Overall approach

The spike trains elicited by the entire pool of N MNs were inferred from a sample of N_r experimentally identified spike trains with a 4-step approach displayed in Fig 1. The N_r experimentally-identified MNs were allocated to the entire MN pool according to their recorded force recruitment thresholds (step 1). The common synaptic input to the MN pool was estimated from the experimental spike trains of the N_r MNs (step 2) and was linearly scaled for simplicity to the total postsynaptic membrane current I(t) responsible for spike generation. A cohort of N_r single-compartment leaky-and-fire (LIF) models of MNs, the electrophysiological parameters of which were mathematically determined by the unique MN size parameter S_i, transformed the input I(t) to simulate the experimental spike trains after calibration of the S_i parameter (step 3). The distribution of the N MN sizes S(j) in the entire MN pool, which was extrapolated by regression from the N_r calibrated S_i quantities, scaled the electrophysiological parameters of a cohort of N LIF models. The N calibrated LIF models predicted from I(t) the spike trains of action potentials elicited by the entire pool of N virtual MNs.

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Fig 1.

Four-step workflow predicting the spike trains of the entire pool of N MNs (right figure) from the experimental sample of N_r MN spike trains (left figure). Step (1): according to their experimental force thresholds , each MN, ranked from i = 1 to i = N_r following increasing recruitment thresholds, was assigned the location in the complete pool of MNs (i→N_i mapping). Step (2): the current input I(t) common to the MN pool was derived from the N_r spike trains . Step (3): using I(t) as input, the size parameter S_i of a cohort of N_r leaky-and-fire (LIF) MN models was calibrated by minimizing the error between predicted and experimental filtered spike trains. From the calibrated S_i and the MN i→N_i mapping, the distribution of MN sizes S_j in the entire pool of virtual MNs was obtained by regression. Step (4): the S_j distribution scaled a cohort of N LIF models which predicted the MN-specific spike trains of the entire pool of MNs (right). The approach was validated by comparing experimental and predicted spike trains (Validation 1) and by comparing normalized experimental force trace (Left figure, green trace) with normalized effective neural drive (Validation 2). In both figures, the MN spike trains are ordered from bottom to top in the order of increasing force recruitment thresholds.

https://doi.org/10.1371/journal.pcbi.1010556.g001

The following assumptions were made. (1) The MU pool is idealized as a collection of N independent MUs that receive a common synaptic input and possibly MU-specific independent noise. (2) In a pool of N MUs, N MNs innervate N muscle units (mUs). (3) In our notation, the pool of N MUs is ranked from j = 1 to j = N, with increasing recruitment threshold. For the N MNs to be recruited in increasing MN and mU size and recruitment thresholds according to Henneman’s size principle [20,48–53], the distribution of morphometric, threshold and force properties in the MN pool follows where S is the MN surface area, I^th the MN current threshold for recruitment, IR the MU innervation ratio defining the MU size, F^th is the MU force recruitment threshold, and is the MU maximum isometric force. (4) The MN-specific electrophysiological properties are mathematically defined by the MN size S [42]. This extends the Henneman’s size principle to:

Where C is the MN membrane capacitance, R the MN input resistance and τ the MN membrane time constant.

Experimental data

The four sets of experimental data used in this study, named as reported in the first column of Table 1, provide the time-histories of recorded MN spike trains and whole muscle force trace F(t) (left panel in Fig 1), and were obtained from the studies [26,27,54,55], as open-source supplementary material and personal communication, respectively. In these studies, the HDEMG signals were recorded with a sampling rate f_s = 2048Hz from the Tibialis Anterior (TA) and Gastrocnemius Medialis (GM) human muscles during trapezoidal isometric contractions. As displayed in Fig 2, the trapezoidal force trajectories are described in this study by the times reported in Table 1, as a zero force in [t_tr0; t_tr1], a ramp of linearly increasing force in [t_tr1; t_tr2], a plateau of constant force in [t_tr2; t_tr3], a ramp of linearly decreasing force in [t_tr3; t_tr4], and a zero force in [t_tr4; t_tr5].

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Fig 2. Definition of times that describe the trapezoidal shape of the muscle isometric contraction.

https://doi.org/10.1371/journal.pcbi.1010556.g002

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Table 1. The four experimental datasets processed in this study.

N_r spike trains are identified per dataset during trapezoidal contractions of the Tibialis Anterior (TA) or Gastrocnemius Medialis (GM) muscles. The trapezoidal force trace is described by times in seconds up to a dataset-specific level of maximum voluntary contraction (%MVC).

https://doi.org/10.1371/journal.pcbi.1010556.t001

The HDEMG signals were decomposed with blind-source separation techniques and N_r MN spike trains were identified. In this study, the experimental N_r MNs were ranked from i = 1 to i = N_r in the order of increasing recorded force recruitment thresholds , i.e. . The sample time of the k^th firing event of the i^th identified MN is noted as , and the binary spike train of the i^th identified MN was mathematically defined as: (Eq 1)

The train of instantaneous discharge frequency IDF_i(t) of the i^th identified MN was computed between firing times and as: (Eq 2)

The IDFs were moving-average filtered by convolution with a Hanning window of length 400ms [56], yielding the continuous filtered instantaneous discharge frequencies (FIDFs) for all N_r identified MNs.

Approximation of the TA and GM MU pool size

The typical number N of MUs was estimated for the TA muscle from cadaveric studies [57], statistical methods [58], decomposed-enhanced spike-triggered-averaging (DESTA) approaches [59–64], and adapted multiple point stimulation methods [65] in 20-80-year-old human subjects. Because of method-specific limitations [66], results across methods varied substantially, with estimates for N of, respectively, 445, 194, 190, 188 and 300 MUs for the TA muscle. DESTA methods systematically underestimate the innervation ratio due to the limited muscle volume covered by the surface electrodes. Cadaveric approaches rely on samples of small size and arbitrarily distinguish alpha from gamma MNs. Twitch torque measurements are an indirect method for estimating N. Accounting for these limitations, we estimated the potentially conservative N_TA = 400 MUs in a typical adult TA muscle. Assuming 200,000 fibres in the TA muscle [67,68], N_TA = 400 yields a mean of 500 fibres per TA MU, consistently with previous findings [68]. In two cadaveric studies [57,69], the estimate for the GM was N_GM = 550 MUs, which is consistent with N_TA = 400 as the GM muscle volume is typically larger than TA’s [70]. A sensitivity analysis on the performance of the method presented in Fig 1 upon variations on the value of N_TA was included in this study.

Step (1): MN mapping

In the first step of the approach overviewed in Fig 1, the N_r experimentally identified MNs were allocated to the entire pool of N MNs according to their recorded force recruitment thresholds . Three studies measured in the human TA muscle in vivo the force recruitment thresholds F^th of MUs, given as a percentage of the maximum voluntary contraction (%MVC) force, for 528 [64], 256 [71], and 302 [72] MUs. Other studies investigated TA MU pools but reported small population sizes [73] and/or did not report the recruitment thresholds [74–76].

