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Planck, M. Pond, S. Pumphrey, R. Rashevsky, N. Mathematical Biophysics. Revised Edition. Chicago: University of Chicago Press. Note that classification algorithms could be used prior to binarization to identify different MUs, and thereby estimate the CVs of individual MUs.
The shapes, intensity, and position of each MUAP could be used for this classification. We propose this idea as a potential future work. The performance of the proposed IPE algorithm was quantitatively and qualitatively evaluated by comparison with the MLE algorithm proposed by Farina et al. IPE, however, does not require a choice of initial CV value, nor a priori knowledge about propagation direction; in fact, IPE estimates the CV for both directions of propagation, and does not require any kind of initialization.
A total of synthetic signals were generated, using the simulation method proposed by Farina and Merletti [ 34 ]. Each simulated signal contains seven differential channels. All signals were simulated far from tendinous regions and innervation zones, and based on a single narrow IZ, so that the simulated MUs are innervated at the same location. The simulated IZ was positioned far from the electrodes. The sampling rate was Hz per channel. The simulation was based on a set of 8 linear electrodes contacts with 5 mm IED.
The total duration of each signal was 3 s. These signals were separated into eighteen groups of 50 synthetic signals, which were used to compare the two algorithms. The results were then quantitatively and qualitatively compared between algorithms, and with the true CV for each signal, by means of scatter plots and root mean squared error RMSE levels.
This may be particularly useful in studies on muscle fatigue, in which the temporal behavior of the S-EMG estimators including CV during an exercise is evaluated [ 35 ]. The results were evaluated by comparing RMSE levels. The window length used were 0. We also analyzed the influence of the number of channels used to estimate the CV for both algorithms.
The number of channels was varied between 4 and 7. This evaluation was performed at two different noise levels: noise-free and 16 dB SNR. If the RMSE is higher than 0. Twenty-three females and eighteen males volunteered to participate in the study. Due to withdrawal, hormonal problems, failing channels, fail in sustain isometric strength level and high levels of noise in the S-EMG recordings, only sixteen female volunteers All subjects were right-handed and had no known neurological disease.
All female subjects had regular menstrual cycles, did not practice regular exercise, and were not using any medication or hormonal contraception for at least 6 months. Male volunteers also did not practice regular exercise. All volunteers read and signed an informed consent form. The same experimental protocol was executed with all the volunteers. Each female subject performed the experimental protocol in four sessions, with a one week interval between sessions.
Male volunteers underwent a single session. Each volunteer sat on a chair, specially adapted to secure the elbow, so that the only possible movement of the arm was isometric elbow flexion Fig. This was done to minimize muscle contractions that could impact the results of the experiment. Experimental setup. The load cell was attached to an adjustable cable, and placed next to the chair.
The EMG 16 uses a 4th-order bandpass Bessel filter with passband between 10 and Hz, and provides a maximum sampling frequency of Hz. The amplification gain was set to 1. However, for female volunteers, the MVC measured during the first session was adopted as the MVC for all sessions; MVC values of other acquisitions were measured only to ensure the repeatability of the protocol in all sessions.
Strong verbal encouragement was used for each MVC. After each MVC estimation, the subject rested for 1 min. Three-second signal acquisitions were used for mapping the ideal position. After this mapping procedure, the volunteer rested for two minutes.
After mapping, S-EMG signals were acquired with a semi-disposable linear array of eight surface electrodes 5 mm inter-electrode distance, OT Bioelettronica snc, Italy , which was placed on the optimal region between IZ and tendon regions of the biceps brachii short head Fig. The skin was cleaned and conductive gel was applied between the skin and each electrode.
The differential configuration was used, resulting in a 7-channel S-EMG signal for each acquisition. A reference electrode was placed on the right wrist. The sampling rate was Hz, and an the analog gain was set to A total of 27 signals were selected from those measured during the s contractions signals from the IZ mapping procedure were not used for CV estimation tests.
Signals that presented poor quality in at least one channel—60 Hz interference, movement artifacts, loss of electrode contact intermittent or constant —were discarded. The signals were not segmented i. In these experiments, the true CV is unknown. Thus, CV estimates obtained with the proposed method were compared with those obtained with the MLE algorithm, by means of correlation coefficient, linear regression, and scatter plots analysis.
In Fig. The black diagonal dashed lines represent the identity line; therefore, if the dots are grouped near the diagonal line, this indicates that the two algorithms provided similar CV estimates.
The black diagonal dashed lines are the identity lines. Tables 1 and 2 present the same results in a different fashion, by evaluating the effects of the noise level and force level on the RMSE for each algorithm, and for each CV value.
