Estimation of the strength of common drive to motor neurons

Transcription

1 1 Estimation of the strength of common drive to motor neurons Klaus Mayntzhusen, Martin Nøhr Nielsen and Jacob Koch Pedersen 7th semester Biomedical Engineering and Informatics students at Department of Health Science and Technology, Aalborg University Index Terms Common drive, motor unit, firing rate, principal component analysis, PCA. The author s names are in alphabetical order. MU i Supra spinal centers ni (t) Freq [Hz] Motor cortex + Cerebellum Brain stem Spinal al cord Interneuron MU 2 Common drive to MU pool s (t) + Freq [Hz] n2 (t) Alpha motor neurons MU 1 n1 (t) + Freq [Hz] Abstract When performing voluntary movement, motor neurons in the spinal cord receive a common signal, termed common drive, from the supra spinal centers. Although this common input cannot be directly measured, it is reflected in common oscillations of motor neuron outputs because of the well-established relations between discharge rate and current injected in the motor neuron. Previously the common oscillations in motor neuron discharge patterns have been only analyzed between pairs of motor units (MUs) with cross correlation analysis [1]. In this study, a new method is developed to extract the common drive signal, and to estimate the strength of common drive in large populations of MUs. Intramuscular EMG signals were recorded from the abductor digiti minimi muscle of three subjects (ages 21, 23, 25) during isometric abductions of the fifth finger, right hand. A custom made restraining and force measuring apparatus was developed for measuring the isometric force in both horizontal and vertical direction. The EMG recordings were decomposed into their constituent motor unit action potential trains with an algorithm previously developed [2]. The point processes representing the discharge patterns of the identified MUs were low-pass filtered (4 ms moving average filter). The filtered discharge patterns were decomposed into principle components (principal component analysis). The linear combination of the principal components and the mean firing rates of each MU were the extracted as the common drive signal. The strength of the first principal component was computed as a new index of the strength of common drive. The frequency analysis of the principal component provided novel information of the harmonics in the common drive signal. The strength of common drive computed by the proposed method, was 58.12%, 67.17%, and 55.73% in the three subjects. The strength of the second component was 7.99%, 11.45%, and 13.11%, indicating that one single component well described the common behavior of motor neuron outputs. The average peak of the cross-correlation between filtered discharge patterns of all the pairs of MUs was 48.13%, 58.16%, and 53.69%. The SD of estimation of the strength of common drive from 3-s epochs was (average over all subjects).1 and.12 for the proposed and previous method, respectively. The study proposes a new way of analyzing the common drive signal in populations of motor neurons during voluntary contractions. The method allows the estimation of the strength of common drive and the extraction of the common drive signal as the first principal component of the motor neuron outputs and may extend our understanding of neural control of movement. Motor unit 1 Motor unit 2 Muscle A B Fig. 1. A illustrates the summation between the common drive to the MU pool and the specific drive to the different MUs, not shared by other MUs. Thus the single MUs is not only driven by the common drive, but also by other signals, interpreted as "noise" in this context. B illustrates the origin of the common drive to motor neurons, the supra spinal centers, and the signal path through the spinal cord to the single muscle. I. I NTRODUCTION rom the motor cortex, through the brain stem, and to the α-motor neurons exists a link, which provides direct cortical control of muscle activity. The descending signal from the brain stem can reach the same interneurons, which enables the motor units (MUs) of the same motor pool to receive the same signal, as illustrated in figure 1B. This common signal is termed the common drive [3]. The phenomenon of common drive suggests that all the MUs, that makes up a muscle, are affected in an uniform fashion. This uniform fashion is the result of affecting the entire pool, of which the MUs belong, instead of an individually control of the MUs during a contraction [3]. Still the individual MUs has unique firing patterns, as a result of differently responds to the common signal and specific signals, interpreted as noise, to the single MUs not shared by others, see figure 1A. These different responds depend on the specific architecture of the pool and the inherent properties of the individual elements such as the motor neurons [4] [3]. Common drive is considered as a relative simple strategy of the central nerve system for controlling the MUs. E.g. the timing of activation of agonists and antagonists muscles is intrinsic to the spinal circuit and thus the descending signals themselves need not to be timed as precisely [5]. The common drive cannot be directly measured in the spinal cord, hence it is exploited that the oscillations of the common drive from the supra spinal centers, is reflected in the com- F