We digitized the scatter plot in Fig 3 in [64] using the online tool WebPlotDigitizer [77]. The normalized MU population was partitioned into 10%-ranges of the values of F^th (in %MVC), as reported in [71,72]. The distributions obtained from these three studies were averaged. The normalized frequency distribution by 10%MVC-ranges of the F^th quantities hence obtained was mapped to a pool of N MUs, providing a step function relating each j^th MU in the MU population to its 10%-range in F^th. This step function was least-squares fitted by a linear-exponential trendline (Eq 3) providing a continuous frequency distribution of TA MU recruitment thresholds in a MU pool that reproduces the available literature data.

(Eq 3)

Simpler trendlines, such as [29], returned fits of lower r² values. According to the three studies and to [20], a Δ_F = 120-fold range in F^th was set for the TA muscle, yielding F^th(N) = 90%MVC = Δ_F∙F^th(1), with F^th(1) = 0.75%MVC. Finally, the equation was solved for the variable N_i for all N_r identified MUs for which the experimental threshold was recorded. The N_r identified MUs were thus assigned the locations of the complete pool of N MUs ranked in order of increasing F^th:

When considering a 100-ms electromechanical delay between MN recruitment time and onset of muscle unit force, the mapping did not substantially change. It was therefore simplified that a MN and its innervated muscle unit were recruited at the same time, and the mapping derived for muscle units was extrapolated to MNs.

Considering that typically less than 30 MUs (5% of the GM MU pool) can be currently identified by HDEMG decomposition in GM muscles [27], and that the few papers identifying GM MUs with intramuscular electrodes either did not report the MU F^th [78–80] or identified less than 24 MUs up to 100% MVC [81,82], a GM-specific F^th(j) distribution could not be obtained from the literature for the GM muscle. The F^th(j) distribution obtained for the TA muscle was therefore used for the simulations performed with the GM muscle, which is acceptable as an initial approximation based on visual comparison to the scattered data provided in these studies.

In the second step of the approach in Fig 1, the common synaptic input to the MN pool was first estimated from the experimental data. The cumulative spike train (CST) was obtained as the temporal binary summation of the N_r experimental spike trains .

(Eq 4)

The effective neural drive (eND) to the muscle was estimated by low-pass filtering the CST in the bandwidth [0; 10]Hz relevant for force generation [47]. As the pool of MNs filters the MN-specific independent synaptic noise received by the individual MNs and linearly transmits the common synaptic input (CSI) to the MN pool [31,83], the CSI was equalled to the eND in arbitrary units: (Eq 5)

The common synaptic control (CSC) signal was obtained by low pass filtering the CSI in [0; 4]Hz.

This approach, which estimates the CSI from the N_r experimental spike trains is only valid if the sample of N_r MNs is ‘large enough’ and ‘representative enough’ of the complete MN pool for the linearity properties of the population of N_r MNs to apply [83]. To assess if this approximation holds with the N_r MNs obtained experimentally, the following two validations were performed. (1) The coherence , averaged in [1; 10]Hz, was calculated between two cumulative spike trains and computed from two complementary random subsets of MNs. This was repeated 20 times for random permutations of complementary subsets of MNs, and the values were average yielding . The coherence between the complete experimental sample of N_r MNs and a virtual sample of N_r non-identified MNs was finally estimated by reporting the pair similarly to Fig 2A in [47]. (2) The time-histories of the normalized force and common synaptic control , which should superimpose if the linearity properties apply (see Fig 6 in [83]), were compared with calculation of the normalized root-mean-square error (nRMSE) and coefficient of determination r². If , r²>0.7 and nRMSE<30%, it was assumed that the sample of N_r MNs was large and representative enough of the MN pool for the linearity properties to apply, and the eND was confidently assimilated as the CSI to the MN pool in arbitrary units. It must be noted that if , the linearity properties do not fully apply for the sample of N_r MNs, and the CSI computed from the N_r MNs is expected to relate to the true CSI with a coherence close but less than .

The CSI hence obtained reflects the net excitatory synaptic influx common to the MN population, which triggers the opening of voltage-dependent dendritic channels and the activation of intrinsic membrane currents, such as persistent inward currents (PICs), which contribute to the total dendritic membrane current I(t) responsible for spike generation. As the single-compartment LIF model considered in this study does not describe the dendritic activity, the PIC-related nonlinearity in the CSI−I transformation was excluded from the model and I(t) was simplified to be linearly related to CSI with a constant gain G across the MN pool. To avoid confusion, I(t) is referred as ‘current input’ in the following. The limitations of this simplification are investigated in the Limitations Section (Limitation 3). To identify G, it must be noted that the CSI, which was computed from a subset of the MN pool, does not capture the firing activity of the MNs that are recruited before the smallest identified MN, which is recruited at time . It non-physiologically yields . Accounting for this experimental limitation, I(t) was defined to remain null until the first identified MN starts firing at , and to non-continuously reach at : (Eq 6)

In (Eq 6), the rheobase currents of the first and last identified MNs and were estimated from a typical distribution of rheobase in a MN pool , obtained as for the distribution of F^th, from normalized experimental data from populations of hindlimb alpha-MNs in adult rats and cats in vivo [84–92]. It is worth noting that the experimental I^th were obtained by injecting current pulses directly into the soma of identified MNs, thus bypassing the dendritic activity and most of the aforementioned PIC-generated nonlinear mechanisms [40]. A Δ_I = 9.1-fold range in MN rheobase in [3.9; 35.5]nA was taken [21,42], while a larger Δ_I is also consistent with the literature ([42], Table 4) and larger values of I^th can be expected for humans [21].

Step (3): LIF model – parameter tuning and distribution of electrophysiological properties in the MN pool

In the third step of the approach in Fig 1, single-compartment LIF MN models with current synapses are calibrated to mimic the discharge behaviour of the N_r experimentally identified MNs.

MN LIF model: description

A variant of the LIF MN model was chosen in this study for its relative mathematical simplicity and low computational cost and its adequacy in mimicking the firing behaviour of MNs. The LIF model described in (Eq 7) describes the discharge behaviour of a MN of rheobase I^th and input resistance R as a capacitor charging with a time constant τ and instantaneously discharging at time ft when the membrane voltage V_m meets the voltage threshold V_th, after which V_m is reset and maintained to the membrane resting potential V_rest for a time duration called ‘inert period’ IP in this study. For simplicity without loss of generalisation, the relative voltage threshold was defined as ΔV_th = V_th−V_rest>0, and V_rest was set to 0.

The model is described by the following set of equations: (Eq 7)

The differential equation was solved with a time step dt≤0.001s as: (Eq 8)

This model includes 5 electrophysiological parameters: R, C, τ, ΔV_th and IP. The passive parameters R and C were mathematically related to the MN surface area as and C = C_m∙S after an extensive meta-analysis of published experimental data on hindlimb alpha-MNs in adult cats in vivo [42], that sets the constant value of C_m to 1.3μF∙cm², supports the equality τ = RC and the validity of Ohm’s law in MNs as , setting the constant value ΔV_th = 27mV in this study. The model was thus reduced to the MN size parameter S and to the IP parameter.

