Motor-Unit Pool Model of Continuous and Discrete Force Variability

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439 Motor Control, 2011, 15, 439-455

original research

The authors are with the Department of Kinesiology, The Pennsylvania State University, University Park, PA.

Motor-Unit Pool Model of Continuous and Discrete Force Variability

Xiaogang Hu and Karl M. Newell

Department of Kinesiology, The Pennsylvania State University The purpose of this study was to investigate the mechanisms contributing to the different scaling functions between force and force variability in continuous and discrete isometric forces. Muscle forces were simulated with the Fuglevand et al.

(1993) model of motor unit recruitment and rate coding, and a range of recruitment and firing properties were manipulated. The influence of time-to-peak force on the discrete force variability was also examined. The results revealed that the peak firing rate, the synchrony between motoneurons, and the recruitment range contributed to the different variability functions in continuous and discrete forces. The shorter time-to-peak force led to higher variability in the peak force.

The findings show that the model can produce the distinct properties of the force variability scaling functions in continuous and discrete forces. The simulation results provide preliminary insight into the neuromuscular mechanisms of the different force variability functions in continuous and discrete isometric forces.

Keywords: motor control; isometric force; force variability; motor unit

The force produced by effectors in isometric contractions exhibits variability that scales with force level. However, the specific proportional relation between force and force variability tends to vary in different isometric force production tasks. In continuous constant force production, the variability (SD) increases at an accelerating rate (quadratic scaling) with force level (Enoka, Burnett, Graves, Kornatz, & Laidlaw, 1999; Hong, Lee, & Newell, 2007; Slifkin & Newell, 1999), although linear (Jones, Hamilton, & Wolpert, 2002) and square-root scaling (Barry, Pascoe, Jesunathadas, & Enoka, 2007) have also been found in hand muscles. In contrast, for discrete force pulse production, the trial-to-trial peak force or impulse variability (SD) increases at a decelerating rate (square root scaling) with force level (Carlton, Kim, Liu, & Newell, 1993; Carlton & Newell, 1993; Christou &

Carlton, 2001, 2002; Schmidt, Sherwood, Zelaznik, & Leikind, 1985; Sherwood, Schmidt, & Walter, 1988). The mechanisms that produce the contrasting scaling relations in discrete and continuous isometric tasks have not been addressed and this is our focus here.

It is well established that force output is mediated by properties of motor unit recruitment and rate coding. They include: number of units recruited, rate coding of

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440 Hu and Newell

motor units, and synchrony of motor units (Harrison, 1983; Milner-Brown, Stein,

& Yemm, 1973; Moore, Perkel, & Segundo, 1966). The recruitment order of the motor units follows similar patterns in different types of contractions (Burke, 2002;

Desmedt & Godaux, 1978; Thomas, Ross, & Calancie, 1987); however, the range of recruitment tends to vary in different tasks and muscles groups (Cope, Sokoloff, Dacko, Huot, & Feingold, 1997; Desmedt & Godaux, 1977; Kukulka & Clamann, 1981). Therefore, variations in motor unit recruitment range may change the variability scaling functions in the force output. The firing rate of the motoneurons in fast ramp tracking contractions or ballistic contractions is higher than that in steady-state or slow changing contractions (Desmedt & Godaux, 1977; Grimby &

Hannerz, 1977). Higher firing rates of motoneurons also attenuate the fluctuations of motor unit force (Fuglevand, Winter, & Patla, 1993), which effectively reduces the variability in muscle force output. Thus, the distinct firing rates of motor units during the continuous and discrete isometric contractions may contribute to the different variability scaling functions in force output.

The firing rate properties between motor units are correlated due to common synaptic input to the motoneurons (Nordstrom, Fuglevand, & Enoka, 1992; Sears

& Stagg, 1976). Several experimental and simulation studies have shown that the temporal synchronization of motor units increases the amount of variability in continuous force output and EMG (McAuley, Rothwell, & Marsden, 1997; Sem- mler & Nordstrom, 1998; Yao, Fuglevand, & Enoka, 2000). The magnitude of the synchronization is also task dependent (Bremner, Baker, & Stephens, 1991a, 1991b;

Schmied, Ivarsson, & Fetz, 1993). Thus, the force variability scaling functions due to the synchrony between motor units may also be influenced by the type of muscle contractions.

In a motor-unit pool, the number of motor units varies with the muscle group (McComas, 1998). Although the size of motor-unit pool has been shown to correlate with the force variability, the force variability scaling function is invariant with different motor-unit pool size (Hamilton, Jones, & Wolpert, 2004; Jones et al., 2002).