This is due to local minima problems associated with the Newton—Raphson method, which is used in the MLE algorithm. The proposed IPE method does not use iterative optimization methods, therefore it is not susceptible to this type of error. Another important aspect we observed is the fact that the applied force level seems to have little effect on the performance of both the IPE and the MLE methods, specially for high SNRs.
In fact, Tables 1 and 2 show that for signals with 30 and 20 dB, there is no difference between the measured root mean squared errors for different force levels, within the considered numeric precision. In the case of signals with 12 dB, there were different RMSE for the MLE only for different tested force levels, but the error sometimes increased and sometimes decreased with the force level. We believe that the local minimum problem previously mentioned has an impact on the MLE performance in this high-noise scenario, and we did not identify a monotonic relation between the force level and the measured error.
The results in Fig. We now evaluate the influence of window length and number of channels on CV estimation. Overall, the performance of the MLE method was slightly superior to that of the proposed algorithm, which performed better than MLE only for windows lengths of 0. As expected, the use of long windows is more advantageous when the SNR is low.
At high SNR, the window length may be decreased in order to improve the temporal resolution. Similarly, results are shown only for noise free and 16 dB SNR signals. For noise free signals Fig. In all cases, it is evident that the use of 5 or more channels is recommended to avoid RMSE higher than the high-goodness threshold.
Five more signals were evaluated with both algorithm but due to outlier results on the MLE method, this number was reduced to 22 and the outliers were evaluated separately.
The green dashed line is the regression line and the red dashed line is the identity line. The results show a strong linear correlation between the two algorithms correlation coefficient was 0.
We then used this equation to predict the delay values of a novel dataset produced with the same stimulation rate. The fit produced a coefficient of determination of 0.
The history-dependence of conduction delay in experiments can therefore be predicted from the spike voltage trajectories without any need for computational modeling. B Data in panel A plotted as a function of F inst. The classic description of ionic mechanisms underlying spike generation and conduction was based on only two voltage-gated currents to describe the membrane behavior of the squid giant axon Hodgkin and Huxley, a , b.
This relatively simple model has dominated the common perception of axonal spike propagation, even though ionic mechanisms in most axons are more complex, as multiple currents with very different properties can be present Krishnan et al. Later theoretical work examined in detail the properties determining conduction velocity Muratov, ; Tasaki, , but only for single spikes and considering only a limited set of properties.
We show that these models fail to capture the history-dependence of conduction velocity during repetitive activity. Such history-dependence is evident in findings that changes in axonal excitability and conduction velocity can occur at timescales far exceeding single inter-spike intervals, and can substantially alter the temporal pattern of spikes during propagation Weidner et al. It is important to understand the mechanisms underlying history-dependence of spike propagation, because changes in temporal patterns potentially play an important role in shaping the neural code Izhikevich, ; Bucher and Goaillard, ; Bucher, We describe the history-dependence of conduction delay in a biophysically realistic model axon designed to match properties of the PD motor axon of the crustacean stomatogastric nervous system.
The model captures well how delay changes as a function of prior spiking in this axon, at two different timescales Ballo et al. At the slow timescale STS effect , repetitive activity leads to cumulative hyperpolarization and conduction slowing.
Over the course of minutes, mean delay and variability of delay increases. Direct experimental evidence of the involvement of the pump in activity-dependent dynamics for the PD axon is missing because of the difficulty that pharmacological block of the pump also interferes with its overall role in maintaining a functional membrane potential Ballo and Bucher, However, other slow outward currents have not been identified in the PD axon Bucher, unpublished results.
K v 7 channels that produce M-type currents Dubois, ; Baker et al. Such currents are thought to contribute to hyperpolarization and reduced excitability and may have cumulative effects during repetitive activity Baker et al.
An interesting difference between the pump current and currents due to ion channel conductances is that the pump current does not depend on a driving force. Equivalently, an ionic current would change the input conductance of the neuron, whereas the pump current does not. The same would be true for Cl - conductances and E Cl. We chose the E K value near the resting potential because spikes in PD axon recordings show an absence of fast after-hyperpolarization undershoot , and we could only replicate this in the model with a less negative E K.
An alternative approach would have been to model E K as a dynamic variable and allow it to transiently change to more depolarized values over the course of a single spike. However, our results demonstrate that the increase of variability of delay was dependent on hyperpolarization, as a mere increase in conductance only increased the mean delay.
The dependence of the STS effect on hyperpolarization is further highlighted by the role of I h Figure 5 , which is thought to play an important role in preventing spike failures in many axons because its inwardly rectifying properties can balance activity-dependent hyperpolarization Baker et al.