MN LIF model: the IP parameter

Single-compartment LIF models with no active conductances receiving current inputs can predict non-physiologically large values of MN firing frequency (FF) because of a linear FF-I gain at large but physiological current inputs I(t) [44]. The saturation in the FF-I relation typically observed in the mammalian motor unit firing patterns is primarily mediated by the voltage-dependent activity of the PIC-generating dendritic channels [40], that was overlooked in the CSI−I transformation in Step (2), and which decreases the driving force of the synaptic current flow as the dendritic membrane depolarizes. While a physiological modelling of this saturation could be achieved with a LIF model with conductance synapses, as discussed in Limitation 3 in the Limitations section, this nonlinear behaviour could also be captured in a simple approach by tuning the phenomenological IP parameter in (Eq 7). Considering a constant supra-threshold current I≫I^th input to a LIF MN, the steady-state firing frequency FF predicted by the LIF model is: (Eq 9)

As I and C typically vary over a 10-fold and 2.4-fold range respectively [42], the FF predicted by the LIF model is dominantly determined by the value of as the input current increasingly overcomes the MN current threshold: .

In the previous phenomenological models of MNs [9,10,29], a maximum firing rate FF^max was defined and a non-derivable transition from FF(I) to constant FF^max was set for increasing values of I(t). Here, IP was integrated to the dynamics of the LIF model and was derived from experimental data to be MN-specific in the following manner. For each of the N_r identified MNs, the time-course of the MN instantaneous discharge frequencies (IDFs) was first smoothed, as performed in [55], with a sixth-order polynomial trendline (IDF_trend) to neglect any unexplained random noise in the analysis. The mean M of the trendline values during the plateau of force was then obtained. If , i.e. if the MN reached during the ramp of I(t) in an IDF larger than 90% of the IDF reached one second before the plateau of force, the MN was identified to ‘saturate’. Its IP parameter was set to , which constrains the MN maximum firing frequency for high input currents to . A power trendline IP(j) = a∙j^b was finally fitted to the pairs of saturating MNs and the values of the non-saturating MNs were predicted from this trendline. To account for the residual variation in FF observed to remain at high I(t) due to random electrophysiological mechanisms, the IP parameter was randomized at each firing time, taking the value IP+o, where o was randomly obtained from a normal gaussian distribution of standard deviation.

MN LIF model: MN size parameter calibration

The remaining unknown parameter - the MN size S - defines the recruitment and firing dynamics of the LIF model. The size S_i of the i^th identified MN was calibrated by minimizing over the time range (Table 1) the cost function J(S_i) computed as the root-mean-square difference between experimental and LIF-predicted filtered instantaneous discharge frequencies: (Eq 10)

To assess how well the calibrated LIF models can replicate the available experimental data, the normalized RMS error (nRMSE) (%) and coefficient of determination r² between and , and the error in seconds between experimental and predicted recruitment times were computed for the N_r MNs. Finally, a power trendline in (Eq 11) was fitted to the pairs , and the continuous distribution of MN sizes in the entire pool of N MNs was obtained.

(Eq 11)

Δ_S = 2.4 was taken in [42]. The S(j) distribution defines the continuous distribution of the MN-specific electrophysiological properties across the MN pool ([42], Table 4).

MN LIF model: parameter identification during the derecruitment phase

The time-range over which the MNs are being derecruited was not considered in the S calibration in (Eq 10) because the MN’s current-voltage relation presents a hysteresis triggered by long-lasting PICs, as discussed in [40,93,94]. This hysteresis, which leads MNs to be derecruited at lower current threshold than at recruitment, can be phenomenologically interpreted in the scope of a single-compartment LIF model with no active conductance as an increase in the ‘apparent’ MN resistance R during derecruitment to a new value R^d. To determine R^d, a linear trendline () was fitted to the association of experimental MN recruitment I^th and recruitment I^dth current thresholds for the N_r recruited MNs, and the distribution of MN input resistance was increased over the derecruitment time range to: (Eq 12)

As reviewed in [20,40], the current-voltage hysteresis also explains the typically lower MN discharge rate observed at derecruitment than at recruitment. For purely modelling perspectives, this phenomenon was phenomenologically captured by increasing over the time-range the value of the C_m property, which is the main parameter influencing the precited MN FF in (Eq 7) for current inputs close to I^th. The simulated traces were iteratively simulated for the N_r MNs over the time-range with the N_r S_i−IP_i−calibrated LIF models obtained from Step (3), for 0.1μF∙cm² incremental changes in the value of the membrane specific capacitance . The results were compared to the traces with and r² values. The ‘apparent’ C_m value returning the lowest output for (Eq 13) was retained and was renamed .

(Eq 13)

In the following, the individual spike trains were predicted with C_m over the time range and over the time range.

It must be noted that, although this approach can accurately capture the observed MN nonlinear input-output behaviour and is coherent with the modelling constraints imposed by the LIF formulation in (Eq 7), as the C_m and R parameters mainly affect the discharge and recruitment properties of the LIF model respectively in this model, the R and C_m parameters are passive MN properties which remain independent from the MN synaptic and membrane activity in actual MNs, while the R-to-R^d and C_m-to- transformations are merely phenomenological interpretations of complex neurophysiological mechanisms, and do not provide any insights into the working principles underlying MN recruitment. This limitation is further discussed in Limitation 3.

Step (4): Simulating the MN pool firing behaviour

The firing behaviour of the complete MN pool, i.e. the MN-specific spike trains sp_j(t) of the N virtual MNs constituting the MN pool, was predicted with a cohort of N LIF models receiving the common synaptic current I(t) as input. The inert period IP_j and MN size S_j parameters scaling each LIF model were obtained from the distributions IP(j) and S(j) previously derived at Step (3).

Validation

Application to MN-driven muscle modelling

The N MN-specific spike trains predicted in step (4) were input to a phenomenological muscle model to predict the whole muscle force trace in a forward simulation of muscle voluntary isometric contraction. As displayed in Fig 3, the muscle model was built as N in-parallel Hill-type models which were driven by the simulated spike trains and replicated the excitation-contraction coupling dynamics and the contraction dynamics of the N MUs constituting the whole muscle. In brief, the binary spike train triggered for each j^th MU the trains of nerve and muscle fibre action potentials that drove the transients of calcium ion Ca²⁺ and Ca²⁺-troponin complex concentrations in the MU sarcoplasm, yielding in a last step the time-history of the MU active state a_j(t). The MU contraction dynamics were reduced to a normalized force-length relationship that scaled nonlinearly with the MU active state [95] and transformed a_j(t) into a normalized MU force trace . The experiments being performed at constant ankle joint (100°, 90° being ankle perpendicular to tibia) and muscle-tendon length, it was simplified, lacking additional experimental insights, that tendon and fascicle length both remained constant during the whole contractile event at optimal MU length . The dynamics of the passive muscle tissues and of the tendon and the fascicle force-velocity relationships were therefore neglected. Finally, the MU-specific forces f_j(t) were derived with a typical muscle-specific distribution across the MU pool of the MU isometric tetanic forces [64,71,74]. The whole muscle force was obtained as the linear summation of the MU forces .