2 2 mon oscillations of the motor neuron outputs because of the well-established relations between discharge rate and current injected in the motor neuron. Thus common drive can be estimated through measurements of the motor neuron output. Previously the common oscillations in motor neuron discharge patterns have been only analyzed between pairs of MUs using cross correlation analysis [3]. To extend our understanding of the common drive, the aim of this present work was to develop a new method to extract the common drive signal from large populations of MUs and in addition estimate the strength of common drive during voluntary contractions. II. METHODS Three healthy subjects (ages 21, 23, 25) participated in the study. Each subject read and signed an informed consent form approved by The Ethic Committee, approval number (N-2719), prior to participating in the experiment. The inclusion criterion for the subjects was an age of 18 to 45 years. Isometric constant-force abductions of the fifth finger, at the right hand were performed in this study. The abductor digiti minimi muscle was studied, due to the fact that the cortical cells involved in hand muscle movements are known to have more monosynaptic connections, hence the origin of the extracted common drive was believed to be supraspinal. Furthermore the abduction of the fifth finger was controlled solely by this muscle. A. Data collection The subject was comfortably seated at a chair, with the right hand supinated and right forearm placed horizontally and supported, in a custom made restraining and force measuring apparatus placed at a table, see figure 2. The apparatus was placed at a height of centimeters above the subject s thighs. To ensure isometric abductions, the right forearm and the second, third, and fourth finger at the right hand were immobilized by fixing them by use of straps placed around the upper forearm, the wrist and around the three fingers. The three fingers were fixed in flexed position, to reduce their contribution to the abduction of the fifth finger. The outermost segment of the fifth finger was fixed in an u-shaped mounting mounted on a force transducer, which was attached at a greater mounting at the apparatus. The force transducer was measuring forces in the directions of extension/flexion of the fifth finger (vertical axis), to obtain a measure of the level of isolated abduction of the fifth finger. Furthermore the u- shaped mounting was placed in contact with a second force transducer, also attached to the greater mounting. This force transducer was measuring the forces in the abduction direction (horizontal axis), to obtain a measure of the level of abduction. Both transducers had a maximum load of N. The maximal voluntary contraction (MVC) level for the subjects, by performing abduction of the fifth finger at the right hand, was determined by choosing the greater of two attempts at generating the maximal voluntary effort. Both of these attempts lasted three seconds and were two minutes apart in time. The MVC amplitudes were recorded and served as Fig. 2. Arrangement of the restraining and force measuring apparatus. Note the place for insertion of the electrodes in the abductor digiti minimi muscle. At the inserted figure the placements of the two force transducers are illustrated. controls in the further experiment. An oscilloscope in front of the subject was then marked at the levels of 5% and 1% abduction MVC, to provide visual feedback in the further experiment. Both the force and the EMG signal were sampled at 1 khz and recorded through the experiments. The force was amplified by a factor of two, ensuring higher resolution of the feedback. Three bipolar fine-wire electrodes were inserted into the belly of the abductor digiti minimi muscle, with a transversal distance between them of three millimeters. One electrode was inserted into the centre of the belly of the muscle, the second was placed dorsal proximal to the first and the third electrode placed palmar distal to the first electrode. All the electrodes were inserted into a depth of four millimeters. An electrode consisted of two Teflon-insulated fine wires (.11 mm core diameter) threaded through the lumen of a 25-gauge disposable needle. The positions of the electrodes were controlled by observing that at least four MUs were detected in the intramuscular EMG signals from the abductor digiti minimi muscle. If this was not the case, the needles were re-inserted. The subject was asked to make an abduction of the fifth finger at a level of 5% MVC, and to keep this level for 3 s while recording the EMG signals. The actual force output of the subject was displayed real-time at the oscilloscope, providing visual feedback. Audio feedback of the MUs discharge was provided by the electromyograph. After the contraction of 3 s, the subject rested for one minute before making a 3 s abduction of 1% MVC. The EMG was recorded from three channels using the finewire electrodes. These signals were digitally low pass filtered at 5 khz by the electromyograph. The EMG signals were acquired and decomposed into their constituent motor unit action potential trains (MUAPTs) with an algorithm described by [2]. The decomposition algorithm used template matching, template updating, and MU firing statistics to identify individual MU firing times. Algorithms designed to resolve super positions, such as those that may occur during synchronous