In the PD axon, I h is increased several-fold by dopamine Ballo et al. Our model provides a mechanistic explanation of our previous experimental observation that the balance of I h and pump current plays a dynamic role in setting how sensitive conduction is to the history of activity.
This represents a level of importance in controlling temporal fidelity that goes beyond just preventing spike failures. Previous studies have also used a modeling approach to explore the role of slow accumulation of ionic currents and concentrations in axons. It has long been known that, with increasing spike intervals even well beyond the refractory period, excitability and conduction velocity go through an oscillation of decreases and increases, Bullock, ; Raymond, , collectively termed the recovery cycle.
Such activity-dependent changes are often gauged with paired-pulse stimulations Bucher and Goaillard, , particularly for diagnostic purposes in the context of peripheral neuropathies Bostock and Rothwell, ; Bostock et al.
The dependence on spike interval changes dramatically when axons are stimulated with trains of conditioning pulses instead of simple paired pulses. In our model, prior activity can be mimicked by setting the pump current to different values Figure 6. This suggests that FTS and STS effects are completely separated in their time dependence, meaning that delay depends on the last interval and the mean rate of prior activity, but not on the exact temporal pattern of the last few preceding spikes.
This independence of exact prior pattern is further demonstrated by our finding that paired-pulse stimulation results predict delays from both Poisson Figure 6 and burst Figure 7 stimulations. A previous modeling study on primary afferent C-fibers has also demonstrated an activity-dependent alteration of recovery cycles Tigerholm et al.
In comparison, two equations that accurately describe conduction delay of single spikes fail to capture the dynamics. The Matsumoto-Tasaki equation predicts conduction delay dependent on the change in total membrane conductance elicited by a spike Matsumoto and Tasaki, ; Tasaki and Matsumoto, ; Tasaki, Both equations aim to predict absolute values of delay and include constant biophysical parameters such as diameter and membrane capacitance.
In contrast, our empirical equation describes only how conduction delay changes as a function of activity. These three constants have to be found empirically and may represent stand-ins for actual biophysical parameters.
Interestingly, the gating parameters at the trough potential before a spike and the peak voltage of the spike correlate linearly with these voltages, so that the voltages can be treated as stand-ins for the gating parameters in the empirical fit, which allows us to predict delay solely from voltage trajectories Figure It should be noted that the small level of sensitivity to the various currents is because sensitivity analysis only examines small changes in the parameters of these currents which would not greatly alter the membrane potential.
However, some of these currents, as shown for the pump current and I h , undergo large changes that have substantial effects on delay. Again, these mechanisms act at two different timescales. It should be noted that changes in baseline membrane potential overall can have ambiguous effects on conduction velocity. In the PD axon, relatively slow spike repolarization leads to summation even at moderate frequencies, and spike duration depends on I A activation and inactivation, which vary substantially with the baseline membrane potential Ballo and Bucher, The model presented here aimed to mimic, as faithfully as possible, the behavior of a well described unmyelinated invertebrate axon.
It is plausible as a general mechanism that the gating state of these channels at the onset and peak of the spike acts as a critical determinant of conduction velocity. Different complements of additional voltage-gated or electrogenic transporter currents across axon types may lead to very different dependence of conduction velocity on preceding activity.
Consequently, once the peak and trough voltages of the action potential have been measured, the empirical equation introduced in this study would serve to accurately predict the conduction velocity, independent of the combinations of ionic mechanisms involved. We constructed a conductance-based biophysical axon model to examine the role of different ionic currents in shaping the history-dependence of conduction delay.
The length of the model axon was set to 1 cm, and divided into identical compartments for simulation. In order to apply the finite difference method to solve the model equations numerically, all compartments were assumed to be isopotential during the simulation process.
All axon parameters are listed in Table 1. Ionic currents used in the model are shown in Figure 1A. Different versions of the model contained different combinations of ionic currents currents not included in all versions are shown in brackets :.
In the simplest version, it included only the standard Hodgkin-Huxley-like leak I Leak , fast sodium I Na , and delayed-rectifier potassium I Kd currents. The PD axon expresses two additional voltage-gated ionic currents, a transient potassium current I A , and a hyperpolarization-activated inward current I h Ballo and Bucher, ; Ballo et al. We mimicked DA effects on I h by using different values of voltage half-activation and slope Table 2 that were described experimentally Ballo et al.
However, these changes alone were not sufficient to produce the experimentally-observed effect of I h increase by DA on axonal delay, presumably because of differences in resting membrane potential between our model and the biological neuron. We also tested the effect of a slow potassium current I Ks. To this end, we introduced a conductance with the same voltage dependence as I Kd , but a times slower time constant of activation.