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Fig 3. MN-driven neuromuscular model.

The in-parallel Hill-type models take as inputs the spike trains predicted in steps (1-4) and output the predicted whole muscle force trace . The MU-specific active states a_j(t) are obtained from the excitation-contraction coupling dynamics adapted from [98] and extended to model the concentration of the Calcium-troponin system. The MU normalized forces are computed by the MU contractile elements (CE_i) at MU optimal length and are scaled with values of MU maximum isometric forces to yield the MU force traces f_j(t). The predicted whole muscle force is taken as .

https://doi.org/10.1371/journal.pcbi.1010556.g003

To validate , the experimental muscle force F^exp(t) was first approximated from the experimental force trace F(t), which was recorded at the foot with a force transducer [26]. The transducer-ankle joint and ankle joint-tibialis anterior moment arms L₁ and L₂ were estimated using OpenSim [96] and a generic lower limb model [97]. Using the model muscle maximum isometric forces, it was then inferred the ratio q of transducer force F(t) that was taken at MVC by the non-TA muscles spanning the ankle joint in MVC conditions. The experimental muscle force was estimated as and was compared with calculation of normalized maximum error and r² values against the muscle force predicted by the MN-driven muscle model from the N neural inputs.

The whole muscle force was also predicted using the N_r experimental spike trains as inputs to the same muscle model of N_r in-parallel Hill-type models (Fig 3). In this case, each normalized MU force trace was scaled with the same distribution, however assuming the N_r MNs to be evenly spread in the MN pool. was similarly compared to F^exp(t) with calculation of nME, nRMSE and r² values. To assess the added value of the step (1-4) approach in the modelling of MN-driven muscle models, the (nME, nRMSE, r²) values obtained for the predicted and were compared.

Experimental data

As reported in Table 1, the experimental datasets DTA₃₅ and HTA₃₅ respectively identified 32 and 21 spike trains from the trapezoidal isometric TA muscle contraction up to 35%MVC, HTA₅₀ identified 14 up to 50%MVC, and HGM₃₀ identified 27 from the GM muscle up to 30%MVC. The N_r = 32 MN spike trains, identified in this dataset across the complete TA pool of N = 400 MNs, are represented in Fig 4A in the order of increasing force recruitment thresholds . The N_r MNs were globally derecruited at relatively lower force thresholds (Fig 4B) and generally discharged at a relatively lower firing rate (Fig 4C) at derecruitment than at recruitment.

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Fig 4. Experimental data obtained from HDEMG signal decomposition in the dataset DTA₃₅.

(A) Time-histories of the transducer force trace in %MVC (green curve) and of the N_r = 32 MN spike trains identified from HDEMG decomposition and ranked from bottom to top in the order of increasing force recruitment thresholds F^th. (B) Association between force recruitment and derecruitment threshold, fitted by a linear trendline y = 0.9∙x (r² = 0.85). Identity is displayed as a black dotted line. (C) Time-histories of the MN instantaneous discharge frequencies (IDFs, blue dots) smoothed by convolution with a Hanning window of length 400ms (red curve) and with a sixth-order polynomial trendline (green curve) of the lowest-threshold identified (1^st) MN.

https://doi.org/10.1371/journal.pcbi.1010556.g004

Step (1): MN mapping

The N_r = 32 MNs identified in the dataset DTA₃₅ were allocated to the entire pool of N = 400 MNs according to their recruitment thresholds (%MVC). The typical TA-specific frequency distribution of the MN force recruitment thresholds F^th, which was obtained from the literature and reported in the bar plot in Fig 5A, was approximated (Fig 5B) by the continuous relationship: (Eq 14)

With this distribution, 231 TA MNs, i.e. 58% of the MN pool is recruited below 20%MVC, which is consistent with previous conclusions [20].

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Fig 5.

Distribution of force recruitment thresholds F^th in the human Tibialis Anterior (TA) muscle and mapping of the N_r identified MNs to the complete MN pool. (A) Typical partition obtained from the literature of the TA MN pool in 10% increments in normalized F^th. (B) Equivalent F^th stepwise distribution (black dots) in a TA pool of N = 400 MNs, approximated by the continuous relationship F^th(j) (red curve). The mapping (blue crosses) of the N_r = 32 MNs identified in the dataset DTA₃₅ was obtained from the recorded (Fig 4B). (C) N_r = 32 MNs (red dots) of unknown properties (Left) are mapped (Right) to the complete MN pool from the F^th(j) distribution, represented by the blue dots of increasing sizes. The MNs represented here are numbered from left to right and bottom to top from j = 1 to .

https://doi.org/10.1371/journal.pcbi.1010556.g005

From the F^th(j) distribution, the N_r identified MNs were mapped to the complete MN pool (blue crosses in Fig 5B) according to their recorded force recruitment thresholds (ordinates in Fig 4B). As shown in Fig 5C, the N_r MNs identified experimentally were not homogeneously spread in the entire MN pool ranked in the order of increasing force recruitment thresholds, as two MNs fell in the first quarter of the MN pool, 5 in the second quarter, 18 in the third quarter and 5 in the fourth quarter. Such observation was similarly made in the three other experimental datasets, where no MN was identified in the first quarter and in the first half of the MN pool in the datasets HGM₃₀ and HTA₅₀ respectively (second column of Table 2). In all four datasets, mostly high-thresholds MNs were identified experimentally.

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Table 2. Intermediary results obtained for the datasets DTA₃₅, HTA₃₅, HTA₅₀ and HGM₃₀ from the three first steps of the approach.