3 3 Spinal cord Alpha-motor neurons Right hand Abductor digiti minimi (> 4 MU) Intramuscular EMG 3 bipolar needle electrodes Raw EMG signal Decomposition MUAPT of n MUs Hanning window Mean firing rates Principal component analysis Common drive signal Frequency analysis Common drive signal Amplitude [mv] Pulses/sec Pulses/sec Amplitude MU 1... MU n Frequency [Hz] Fig s EMG was recorded from the selected muscle, abductor digiti minimi. The recorded EMG was decomposed into its constitutional MUAPs. The firing times of each MUAPs were represented by an impulse train (a bar plot). The mean firing rate of each MUAP was computed by passing the impulse train through a 4 ms Hanning window. To avoid the effect of the common DC component, this was eliminated by removing the mean value. The common drive signal was extracted on the basis of principal component analysis. Frequency analysis was applied to the extracted common drive signal. discharges, were included in the decomposition algorithm. To fully decompose all the MUAPTs, a great amount of interaction with the algorithm by the operator was necessary. Earlier studies have shown that with proper use, this technique can provide 95% accuracy [2]. All the MUAPTs analyzed in this study were decomposed and all the interfiring intervals were confirmed by the operator to be correct by accounting for all the action potentials identified and inspecting all the MUAPTs to verify that no unexpectedly long or short firing interval was present. B. Data analysis Proposed method: Figure 3 provides an overview of the proposed method. Initially the recorded 3-s EMG signals were decomposed into its constitutional MUAPs, thus the decomposed data consisted of the firing times for the MUs from all channels. The MUs were numbered consecutive through the channels and it was ensured that only different MUs were analyzed by counting and comparing the number of firings for each MU with the rest, hence to realize if multiple MUs fired virtually simultaneous (firing in an interval of 3 ms of the others). Similar firings lead to a deletion of the multiple MUs. Firing patterns of each MU were represented binary, 1 representing a firing and representing no firing. On the basis of the MUAPTs, the mean firing rate of each MU was calculated as a weighted average by applying a Hanning window of length 4 ms, which corresponded to low pass filtering the impulse train. The Hanning window was normalized in proportion to a rectangular window (in both shape and length), which lead to a normalization factor of the Hanning window of 2.5, hence the window was of the form h(τ) = 2.5(1 + cos(π( τ 4 ))) for τ 4 ms and for τ 4 ms. The fluctuations in the mean firing rates were represented as zero-mean, ensuring that the DC component of the signal (present at Hz) did not influence the following analysis. Recorded EMG signals are not precisely predictable even though past history of the signal and the amplitude values are known. This is due to the always present noise in the acquired signals, but also due to the fact, that the firings of the MUs cannot be totally predicted, hence the EMG signal was considered as a stochastic process. Furthermore it was considered as a wide sense stationary (WSS) process, which implied that both the mean and autocorrelation of the stochastic process were invariant to time shifts. Furthermore ergodicity of the process was assumed, thus the ensemble average could be exchanged with the time average, which was exploited in computation of the cross correlation. The proposed method was based on the previous method [1] and the technique of principal component analysis (PCA), to extract the principal component (common drive signal). From the common drive signal in addition estimation of the strength of common drive to motor neurons and spectral analysis of the common drive signal is enabled. Through PCA, the aim was to clarify the different weights of the common drive to different MUs in a specific muscle. PCA is a statistical technique for finding patterns in data of high dimensions, by reducing multidimensional datasets to lower dimension for further analysis. PCA transforms data and represent it with basic eigenvectors [6]. The computations of these were based on the MU firing rates, which were to be transformed. The transformation was expressed in equation (1). y = A(x µ x ) (1) A is denoting the transformation matrix A = [e 1 e 2...e n ] T. x is denoting the MU mean firing rate data, organized in a matrix, containing m columns for m MUs, and n rows for n data points. µ x = 1 n n i=1 x i is denoting the mean of the firing rates, used for computing the zero mean firing rates. The essential part was the computation of the transformation matrix A which should be computed minimizing the error α. This implied minimizing the expected squared difference between the original matrix x and the estimate of x (ˆx ), seen by simplifying equation (1) by x = x µ x, hence the inverse transformation was x = A 1 y, as expressed in equation (2) [6]. α = E { (x ˆx ) T (x ˆx ) } (2) The m m covariance matrix, C x, was computed to describe the different MU mean firing rates varying from the mean, and to contain the correlation within the input data, x. The eigenvalues λ i of C x, corresponding to the eigenvectors, was computed numerically, expressed as the

