Fig 1.
Schematic representation of our new model of a motor unit pool.
The model consists of three modules. Module 1 converts synaptic input, Ueff, into spike trains of individual motor units. Module 2 turns spike trains into motor unit activation, A, through three-stage process shown below. Stage 1 simulates calcium kinetics driven by action potentials (R). The calcium kinetics is described using five states, [s], [cs], [c], [f] and [cf] with associated rate constants (k1, k2, k3 and k4) between those states. Stage 2 converts [cf] into the intermediate activation, , through a non-linear filter, which describes cooperativity and saturation of calcium binding and cross-bridge formation. Stage 3 introduces an additional first-order dynamics to generate motor unit activation, A, from
. Module 3 describes the contraction dynamics between muscle and a series elastic element and generates tendon force, Fse, as the output. The detail descriptions of each module are given in the text.
Fig 2.
New recruitment scheme of our new model mimics discharge patterns of human tibialis anterior motor units.
A) The distribution of minimal discharge rates of all motor units compared against experimental data from Cutsem et al. [46]. B) The distribution of peak discharge rates of all motor units compared against experimental data from Cutsem et al. [46]. C) The frequency-synaptic input relationship of the selected motor units (n = 10). U_r indicates the level of synaptic input at which all motor units are recruited. Lower-threshold motor units (below 10%of the maximal synaptic input) show rapid acceleration upon recruitment and saturation of their discharge rates. Higher-threshold units (red) linearly increase their discharge rates up to the maximal synaptic input. D) Mean discharge rates of all active units at four different force levels (11, 21, 50, 72 and 93 % of the maximum force)compared against experimental data from Connelly et al. [68].
Table 1.
Model parameters for muscle based on architectural parameters of tibialis anterior muscle [144, 145].
Fig 3.
Population characteristics of our default motor unit pool.
A) Histogram of peak tetanic force of 200 motor units (blue: slow-twitch, red: fast-twitch). The distribution follows an exponential distribution where a large portion of motor units produce relatively smaller tetanic force. B) Peak tetanic forces as a function of recruitment thresholds. The relationship assumes the size-principle (e.g. Henneman, 1957): smaller units get recruited earlier than larger units. C) Histogram of contraction time of 200 motor units. The distribution follows the Rayleigh distribution which spans 20 ms to 85 ms, which closely matched the distribution observed in human tibialis anterior muscle by Cutsem et al. (1993). D) The relationship between peak tetanic force and contraction time. Slow-twitch units (in blue) have slower contraction time and smaller peak tetanic force. Within each fiber type, no correlation between peak tetanic force and contraction time was assumed. E) The relationship between contraction time and stimulus interval at the frequency at which half the tetanic force is achieved f0.5. Consistent with previous experimental data [69, 70], those parameters in our default motor unit pool are highly correlated (r = 0.920). F) The relationship between contraction time and twitch-tetanus ratio. Within each fiber type, twitch-tetanus ratio is positively correlated with contraction time (correlation coefficients, r, are 0.638 and 0.704 for slow-twitch and fast-twitch units, respectively).
Fig 4.
Our new model improves predictions of force production at the individual motor unit level.
(Row A) Output of representative motor units, one slower and one faster, from each model to constant synaptic input to their motoneuron at various frequencies. Note the output of the Fuglevand model is force in arbitrary units, whereas that of our model is motor unit activation (0–1) that is then scaled by peak tetanic force to produce force. (Row B) The output-to-frequency relationship of those same motor units. The shaded area represents the range of simulated discharge rates for those motor unit, which for the Fuglevand model does not correspond to the region of the steepest force-frequency relationship. Also, note that our new model includes the length-dependent output-to-frequency relationship described in [66, 138]. Note that the choice of units is not itself related to model validity, but to display the length-dependence of activation in our model, which was described by Brown et al. [138]. (Row C) The degree of fusion as a function of discharge rates. In the Fuglevand model, the increase in fusion is not monotonic and some units do not approach complete fusion. These issues are corrected in our new model. (Row D) The degree of fusion attained as output levels increase. Compared to the experimental observation from [151], the degree of fusion increases too slowly only to rise abruptly at higher outputs in the Fuglevand model, which is corrected in our new model. The dotted identity line is included for reference.
Fig 5.
Both ‘onion-skin’ and ‘reverse onion-skin’ patterns emerge from our model.