For each dataset are reported (1) the locations in the complete pool of MNs of the lowest- (N₁) and highest-threshold () MNs identified experimentally, (2) the value between the experimental and virtual cumulative spike trains (CST), and the coefficients defining the distributions in the complete MN pool of (3) the inert period (IP) parameter and of (4) the MN size (S). For the TA and GM muscles, N = 400 and N = 550 respectively.

https://doi.org/10.1371/journal.pcbi.1010556.t002

Step (2): Common current input I(t)

To approximate the common synaptic input CSI(t) to the MN pool, the CST and the eND to the MN pool were obtained in Fig 6 from the N_r MN spike trains identified experimentally using (Eq 4) and (Eq 5). After 20 random permutations of complementary populations of MNs, an average coherence of was obtained between -sized CSTs of the DTA₃₅ dataset. From Fig 2 in [47], a coherence of is therefore expected between the CST in Fig 6A and a typical CST obtained with another virtual group of N_r = 32 MNs, and by extension with the true CST obtained with the complete MN pool. The normalized eND and force traces (black and green curves respectively in Fig 6B) compared with r² = 0.92 and nRMSE = 20.0% for the DTA₃₅ dataset. With this approach, we obtained , r²>0.7 and nRMSE<30% for all four datasets, with the exception of HGM₃₀ for which (third column of Table 2). For the TA datasets, the sample of N_r identified MNs was therefore concluded to be sufficiently representative of the complete MN pool for its linearity property to apply, and the eND (red curve in Fig 6B for the dataset DTA₃₅) in the bandwidth [0,10]Hz was confidently identified to be the common synaptic input (CSI) to the MN pool. As observed in Fig 6B, a non-negligible discrepancy, which partially explains why , was however systematically obtained between the eND and force traces in the regions low forces, where mostly small low-threshold MNs are recruited. As discussed in the Discussion Section, this discrepancy reflects the undersampling of small MUs identified from decomposed HDEMG signals and a bias towards mainly identifying the large high-threshold units which are recruited close to the plateau of force. From Fig 2 in [47], it is worth noting that the computed CSI in Fig 6B (red curve) accounts for 60% of the variance of the true synaptic input, which is linearly transmitted by the MN pool, while the remaining variance is the MN-specific synaptic noise, which is assumed to be filtered by the MN pool and is neglected in the computation of the eND in this workflow.

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Fig 6. Neural drive to the muscle derived from the N_r identified MN spike trains in the dataset DTA₃₅.

(A) Cumulative spike train (CST) computed by temporal binary summation of the N_r identified MN spike trains. (B) Effective neural drive – Upon applicability of the linearity properties of subsets of the MN pool, the effective neural drive is assimilated as the synaptic input to the MN pool. The normalized common synaptic input (red), control (black) and noise (blue) are obtained from low-pass filtering the CST in the bandwidths relevant for muscle force generation. The normalized experimental force trace (green curve) is displayed for visual purposes.

https://doi.org/10.1371/journal.pcbi.1010556.g006

As discussed in the Methods section, the normalized CSI (red curve in Fig 6B) was simplified to be linearly related to the total dendritic membrane current I(t). To do so, the scaling factor was determined with a typical distribution of the MN membrane rheobase I^th(j) in a cat MN pool, which was obtained (Fig 7A) from the literature [85,87,88,90,91,99–101] as: (Eq 15)

Using (Eq 6), the time-history of the current input I(t) was obtained (Fig 7B): (Eq 16)

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Fig 7.

(A) Typical distribution of MN current recruitment threshold I^th (N_i) in a cat MN pool according to the literature. (B) Current input I(t), taken as a non-continuous linear transformation of the common synaptic input (red curve in Fig 6B) for the dataset DTA₃₅.

https://doi.org/10.1371/journal.pcbi.1010556.g007

Step (3): LIF model – MN size calibration and distribution

Because of the modelling choices made for our MN LIF model, the MN inert period (IP) and the MN size S parameters entirely define the LIF-predictions of the MN firing behaviour. The IP_i parameters of the N_r MNs in the dataset DTA₃₅ (Fig 8) were obtained from the maximum firing frequency of the 20 MNs identified to ‘saturate’, from which the distribution of IP values in the entire pool of N MNs was obtained: IP(j)[s] = 0.04∙j^0.05. With this approach, the maximum firing rate assigned to the first recruited and unidentified MN is . The IP distributions obtained with this approach for the three other datasets are reported in the fourth column of Table 2 and yielded physiological approximations of the maximum firing rate for the unidentified lowest -threshold MN for all datasets, with the exception of the dataset HTA₅₀, which lacks the information of too large a fraction of the MN pool for accurate extrapolations to be performed.

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Fig 8.

MN Inert Periods (IPs) in ms obtained from the experimental measurements of IDFs in dataset DTA₃₅. The twenty lowest-threshold MNs are observed to ‘saturate’ as described in the Methods and their IP (black crosses) is calculated as the inverse of the maximum of the trendline fitting the time-histories of their instantaneous firing frequency. The IPs of the 12 highest-threshold MNs (red dots) are obtained by trendline extrapolation.

https://doi.org/10.1371/journal.pcbi.1010556.g008

The size parameter S_i of the N_r LIF models was calibrated using the minimization function in (Eq 14) so that the LIF-predicted filtered discharge frequencies of the N_r MNs replicated the experimental , displayed in Fig 9A and 9B (blue curves). As shown in Fig 9C, the recruitment time ft¹ of three quarters of the N_r identified MNs was predicted with an error less than 250ms. The calibrated LIF models were also able to accurately mimic the firing behaviour of the 27 lowest-threshold MNs as experimental and LIF-predicted FIDF traces compared with r²>0.8 and nRMSE<15% (Fig 9D and 9E). The scaled LIF models reproduced the firing behaviour of the five highest-threshold MNs (Ni>300) with moderate accuracy, with Δft¹ and nRMSE values up to -1.5s and 18.2%. Despite modelling limitations discussed in the Discussion section, the global results in Fig 9 confirm that the two-parameters calibrated LIF models can accurately reproduce the firing and recruitment behaviour of the N_r experimental MNs. It must be noted that the Δft¹ and r² metrics were not included in the calibration procedure and the (Δft¹, r²) values reported in Fig 9 were therefore blindly predicted.

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Fig 9. Calibration of the MN size S_i parameter.

(A and B) Time-histories of the experimental (black) versus LIF-predicted (blue) filtered instantaneous discharge frequencies (FIDFs) of the 2^nt (A) and 16^th (B) MNs identified in the DTA₃₅ dataset after parameter calibration. (C) Absolute error Δft¹ in seconds in predicting the MN recruitment time with the calibrated LIF models. The accuracy of LIF-predicted FIDFs is assessed for each MN with calculation of the nRMSE (D) and r² (E) values. (C-E) The metrics are computed for the time range only. The dashed lines represent the Δft¹∈[−250; 250]ms, nRMSE∈[0; 15]%MVC and r²∈[0.8, 1.0] intervals of interest respectively.

https://doi.org/10.1371/journal.pcbi.1010556.g009

The cloud of N_r pairs of data points (black crosses in Fig 10A), obtained from MN mapping (Fig 5B and 5C) and size calibration for the dataset DTA₃₅, were least-squares fitted (red curve in Fig 10A, r² = 0.96) by the power relationship in (Eq 17), with Δ_S = 2.4: (Eq 17)

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Fig 10. Reconstruction of the firing behaviour and recruitment dynamics of the complete MN pool.