4 Amplitude [mv] 4 eigenvalue problem as expressed in equation (3), the eigenvalue λ i corresponding to the i th eigenvector e i, which were computed on the basis of the eigenvalues [6]. MU 1 MU 2 MU 3 (C x λi) e i = (3) The eigenvectors e i were arranged as row vectors in A of equation (1), followed by the dimensionality reduction, to maintain the principle component representing most of the variance in the mean firing rates of the MUs. Hence the eigenvectors and eigenvalues provided information about the pattern in the data. Variance of the data was in this context perceived as essential information about the common drive, and the eigenvector, e i, with the highest corresponding eigenvalue, λ i, was the principle component of the mean firing rates of the MUs, which described most of the variance. Both the largest and second largest eigenvalue and corresponding eigenvectors were evaluated, due to the importance of variance distribution between the different eigenvectors. A high common drive would be reflected as a significant difference in the size between the largest and second largest eigenvalue. The most common firing rate signal, the common drive signal, was represented by the linear combination of the principal components multiplied by the different MU firing rates. The cross correlation between the common drive signal and the MU firing rate data was computed, and described how the single motor unit firing rate data was correlated with the common drive signal. Hence this revealed to which extend, the single MUs was driven by the common drive signal. MUs with a high correlation were believed to be more driven by the common drive signal, than the MUs with a low correlation. Varying correlation between the MU firing rates and the common drive signal would indicate different interpretation of the common drive by different MUs. Cross correlation is a standard method of estimating the degree, to which two time series are correlated. It determines the degree of similarity between two signals, 1 indicating complete correlation, -1 complete anti correlation, and, no correlation. Spectral analysis of the common drive signal, may reveal dominant frequency components, thus determining the frequency properties of the common drive. Spectral analysis of the principle components, the common drive signals, were implemented by estimation of the power spectrum using a nonparametric estimation by the periodogram method, average of periodograms across time (Welchs method). Thus no assumptions were made with respect to the stochastic process under study. For each epoch the common drive signal were subtracted the mean, to remove any trend. Zero padding were used to achieve adequate resolution to resolve any close positioned peaks. The strength of common drive, CD strength, was computed as the eigenvalues as the percentage of the largest eigenvalue, λ max, compared to the sum of all eigenvalues, N n=1 (λ n), as expressed in equation (4). CD strength = λ max 1 N n=1 (λ n) (4) 4 ms 4 ms 4 ms Fig. 4. An example of three distinct identified action potential from three MUs in subject 2 during 1% MVC, identified by decomposition. Previous method: This method for computing the strength of common drive was based on the basis of a previous developed method [1]. The initial steps, from recordings of the EMG to the computation of the mean firing rates, was also adapted in the proposed method, thus these steps are represented in figure 3. Unlike previous works using the method [1], the common drive was computed through the entire 3-s recorded signal. The computation of the strength of common drive between pairs of MUs was based on cross correlation. The cross correlations between the mean firing rates (zeromean) for all possible pairs of MUs were computed. Thus the maximum of the cross correlation was a measure of the strength of the common drive. The variability of the common drive was computed as the SD of the mean of all cross correlations in all of the 3-s epochs, through the entire 3 s signal. Comparison: To express the variability in the computations of common drive, the mean of the cross-correlation of pair wise mean firing rates for MUs were computed for 3-s epochs, and for the number of epochs, the mean ± SD of these were computed. A likewise measure was computed in the proposed method by computing the mean ± SD of the strength of common drive through all 3-s epochs. In both methods, the values for the three subjects were averaged. Hence by comparing the variability of the two methods, the stabilization in the computation of the strength of common drive between the two methods could be evaluated. III. RESULTS Proposed method: Four EMG signals from three subjects were recorded, three were decomposed and 7, 5, and 11 active MUs were found in the three subjects during the isometric contractions at 5%, 1% and 1% MVC respectively. Figure 4 shows an example of three decomposed action potentials of three different identified MUs, from subject 2 exerting 1% MVC. On the basis of the point process, the mean firing rates were calculated. Figure 5 shows the mean firing rates, in subject 2 exerting 1% MVC. The exerting % MVC, and the extracted common drive were superimposed on the figure. The horizontal force for the three subjects was recorded and the dispersion was expressed by the coefficient of variation (CV). CV is expressed in equation (5).