This leads to the hypothesis that the size of motor-unit pool may not contribute to the change of variability scaling functions. During isometric contractions, the variability in interspike intervals (ISI) also contributes to force output variability (Jones et al., 2002; Laidlaw, Bilodeau, & Enoka, 2000; Moritz, Barry, Pascoe, &

Enoka, 2005). However, the variability in ISI (constant SD, constant coefficient of variation (CV), or exponentially decreasing CV with force level) has not been shown to alter the force variability scaling functions (Hamilton et al., 2004; Jones et al., 2002; Moritz et al., 2005), and the specific scaling relation was invariant at different amount of variability in ISI. Thus, the variability in ISI may not contribute to the contrasting variability functions in continuous and discrete contractions.

Fuglevand and colleagues (Fuglevand et al., 1993) have developed a motor-unit pool model of force production, and the model has been updated to include firing synchrony between motoneurons (Taylor, Steege, & Enoka, 2002; Yao et al., 2000).

The model includes the distributional properties of motor unit recruitment, firing rate, and synchronization among motor units. Simulation studies have used this model to examine the functional significance of motor unit synchronization (Yao et al., 2000), contribution of motor unit properties to force fluctuations (Moritz et al., 2005; Taylor, Christou, & Enoka, 2003), sources of isometric force variability (Jones et al., 2002), and variations of force variability at different muscle groups

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Continuous and Discrete Force Variability 441

(Hamilton et al., 2004), including between young and old populations (Barry et al., 2007). However, the force production tasks used in these studies were constant isometric contractions and the variability of discrete force pulse contractions has not been modeled.

Through simulations with the motor-unit pool model (Fuglevand et al., 1993), the purpose of this study was to investigate the mechanisms that contribute to the distinct scaling relations between force and force variability in continuous and discrete contraction tasks. Specifically, we sought to identify the properties of the motor-unit pool that lead to the different scaling functions of force variability in continuous and discrete contractions. The excitatory drive to the motoneuron was modeled to produce the continuous and discrete force output. The recruitment range and firing properties of the motor-unit pool were manipulated. Two properties of motor unit firing were examined in this study: the peak firing rate of motor units and synchronization of motor units. Finally, it has been shown that the variability in discrete force production is also influenced by the time-to-peak of the force impulse (Carlton & Newell, 1993; Freund & Budingen, 1978; Ghez & Vicario, 1978; Schmidt, Zelaznik, Hawkins, Frank, & Quinn, 1979). To further investigate the validity of the motor-unit pool model for discrete force, we also examined the relation between the time-to-peak force and the variability in the simulated peak forces over trials.

Methods

Motor-Unit Pool Model

Our simulations of isometric force variability used a model of motor unit recruitment and rate coding (Fuglevand et al., 1993). The original model outputs comprised force output and surface EMG, whereas we used only the portions of the model for force output. All the values of the variables were set at values that were the same as the original model, except the variables manipulated here. The method of motor unit synchronization was adopted from Taylor et al. (2002). Briefly, the model consisted of a pool of 120 motor units with systematic variations in recruitment threshold, firing rate patterns, synchronization characteristics, and amplitudes and durations of twitch forces. A prescribed excitatory drive (E) was used as the model input to produce the desired force output. In the simulations, E was assumed to have uniform distribution across the motoneurons.

The range of twitch forces was set at 100 fold, with motor unit 1 being the smallest (and thus first recruited) and having a twitch force of 1 arbitrary unit (au), and motor unit 120 being the largest and having a twitch force of 100 au. The contraction time for the motor units ranged from 30 (largest motor unit) to 90 ms (smallest motor unit). Given the force variability scaling function was invariant with different amount of variability in ISI of the motoneurons (Hamilton et al., 2004; Jones et al., 2002), the coefficient of variation (CV) in the ISI was adopted from the original model (20%). The force was simulated at 100 Hz, and the dura- tion for each force level was 11 s for continuous force and 1 s for discrete force.

Continuous Force. The force output comprised 9 force levels in response to 9 steady state levels of E from 10% to 90% of maximum E with 10% increment.

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During the initial 0.2 s, the excitatory drive increased linearly from 0 to the corresponding E value and was then maintained constant at the peak value for the rest of the simulation. Figure 1A illustrates the 9 levels of excitatory drive and only the initial 0.5 s of E are shown to get better illustrations. Figure 1C shows the force output of 11 s that resulted from the excitatory drives in Figure 1A.