(A) The N_r calibrated MN sizes (black crosses) are lest-squares fitted by the power trendline , which reconstructs the distributions of MN sizes in the complete MN pool. (B) The S(j) distribution determines the MN-specific R and C parameters of a cohort of N LIF models, which is driven by the current input I(t) and predicts (C) the spike trains of the N virtual MNs constituting the complete MN pool. (D) The MN instantaneous discharge frequencies were computed from the simulated the spike trains and were smoothed with a sixth-order polynomial [55]. One out of ten IDFs is displayed for clarity. The blue-to-red gradient for small-to-large MNs shows that the output behaviour of the reconstructed MN population is in agreement with the Onion-Skin scheme.

https://doi.org/10.1371/journal.pcbi.1010556.g010

As reported in the last column of Table 2, the minimum MN size (in m²) obtained by extrapolation of this trendline was in the range [1.20, 1.73]∙10⁻¹mm² for the datasets DTA₃₅, HTA₃₅ and HGM₃₀, with expected maximum MN size (in m²) in the range [2.88, 4.15]∙10⁻¹mm² which is consistent with typical cat data [42,102–107], even if slightly in the lower range. The low value for the c coefficient observed in Table 2 for the TA₅₀ and GM₃₀ datasets suggests that the typical skewness in MN size distribution reported in the literature [42] was not captured for these two datasets, in which low-threshold MNs in the first quarter of the MN population were not identified from HDEMG signals.

As displayed in Fig 9A and 9B, the phenomenological adjustment of the R and C_m parameters described in (Eq 12) and (Eq 13) successfully captured the main effects of the hysteresis mechanisms involved with MN derecruitment, where MNs discharge at lower rate and stop firing at lower current input than at recruitment. In all four datasets, was parabolic for incremental values of C_m and a global minimum was found for and 2.2μF∙cm² for the datasets DTA₃₅, HTA₃₅, HTA₅₀ and HGM₃₀, respectively. These values for the parameter were retained for the final simulations (Step (4)) over the time range.

Step (4): Simulating the MN pool firing behaviour

As displayed in Fig 10B for the dataset DTA₃₅, the S(j) distribution determined the MN-specific electrophysiological parameters (input resistance R and membrane capacitance C) of a cohort of N = 400 LIF models, which predicted from I(t) the spike trains of the entire pool of N MNs (Fig 10C). As displayed in the plot of smoothed FIDFs in Fig 10C, the output behaviour of the reconstructed MN pool agrees with the Onion-skin scheme [108] where lower-threshold MNs start discharging at higher frequencies and reach higher discharge rates than larger recruited units.

Validation

The four-step approach summarized in Fig 1 is detailed in Fig 11 and was validated in two ways.

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Fig 11. Detailed description of the 4-step workflow applied to the DTA₃₅ dataset.

The firing activity of a fraction of the MN pool is obtained from decomposed HDEMG signals. These N_r = 32 experimental spike trains provide an estimate of the effective neural drive to muscle and explain most of the MN pool behaviour (coherence = 0.8). From a mapping of the N_r identified MNs to the complete MN pool (Step (1)), literature knowledge on the typical distibution of I^th in a mammalian MN pool, and using the linearity properties of the population of N_r MNs, the current input I(t) is estimated (Step (2)). I(t) is input to N_r LIF models of MN to derive, after a one-parameter calibration step minimizing the error between experimental and LIF-predicted FIDFs, the distribution of MN sizes across the complete MN pool (Step (3)). The distribution of MN sizes, which entirely describes the distribution of MN-specific electrophysiological parameters across the MN pool, scales a cohort of N = 400 LIF models which transforms I(t) into the simulated spike trains of the N MNs of the MN pool (Step (4)). The effective neural drive to muscle is estimated from the N simulated spike trains.

https://doi.org/10.1371/journal.pcbi.1010556.g011

Validation 1.

The simulated spike trains were validated for the N_r MNs by comparing experimental and LIF-predicted FIDFs, where the experimental information of the investigated MN was removed from the experimental dataset and not used in the derivation of the IP(j) and S(j) distributions of the synaptic current I(t). For the N_r MNs of each dataset, Fig 12 reports the absolute error Δft¹ in predicting the MN recruitment time (1^st row) and the comparison between experimental and LIF-predicted filtered instantaneous discharge frequencies (FIDFs) with calculation of nRMSE (%) (2^nd row) and r² (3^rd row) values over the time range. In all datasets, the recruitment time of 45-to-65% of the N_r identified MNs was predicted with an absolute error less than Δft¹= 250ms. In all datasets, the LIF-predicted and experimental FIDFs of more than 80% of the N_r MNs compared with nRMSE<20% and r²>0.8, while 75% of the N_r MN experimental and predicted FIDFs compared with r²>0.8 in the dataset HGM₃₀. These results confirm that the four-step approach summarized in Fig 1 is valid for all four datasets for blindly predicting the recruitment time and the firing behaviour of the N_r MNs recorded experimentally. The identified MNs that are representative of a large fraction of the complete MN pool, i.e. which are the only identified MN in the range of the entire MN pool, such as the 1^st MN in the dataset DTA₃₅ (Fig 5C), are some of the MNs returning the highest Δft¹ and nRMSE and lowest r² values. As observed in Fig 12, ignoring the spike trains of those ‘representative’ MNs in the derivation of I(t), IP(j) and S(j) in steps (2) and (3) therefore affects the quality of the predictions more than ignoring the information of MNs that are representative of a small fraction of the MN population. In all datasets, nRMSE>20% and r²<0.8 was mainly obtained for the last-recruited MNs (4^th quarter of each plot in Fig 12) that exhibit recruitment thresholds close to the value of the current input I(t) during the plateau of constant force in the time range . In all datasets, the predictions obtained for all other MNs that have intermediate recruitment thresholds and are the most identified MNs in the datasets, were similar and the best among the pool of N_r MNs.

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Fig 12. Validation of the MN recruitment and firing behaviour predicted with the 4-step approach (Fig 1) for the N_r MNs experimentally identified in the four datasets DTA₃₅, HTA₃₅, HTA₅₀ and HGM₃₀ described in Table 1.

For the validation of each i^th predicted MN spike train, the experimental information of the i^th identified MN was omitted when deriving the current input I(t) and the IP(j) and S(j) distributions (steps 2 and 3), from which the IP_i and S_i parameters are obtained without bias for the i^th MN. The spike train of the i^th MN is then predicted with an IP_i, S_i-scaled LIF model receiving I(t) as input. The absolute error in predicting the experimental recruitment time is reported for each of the N_r MNs in the first row of the figure. Experimental and filtered instantaneous discharge frequencies and , computed from and are compared with calculation of nRMSE (%) and r² in the second and third rows of the figure. The dashed lines represent the Δft¹∈[−250; 250]ms, nRMSE∈[0; 15]%MVC and r²∈[0.8, 1.0] intervals of interest respectively.

https://doi.org/10.1371/journal.pcbi.1010556.g012

Validation 2.