5 Amplitude Mean firing rate [Hz] Fig. 5. The top figure shows the mean firing rates of 11 MUs through the entire 3 s EMG signal from subject 2. The mean firing rate were computed by low-pass filtering the point process from the decomposed data, using a 4 ms Hanning window. The firing rates varied from 8 Hz to 16 Hz. The lower figure shows the mean firing rate in a five second interval with the force (pink) and the extracted common drive (black). Cv = σ µ Frequency [Hz] Fig. 6. Spectrum analysis of the common drive signal from subject 2. Clear peaks was revealed below.5 Hz, and between 1. Hz and 1.5 Hz. % MVC Mean firing rate [Hz] 3 (5) For the three subjects respectively, the CV was 4.35%, 2.33%, and 6.78%. The vertical force was by all three subjects kept steady and at a minimum. The strength of common drive estimated in the three subjects, were computed to 58.12%, 67.17%, and 55.73% respectively. The strength of the second component in the three subjects was 7.99%, 11.45%, and 13.11% respectively. The extracted common drive were plotted in figure 5 as the solid black line superimposed on the mean firing rates. The result of the strength of common drive from 3-s epochs signals from the three subjects were (mean ± SD) 58.86%±8.41, 6.8%±8.77, and 49.65%±13.2. Figure 6 shows the spectrum analysis of the common drive signal from subject 2. The spectrum analysis revealed a clear peak below.5 Hz in all three subjects. In addition another peak was noted between 1. Hz and 1.5 Hz for all three subjects. The variability of the proposed method was 1.7% (SD) illustrated in figure 8. Previous method: On the basis of the zero mean 3 s mean firing rates from three subjects, the cross correlations between pairs of 7, 5, and 11 MUs, were computed and the mean of the maximum values of the cross correlations in each subject, were computed, expressing the strength of common drive, to 48.13%, 58.16% and 53.69%, respectively. The cross correlation of 11 MUs for one subject were plotted in figure 7. The cross correlation of 3-s epochs signal showed a common drive (mean ± SD) of 58.1%±11.61, 55.82%±.49, and 58.85%±8.4 on the three subjects respectively. The variability of the method was 11.83% (SD), as illustrated in figure 8. The variability between all possible combinations of pairwise cross correlations through the 3 s signal in all 3-s epochs was 17.64% (SD). IV. D ISCUSSION In the present work a new method is developed for extracting the common drive signal from large populations of MUs during voluntary contractions, and in addition estimating the strength of common drive. The proposed method is developed on the basis of an experimental setup and PCA, by which the common drive signal is extracted as the first principal component of the mean firing rates computed from the output of the motor neurons. The custom made restraining and force measurement apparatus is optimized for the current setting, and in addition it is well suited for similar experiments, for instance fatigue experiments ensuring that the subjects exerts a limited vertical force, such that fatigue can be reached only by exerting a horizontal force. The EMG recordings of the three subjects showed a great variability in the number of active MUs at each level of MVC, hence one subject exerts 5% MVC and the other two subjects exerts 1% MVC, due to a lack of active MUs at 5% MVC of

6 Correlation Correlation Time lag Fig. 7. Normalized cross correlation between the mean firing rates of 11 MUs (1 pairs of MUs). In the top the function is shown for lags up to ±3 s, and the lower plot shows the function for lags up to ±1 s. The peak of the cross correlation between MU 1 and MU 2 was Common drive [%] Previous method Proposed method Fig. 8. To express the variability in the computations of the strength of common drive with the previous method, the mean of the cross-correlation of pairwise mean firing rates for MUs were computed for 3-s epochs, and for the number of epochs, the mean ± SD of these, through all epochs, were computed. A likewise measure was computed in the proposed method by computing the mean ± SD of the strength of common drive through all 3-s epochs. the two subjects. The force acquired in the horizontal direction, is stable through the 3 s for the three subjects, and likewise is the vertical force, which in addition was minimal, and was used to minimize the vertical force. It is observed that the decomposition has crucial effect on the analysis of common drive, both on the previous and the proposed method. One have to focus on the identified MUs, and only include those MUs in the data which are consistently firing through the entire period. Furthermore attention to the firings of the MUs, which fires through the entire period, is needed to ensure identification of the periodically