Discrete Force. The same 9 excitation levels as in the continuous force were chosen to produce the discrete force pulse. The time-to-peak of E was adjusted such that the time-to-peak of force output was slightly longer at higher peak force (Freund & Budingen, 1978; Ghez & Vicario, 1978; Newell & Carlton, 1985, 1988).

Specifically, the time-to-peak of E for the 10% of maximum E was set at 0.04 s, and it was increased by 0.01 s for each consecutive E level, which yielded a time- to-peak of 0.12 s for the 90% of maximum E. With the specified time constraints, the excitatory drive increased linearly from 0 to the corresponding peak E, and it was then decreased to 0 for the remainder of the simulation. The excitation drive settings resulted in the time-to-peak force output ranging approximately from 0.2 to 0.3 s. There is a nonzero contraction time in the twitch forces of each motor unit; therefore the time-to-peak of the force output comes later than the time-to- peak of the excitation drive. Figure 1B illustrates the 9 excitatory drives (only the initial 0.25 s of E are plotted for illustration purpose) and the corresponding force output of 1 s is shown in Figure 1D.

Range of Recruitment and Peak Firing Rate. Studies have shown that new motor units are still recruited at up to 90% of MVC that leads to a recruitment range around 70 fold (De Luca, LeFever, McCue, & Xenakis, 1982; Kukulka &

Clamann, 1981; Oya, Riek, & Cresswell, 2009), and a recruitment range of 12 fold was also found (Powers & Binder, 1985; Yao et al., 2000). To examine the effect of recruitment range on force variability, the range of recruitment was varied from narrow to wide. The narrow range of recruitment was set at 20 fold compared with a moderate range of 40 fold and a wide range of 70 fold. The recruitment was calculated in excitatory units. For example, when the range of recruitment was set at 40 fold, the first motor unit was recruited at excitation drive of 1 au and the last motor unit was recruited at excitation drive of 40 au. Studies have shown that the Peak Firing Rate (PFR) varied from as low as 15 Hz up to 45 Hz depending on the muscle and the task performed (Hannerz, 1974; Kukulka & Clamann, 1981; Oya et al., 2009; Romaiguere, Vedel, & Pagni, 1993). Three sets of PFR were used in the simulations. The low PFR ranged from 25 Hz for the first motor unit to 15 Hz for the 120th motor unit, the moderate PFR ranged from 35 to 25 Hz, and the high PFR ranged from 45 to 35 Hz. During the simulations, a moderate synchrony level (i.e., 10% of the action potentials of other active motor units were synchronized with 10% of action potential of the reference motor unit) was used (Taylor et al., 2002).

When the range of recruitment was varied, a moderate PFR range (35–25 Hz) was chosen, and when the PFR was varied, the range of recruitment was set at 70 fold.

Motor Unit Synchronization. Three synchronization levels (no synchrony, moderate synchrony, and high synchrony) were selected in the simulations. The moderate synchrony involved a selection of 10% of the action potentials discharged by the reference motor unit and 25% of spikes in the reference motor unit in the high synchrony (Taylor et al., 2002). The range of recruitment was set at 70 fold

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(Enoka & Fuglevand, 2001; Kukulka & Clamann, 1981). The peak firing rate ranged from 25 to 35 Hz in the continuous force. Since firing rate influences the maximum force capacity (Fuglevand et al., 1993), the peak firing rate ranging from 30 to 40 Hz in the discrete force was chosen, which matched the MVC (around 18000 AU) in the continuous force.

Time Constraints on Peak Force Variability. In discrete force production, the peak force variability has been shown to be inversely scaled with the time-to-peak force constraint (Carlton & Newell, 1993; Newell & Carlton, 1988). In the current simulation, four peak force levels (approximately 20%, 40%, 60%, and 80% of Figure 1 — Excitatory drive (E) and force output of the continuous and discrete force. A: E for the continuous force increased linearly to the specified % of maximum levels at 0.2 s and then maintained constant at the specified levels (only the initial 0.5 s of E are shown). B: E for the discrete force first increased linearly from 0 to the corresponding peak value and then decreased to zero for the remaining of the trial (only the initial 0.25 s of E are shown). The Time-to-Peak of E was set from 0.04 s to 0.12 s with 0.01 s interval for the 9 excitation levels. C: One simulation of the force output in response to each excitatory drive shown in A. D: Twenty simulations of the discrete force pulse in response to each excitatory drive shown in B.

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MVC) were chosen with each having 3 time-to-peak forces (approximately 200 ms, 300 ms, and 400 ms). In the discrete force, since the desired peak force depended on the peak amplitude and the time-to-peak of the excitatory drive, and the force output and the excitatory drive did not follow a linear relation in the motor-unit pool model (see Figure 6C in Fuglevand et al., 1993), an inverse calculation was performed to find the required amplitude and time-to-peak of the excitation drive value for the desired peak force output and time-to-peak force values.