A) The relationship of peak discharge rate vs. recruitment threshold shows a significant positive correlation (r = 0.519, p <0.01). This demonstrates that higher-threshold units tend to show higher peak discharge rates (the reverse-onion scheme feature), without us having explicitly built that in. B) The relationship between the discharge rate at 20% maximum input vs. recruitment threshold shows a significant negative correlation (r = -0.504, p <0.01). In contrast to A), higher-threshold units tend to show slower discharge rates at 20% maximum input, demonstrating the onion-skin pattern as the emergent features of rate coding.
Fig 6.
Increased discharge variability causes an increase in force variability through its interaction with the muscle force generating dynamics.
A) Power spectral density of a motor unit spike train. Note that increased discharge variability introduces low-frequency (<5 Hz) power. B) Power spectral density of motor unit force. Increased discharge variability increases low-frequency force fluctuations while attenuating those associated with unfused tetanic contraction. C) Power spectral density of tendon force. The spatial filtering of motor unit forces selectively attenuates higher-frequency force fluctuations associated with unfused tetanic contraction. D) CoV of force with varying degrees of discharge variability. Increases in discharge variability result in increases in the overall amplitude of force variability.
Fig 7.
The Fuglevand model can overestimate the contribution of motor noise to force variability.
Comparisons between the Fuglevand model and our new model are presented for their single unit responses and population responses. Three simulated conditions are presented as follows: Fuglevand model in gray, the new model without a series-elastic element (SEE) in red and the new model with SEE in blue. A) Mean force of representative motor units in each model as a function the synaptic input to the entire population. B) Mean force of a motor unit population as a function of the synaptic input. C) SD of force normalized to the maximal force for representative motor units, plotted as a function of the synaptic input to the entire population. D) SD of force normalized to the maximal force for the entire population, plotted as a function of mean force levels. E) CoV of force for representative motor units as a function of the synaptic input to the entire population. F) CoV of force for the entire population as a function of mean force levels. Note that our model no longer displays signal-dependence of SD of force due to progressive decreases in SD and CoV of force in individual motor units. It is important to note that addition of a series-elastic element further reduces the amplitude of motor noise across the entire force levels.
Fig 8.
Viscoelastic properties of the contractile element damps high-frequency oscillations associated with discharge rate of motor units.
Power spectra of output force at different synaptic input levels (5%, 20%, 40% and 70%) for our new model without a series-elastic element (SSE) in red and with SEE and blue. Note that addition of SEE substantially reduces power at frequencies >5 Hz (shaded areas).
Fig 9.
Motor noise cannot fully account for the experimentally observed amplitude of force variability nor for the amplitude of motor noise used in a previous theoretical model.
The amplitude of motor noise predicted by our new model is compared to the amplitude of force variability recorded from the tibialis anterior muscle in 11 participants reported by Tracy [133]. To allow for a fair comparison between our result and the experimental data by Tracy [133], the 10-sec hold phase of output force was divided into ten 1-sec segments, the duration of data used in Moritz et al. [37] and Tracy [133]. The force signal in each segment was then linearly de-trended using detrend function in MATLAB and standard deviation was calculated from the de-trended data. CoV was calculated by dividing the standard deviation by the mean force of the original force signal before de-trending. Our prediction is also compared to the SD-mean force relationship observed experimentally by Jones et al. [28] and that assumed in a previous theoretical model by Todorov [12]. Note that the predicted motor noise is smaller than the experimentally measured force variability (a black dotted line) for the entire possible range of force levels. Our prediction deviates substantially from the theoretical SD-force relationship of motor noise observed experimentally by Jones et al. [28] (a green line) and that implemented in a previous models by Todorov [12] (a magenta line). Experimental data presented adapted from Fig 4A in Tracy [133].
Fig 10.
Signal-dependent noise is not the by-product of the motor unit force generation mechanism.
We altered several key parameters/features of our model including a series-elastic element (SEE), the range of peak tetanic force (PR), the range of recruitment thresholds (Ur), the number of motor units in a pool (N) and the recruitment scheme (see the figure keys for the detail). To compute SD and CoV for force, we used the last 8-sec of the hold phase. The force signal was divided into two 4-sec segments and each segment was de-trended by a 2nd order polynomial using detrend function in MATLAB with an order of 2 as done in Jones et al. [28]. A) Mean force in Newtons as a function of synaptic input levels. B) SD of force (% of maximal force) as a function of mean force levels. A dashed black line represents the SD-mean force relationship observed experimentally by Jones et al. [28]. A solid black line represents the SD-mean force relationship used in the theoretical model by Todorov [12]. C) CoV for force as a function of mean force levels. Despite the large changes in these key parameters/features of our model, predicted motor noise still does not follow the expected SD-mean force relationship and its amplitude is substantially smaller than that used in the previous theoretical model [12].