The effective neural drive predicted by the 4-step approach summarized in Fig 1 was validated by comparing for isometric contractions, the normalized eND_N (orange traces in Fig 13) computed from the N predicted spike trains to the normalized force trace (green trace in Fig 13). The eND_N was accurately predicted for the datasets DTA₃₅ and HTA₃₅, with r² = 0.97−0.98 and nRMSE<8% (Table 3). The results obtained from the dataset HGM₃₀ returned r² = 0.89 and nRMSE = 18.2% (Table 3), accurately predicting the eND_N for the positive ramp and first half of the plateau of force and then underestimating the true neural drive (HGM₃₀, Fig 13). The eND_N predicted for the dataset HTA₅₀ returned results of lower accuracy (r² = 0.88 and nRMSE = 15.1%) compared to the three other datasets. A null eND was predicted for one third of the simulation where a muscle force up to 20%MVC is generated, and the eND was overestimated for the rest of the (de)recruitment phase (HTA₅₀, orange trace, Fig 12). As detailed in the discussion, the latter is explained by an inadequate IP(j) distribution (Table 2), due to a lack of experimental information in the dataset HTA₅₀, which returns non-physiological maximum firing rates for the low-thresholds MNs. Two data points (16; 0.032) and (118; 0.037), obtained from the experimental data in the dataset DTA₃₅ with a scaling factor of applied to the IP values, were appended to the list of N_r data points to describe the maximum firing behaviour of the first half of the MN pool for which no MN was identified for the dataset HTA₅₀ (Table 2). A new IP(j) distribution was obtained, returning an improved estimation of the eND_N (HTA₅₀, purple trace, Fig 12) with respectively lower and higher nRMSE and r² metrics (Table 3). With three times lower nRMSE and higher r² values for all four datasets (Table 3), the eND_N predicted from the 4-step approach (orange dotted traces in Fig 13) was a more accurate representation of the real effective neural drive than the (blue dotted traces in Fig 13) computed from the N_r experimental spike trains, especially in the phases of MN (de)recruitment where the real effective neural drive was underestimated by . The predicted eND_N also produced less noise than the trace during the plateau of force. With r²>0.85 and nRMSE<20%, this four-step approach is valid for accurately reconstructing the eND to a muscle produced by a collection of N simulated MN spike trains, which were predicted from a sample of N_r experimental spike trains. It is worth noting that the accuracy of the eND_N predictions, performed in Table 3 and Fig 13 for N_TA = 400 and N_GM = 550 MNs as discussed in the Methods Section, was not sensitive to the size N of the reconstructed MN population. For example, for the DTA₃₅ dataset, the eND_N computed with populations of N = {32, 100, 200, 300, 400} MNs systematically compared to the real effective neural drive with r² = 0.98, and respectively returned nRMSE = 7.7, 6.9, 6.5, 6.1 and 5.9%, with slightly more accurate predictions of eND_N with an increasing N at low force during derecruitment.

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Fig 13.

For the 4 datasets, validation against the normalized force trace (green trace) of the normalized effective neural drive to muscle (eND_N) produced by the complete MN pool (orange trace) and computed from the N MN spike trains predicted with the 4-step approach. For comparison, the directly computed from the N_r identified MNs is also reported (blue dashed trace). For the dataset HTA₅₀, an additional prediction was performed (purple trace), where the IP(j) distribution obtained from step (3) was updated to return physiological maximum firing rates for all N MNs.

https://doi.org/10.1371/journal.pcbi.1010556.g013

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Table 3. For all four datasets, the r² and nRMSE values obtained for the comparison of the time-histories of the normalized experimental force trace against the effective neural drive (1) and (2) eND_N computed from the spike trains of (1) the N_r identified MNs and (2) the N virtual MNs.

The results for HTA₅₀ (1) were obtained with the standard approach, while those for HTA₅₀ (2) were obtained with a revisited IP(j) distribution (see text).

https://doi.org/10.1371/journal.pcbi.1010556.t003

Application to MN-driven muscle modelling

Fig 14 reports for the datasets DTA₃₅ and HTA₃₅, for which the eND_N was the most accurately predicted among the four datasets (Fig 13, Table 3), the time evolutions of the whole muscle force F(t) recorded experimentally (green curve) and the whole muscle forces predicted with the MN-driven muscle model described in Fig 3 using N_r experimental (, blue curves) and N simulated (F_N(t), red curves) spike trains as inputs. The muscle force trace F_N(t) obtained from the N simulated spike trains derived in steps (1-4) compared with the experimental force F(t) with r² = 0.95 and nRMSE = 11% for the dataset DTA₃₅ and r² = 0.97 and nRMSE = 11% for the dataset HTA₃₅. These results validate the approach provided in this study for predicting the whole muscle force as a collection of MU force contributions from the MN-specific contributions to the effective neural drive. The muscle force trace F_N(t) obtained from the N MN spike trains returned more accurate predictions than (r² = 0.88 and nRMSE = 27% for the dataset DTA₃₅ and r² = 0.79 and nRMSE = 33% for the dataset DTA₃₅), in which case the reconstruction of the MN pool in steps (1-4) was disregarded and the N_r experimental spike trains were directly used as inputs to the muscle model in Fig 3.

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Fig 14. Prediction of the experimental force trace F (t) (green curve) with the MN-driven muscle model described in Fig 3.

(blue curve) was predicted with the N_r experimental spike trains input to the muscle model. F_N(t) (red trace) was obtained from the cohort of N experimental spike trains simulated by the four-step approach.

https://doi.org/10.1371/journal.pcbi.1010556.g014

Validation of the approach

The approach was successfully validated both for individual MNs and for the complete pool of N MNs. By blindly scaling the LIF models in steps (1-3) with permutations of N_r−1 input experimental spike trains, the filtered discharge behaviour and the recruitment dynamics of the N_r individual MNs were accurately predicted for the four investigated datasets (Fig 12). The effective neural drive (eND_N) to muscle elicited by the complete pool of N MNs was also accurately predicted by the 4-step approach for the HTA₃₅, DTA₃₅ and HGM₃₀ datasets (Fig 13, Table 3). The latter result suggests that the recruitment dynamics and the normalized distribution of firing rates across the firing MN pool were accurately predicted for the non-identified population of N−N_r MNs. The accuracy of the eND_N predictions was not sensitive to the size N of the reconstructed population, with acceptable eND_N predictions (nRMSE<20%, r²>0.9) obtained with as few as ten distributed MNs. This suggests that the MN mapping (Step 1) and the derivation of the MN properties distribution (Step 3) in Figs 8 and 10A are the key contributions for describing the complete MN pool behaviour from the input experimental data.