All ionic currents were Hodgkin-Huxley type Hodgkin and Huxley, b and were represented by equations of the following form:. As stated above, gating parameters and E rev for I h were based on actual measurements in the PD axon Ballo et al. I Na and potassium current parameters were based on common values and tuned to produce spike shapes that matched those in intracellular PD axon recordings Ballo and Bucher, Of note in these recordings is the absence of an undershoot following a spike.
All gating parameters are listed in Table 2. According to this equation, at steady state, I pump is equal to one third of I Na. These provided somewhat different dynamics but produced very similar qualitative and quantitative results, which for brevity are not shown. Spikes were generated in the axon by applying a stimulus current in the first compartment:. Spikes propagated along the length of the 1 cm axon and were recorded at two sites 0.
Delays measured using action potential peak times produce practically identical results. Delays produced along this 0. Different conduction velocities of consecutive spikes lead to gradually changing intervals over distance. Because axonal excitability changes between spikes depend on interval, conduction delays during repetitive spiking do not scale linearly with distance Kocsis et al.
However, at initial spike frequencies smaller than about Hz, these nonlinearities are relatively subtle and only differentially affect delays over distances of tens of centimeters Moradmand and Goldfinger, ; Bucher and Goaillard, We confirmed that the conduction delay for a longer model axon, measured at the same relative points, scaled reasonably linearly with the length of the axon.
We characterized the history-dependence of conduction delay with three different stimulus regimes that have been used before in the PD axon Ballo et al. First, to describe delay as a function of time and stimulus frequency, we used s Poisson stimulation protocols with different mean rates: 5, 10 and 19 Hz. The last value was chosen to match mean spike frequency in burst stimulation protocols see below.
It should be noted that the Poisson stimulation method is useful in producing a variety of inter-stimulus intervals, but this can be also achieved with other stimulation paradigms that incorporate such a variety, for example through rate adaptation. After a single conditioning stimulus, test stimuli were delivered at different intervals between 10 ms and 10 s.
The change of delay or conduction velocity was then analyzed as a function of stimulus interval. Third, we used a burst stimulation protocol that mimicked normal ongoing rhythmic firing in the PD neuron Ballo and Bucher, ; Ballo et al.
It consisted of bursts at 1 Hz burst frequency, and each burst consisted of 19 spikes ms burst duration in a parabolic instantaneous frequency structure, increasing from 32 Hz to 63 Hz and decreasing back to 32 Hz.
All simulations were run beginning with a s interval of no stimulation to remove any transient and initial condition effects. Two equations have been shown to provide good approximations for conduction velocity of a single spike in the squid giant axon and Hodgkin-Huxley model axon. We tested if these equations yielded good estimates of conduction velocities during repetitive spiking. The first is derived based on boundary matching principles which we will refer to as the Matsumoto-Tasaki equation Matsumoto and Tasaki, ; Tasaki and Matsumoto, ; Tasaki, :.
The second equation is an explicit analytical expression developed for calculating conduction velocity in the Hodgkin-Huxley model Muratov, , which we will refer to here as the Muratov equation:.
To use this equation for repetitive spiking, we used the trough voltage before each spike the local minimum voltage immediately before the spike as a proxy for the resting membrane potential. The sensitivity of conduction delay in the model to parameters was measured by examining how the different attributes that describe the slow- STS and fast-timescale FTS effects depend on small changes in the parameter values.
For the STS effect, these attributes were the mean conduction delay and its coefficient of variation, measured for the final 20 s interval of simulation. For the FTS effect, we used quadratic fits to determine the minimum delay and the frequency at which it occurred, and the curvature of the quadratic fit at this point.
Each new model was subjected to the same Poisson stimulation as the original model and the attributes of the fast and slow timescale effects were measured. We then normalized these measures by their values in the reference model. The resulting four data points, together with the reference value at the origin were fit with a line.
The sensitivity of an attribute to the parameter was defined as the slope of the linear fit. For comparison of model results with experimental results, as well as for testing the empirical method of predicting conduction delay, we used experimental delay data from the PD axon obtained in our previous study Ballo et al.
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
The following individual involved in review of your submission has agreed to reveal their identity: Jeremy Niven Reviewer 2. The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission. In this manuscript the authors examine action potential biophysics and use computational modelling to assess the history-dependence of conduction delay, as motivated by an unmyelinated invertebrate axon.
The manuscript is well-written and well-organized with rationale, logic and motivation clearly presented. The reviewers felt that it should be of interest to neuroscientists from a variety of backgrounds, including those interested in the biophysics and those interested in neural coding and neuropathies. They use parameter screening and NEURON simulations to present findings that can explain the slow time scale via I h and Na-K pump dynamics along with an empirical equation that can be obtained by a nonlinear combination of peak and threshold potentials.
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