]
| Wikiversity has learning resources about Hyper-V |
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