Despite the modelling limitations, discussed in [44] and in the Limitations Section, inherent to single-compartment LIF models with current synapses and related to the phenomenological IP, C^d and R^d parameters, the LIF models scaled following the methodology described in Step (3) can capture the distribution of MN discharging characteristics across the MN population. For example, the firing activity of the reconstructed MN population agrees with the onion-skin scheme (Fig 10D), as expected for slow low-force trajectories in human muscle contraction. While the MN size parameter calibration (Fig 9) only relies on the minimization of the root-mean square difference between experimental and predicted FIDFs, the normalized time-history of the FIDFs (high r² in Fig 12) and the MN recruitment times (low Δft¹ values in Fig 12) of the N_r MNs are both accurately mimicked (Fig 9) and blindly predicted (Fig 12) for all four datasets. Some realistic aspects of the MN neurophysiology are also maintained. For example, the range of MN surface areas [0.15; 0.36]mm² and input resistances [0.5; 4.1]MΩ obtained from the S(j) distribution falls into the classic range of MN properties for cats [21,42], which is consistent with the distribution of rheobase values used in Step (2) that was obtained from cat data. The distribution of MN input resistance in the MN pool, obtained from the calibrated sizes S(j), defines the individual MN rheobase thresholds and predicts that 80% of the human TA MU pool is recruited at 35%MVC (Fig 10C), which is consistent with previous findings in the literature [20,64].

Benefits of the approach

Limitations of the approach and potential improvements

Despite the aforementioned good performance of the four-step workflow, the method presents 4 main limitations, for some of which potential improvements are proposed in the following.

Limitation 3 – LIF MN modelling.

The LIF MN model described from (Eq 7) to (Eq 13) was shown, with credible sets of inter-related parameters S_MN, R, C and τ after [42], to accurately mimic (Fig 9) and blindly predict (Fig 12) most of the key phenomenological features of the N_r firing MNs, including their FIDFs and time of first discharge, as well as nonlinear behaviours such as firing rate saturation (Fig 8) and hysteresis (Fig 4B and 4C). However, this MN model is mostly phenomenological and does not provide a realistic description of the actual mechanisms underlying the dynamics of action potential firing for several reasons. First, this single-compartment approach neglects the activity of the MN dendrites, which account for more than 95% of the total MN surface area and gather most of the post-synaptic receptors and MN PIC-generating channels responsible for the MN nonlinear input-output functions [40]. The inherent difference in membrane voltage dynamics between the dendrites and the soma, also partially mediated by hundred-fold differences in the value of passive electrophysiological parameters such as R which increases with somatofugal distance [125] is therefore neglected in this point model approach. Then, the nonlinear dendritic activity being overlooked, the Common Synaptic Input CSI(t) in (Eq 5), which is the common net excitatory synaptic influx, is non-physiologically assimilated with a constant gain to the post-synaptic total dendritic membrane current I(t) that depolarizes the MN soma and is responsible for spike generation. With this linear CSI−I scaling in (Eq 6), the approach neglects the realistic description of the voltage-driven dynamics of the PIC-generating channels that are responsible for some important nonlinear mechanisms, such as firing rate saturation [41] and hysteresis in the MN’s current-voltage relation [93,94], which dictate the nonlinear CSI−I transformation [40]. In this study, the firing rate saturation, which in real MNs is due to a decrease in driving force for synaptic current flow as the dendrites become more depolarized [41], is captured by the phenomenological IP parameter. The observed hysteresis between the recruitment and decruitment thresholds (Fig 4B) and in the current-firing rate relation between recruitment and derecruitment phases (Fig 4C), explained by an hysteresis in the MN’s current-voltage relation that arises from the activity of long-lasting PICs [20], are successfully but non-physiologically addressed by a tuning during derecruitment of the R and C_m parameters respectively, although the values of the passive property R and of C_m, biological constant for the MN population [126], should be independent from the MN activity [44]. For these reasons, even if the distributions of the S_MN, R, C and τ parameters at recruitment take values consistent with the literature, as discussed, the MN LIF model described in (Eq 7) to (Eq 12) is not suitable for investigating the neurophysiology and the neuromechanics of individual MNs. The phenomenological tuning of the IP, R and C_m parameters, although symptomatic of underlying nonlinear mechanisms, does not provide any insight into the true MN neurophysiology. If a more realistic MN model was considered in lieu of the LIF model, such as multiple-compartments Hodgkin-Huxley-based approaches [127], the four-step workflow would be an adequate tool for testing neurophysiological hypotheses and investigating some mechanisms and properties of the complete MN pool that cannot be directly observed experimentally in conditions of human voluntary contractions. A possible trade-off would be considering a single-compartment LIF model with active conductances [44], in which case (Eq 7) is re-written as (Eq 18), where is the constant MN passive conductance, E₊>ΔV_th is the constant reversal potential for excitatory channels, g(t, CSI) the time-varying total active excitatory synaptic conductance, and I_ps(t, CSI, V_m) = g(t, CSI)∙(E₊−V_m) the post-synaptic total dendritic membrane current induced by the synaptic drive CSI, the driving force of which is decreased by the term (E₊−V_m) when the dendritic membrane depolarizes, as discussed.

(Eq 18)

In the literature, g(t, CSI) = g_max∙T(t, CSI) where g_max is the maximum active conductance of the synapse and T(t, CSI) can be interpreted as the fraction of bound synaptic receptors [30] or of opened ionic channels in the range [0; 1]or as a train of synaptic pulses [44]. Because setting T(t, CSI) = CSI(t), with CSI(t) as defined in Fig 6B, does not meet the requirements in firing rate saturation for large CSI input, the saturation in dendritic ionic channel activation for large CSI [41] must be accounted for and T(t, CSI) should be a nonlinear saturating function of CSI. In this approach, the MN-specific hysteresis and firing rate behaviour could be obtained with a tuning during recruitment and derecruitment and a distribution across the MN pool of the g_max parameter and of the shape parameter of the S function based on the experimental data. Although this approach more realistically describes the neurophysiological mechanisms underlying the nonlinear MN behaviour during discharge events, it is more challenging to scale using the experimental information. Considering that this study focuses on the phenomenological behaviour of individual MNs and on the overall activity of reconstructed MN populations, the LIF model defined in (Eq 7) was judged an adequate trade-off between accuracy and modelling complexity, considering the overall accurate predictions of the MN firing behaviours (Fig 12) and its low computational cost, and no other modelling approaches such as (Eq 18) were pursued.

It must be noted that, whatever the MN modelling approach, the experimental drive to the MN currently considered in the four-step method is the CSI, which disregards the MN-specific synaptic noise SN(t), which is responsible for most of the inter-spike variability (ISV). Considering that the MN pool and the MU neuromechanical mechanisms are expected to filter the MN-specific SN(t), this simplification is adequate for accurate predictions of the eND_N and of the whole muscle force [83]. Sometimes modelled as a random Gaussian-like event [29], the ISV can be obtained as the response to a SN(t) white noise added to the current definition of the synaptic input as CSI(t)+SN(t). The relative proportion of CSI and SN in % of the variance of the total synaptic current can be approximated from Fig 2A in [47]. Accounting for SN(t) might improve the accuracy of the predicted firing behaviour of the largest MNs, for which the largest Δft¹ and nRMSE and lowest r² values were obtained in Fig 12 for all four datasets, and the firing behaviour and recruitment dynamics of which are dominantly dictated by random fluctuations of SN(t).