firing. Thus the % MVC that the subjects were asked to perform was chosen from a practical point of view, due to the lack of possibility to decompose signals above 1-12% MVC, because of the great amount of superimpositions of MUs. This was the likewise the reason why the exerted 1% MVC EMG signal from subject 1, was not decomposed. It has previously been reported that the algorithm, used for decomposition, provides an accuracy of 95%, though based on our experiences this is to a great extent dependent on the operators decomposition experience. In the proposed method the size of the first principle was significantly greater than the second components, implying a high strength of common drive. With a note of caution to the results, due to the limited number of recordings, the variability of the previous and proposed method is compared, showing a slightly difference between the variability of the two methods. In addition the estimated strength of common drive by the proposed method was computed on large populations of MUs, contrary to on pairs of MUs by the previous method. The interpairwise variability of the cross correlations, in the previous method, of 3-s epochs signals is seen to be approximately 18%, which reflects the variations of the strength of common drive between the different pairs of MUs. In addition to previous works [7], [4], [5], [8], [9], and [1], the strength of common drive computed by using the previous method is performed on 3 s EMG signals, and without considering the appearance of the fluctuations in the mean firing rates, of the periods being analyzed. The proposed method for extracting the common drive signal uses the variance as a parameter to express the essential information in the mean firing rates, which reflects the properties of common drive. In relation to the assumptions about the oscillating shape of the common drive signal, the principal component analysis is appropriate as a method to be used. One of the primary objectives is to extract the common drive signal in large populations of MUs, which is obtained by using PCA, and thereby using the weights of the different principal components computed on basis of all MU mean firing rates. Thus the common drive is expressed as the eigenvector with the direction representing most of the variance in the mean firing rates, hence across the low frequency fluctuations (< 2 Hz) of the MU mean firing rates. The peaks in the power spectrum revealed at Hz by spectral analysis were interpreted as the common drive. The peaks in the power spectrum below.5 Hz are not believed to have a physiological interpretation, due to the

7 7 length of the epochs, which were below one period of these oscillations, used for estimating the power spectrum. Though longer recordings and better estimations of the power spectra can provide more clear conclusions. By analysing the power spectrum, several peaks was seen. This indicates that a parametric estimation, using an AR model, can be used for the spectral analysis. Spectral analysis of the common drive could constitute the basis of the finding of the source of common drive. Furthermore, multiple peaks in the power spectrum could reveal more than one common drive. It would be interesting to investigate the properties of the common drive signal in several scenarios using the proposed method, for instance for investigating the effect of fatigue on common drive, or to investigate whether different muscles receives the same common drive. REFERENCES [1] C. J. D. Luca, R. S. LeFever, M. P. McCue, and A. P. Xenakis, Control scheme governing concurrently active human motor units during voluntary contractions. J Physiol., vol. 329, pp , [2] K. C. McGill, Z. C. Lateva, and H. R. Marateb, Emglab: An interactive emg decomposition program, J Neurosci Methods., pp , 25. [3] C. J. D. Luca and Z. Erim, Common drive of motor units in regulation of muscle force, Trends Neurosci., vol. 17, pp , [4] C. J. D. Luca, Control properties of motor units, J Exp Biol., pp , [5] C. J. D. Luca and B. Mambrito, Voluntary control of motor units in human antagonist muscles: Coactivation and reciprocal activation, J Neurophysiol., pp , [6] I. T. Jolliffe, Principal Component Analysis. Springer, 22. [7] J. G. Semmler, M. A. Nordstrom, and C. J. Wallace, Relationship between motor unit short-term synchronization and common drive in human first dorsal interosseous muscle, Brain Res., pp , [8] C. Sauvage, M. Manto, A. Adam, R. R. P. Jissendi, and C. J. D. Luca, Ordered motor-unit firing behavior in acute cerebellar stroke, J Neurophysiol., pp , 26. [9] C. J. D. Luca and Z. Erim, Common drive in motor units of a synergistic muscle pair, J Neurophysiol., pp , 21. [1] Z. Erim, C. J. D. Luca, K. Mineo, and T. Aoki, Rank-ordered regulation of motor units, Muscle Nerve., pp , 1996.