Data Analysis

Variability Estimation. When the recruitment and firing properties of the motor units were manipulated, the force generation capacity (i.e., MVC) of the model also changed; therefore the variability was calculated as a function of absolute force instead of the MVC. Because little variation for repeated simulation was observed in the continuous force at each force level, one trial was simulated at each force level, and the last 10 s was used to calculate the standard deviation (SD) of the force. In the discrete force, 200 trials were simulated at each force level. The average of the maximum sampled force and 2 samples prior and after the maximum sample was calculated as the peak force. The SD of the peak force was calculated over the 200 trials. Three functions (linear: SD = a1 ∙ F + a2, quadratic:

SD = a1 ∙ F² + a2 ∙ F + a3, and square-root: SD = a1 ∙ F^0.5 + a2 (SD: SD of force, F: force, a1,a2,a3: coefficients)) were fit to the force versus force variability relations using the “lsqcurvefit” function in Matlab. The norm (square root on the sum of squared values) of the residuals between the fitted functions and the simulated data were calculated. All the force simulations and data analyses were performed using Matlab version 7.7 (The MathWorks, Inc).

Results

Influence of Motor Unit Recruitment and Peak Firing Rates on Force Variability.

Force variability as a function of the three ranges of recruitment threshold is shown in Figure 2A for the continuous force and Figure 2B for the discrete force. For the continuous force task the narrow range of recruitment led to higher variability, especially at the moderate force levels. However, the level of variability at the three levels of range of recruitment tended to converge at the high force levels. The different recruitment ranges also led to distinct scaling functions of force variability in the continuous task, whereas the same qualitative function was evident at all the ranges of recruitment in the discrete task.

The fitted residuals for the continuous force (Table 1) revealed a quadratic scaling at the narrow range of recruitment, a linear scaling at the moderate range of recruitment and a square-root scaling at the wide range of recruitment. The best fittings are plotted in Figure 2A. The fitted residuals for the discrete force (Figure 2B) showed that the variability was higher at the narrow range of recruitment and was more prominent at higher force levels. However, the scaling function (square- root) held for all the recruitment ranges as shown in Table 1.

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The comparisons of the force variability at the three sets of peak firing rate (PFR) are illustrated in Figure 2C for the continuous force and Figure 2D for the discrete force. In the continuous force (Figure 2C), the lower PFR resulted in higher force variability, that was more pronounced at higher force levels. The scaling functions (Table 2) changed from quadratic (low and moderate PFR) to linear (high PFR) with increasing PFR. In the discrete force (Figure 2D), the level of variability diverged at 50% excitation (approximately 9500 arbitrary unit of force); however, the differences were not as pronounced as in the continuous force.

Figure 2 — Variability in forces at different recruitment ranges and peak firing rates (PFR). A: SD of continuous force at three recruitment ranges, narrow range of recruitment (20) shown as open triangles, moderate range of recruitment (40) as open circles, and wide range of recruitment (70) as open squares.

B: SD of discrete peak force at three recruitment ranges, narrow recruitment range (20) shown as filled triangles, moderate recruitment range (40) as filled circles, and wide recruitment range (70) as filled squares. C: SD of continuous force at three PFR levels, low PFR (15–25 Hz) shown as open triangles, moderate PFR (25–35 Hz) as open circles, and high PFR (35–45 Hz) as open squares. D: SD of discrete peak force at three PFR levels, low PFR (15–25 Hz) shown as filled triangles), moderate PFR (25–35 Hz) as filled circles, and high PFR (35–45 Hz) as filled squares. The best fitting functions shown in Table 1 and 2 are plotted.

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446

Table 2The Norm of Residuals of Fitting Functions in SD of Forces at Different Peak Firing Rates (PFR) in Hz PFRContinuousDiscrete LinearQuadraticSquare-rootLinearQuadraticSquare-root Low (15–25)27.827.18*68.209.094.02*35.72 Moderate (25–35)19.4910.19*52.1217.0717.2713.02* High (35–45)11.52*11.5522.9725.1225.6110.40* * denotes the lowest norm of residual over the three fitting functions Table 3The Norm of Residuals of Fitting Functions in SD at Different Synchronization Levels in % of Action Potentials. SynchronizationContinuousDiscrete LinearQuadraticSquare-rootLinearQuadraticSquare-root No (0%)12.067.45*38.2620.6220.697.01* Moderate (10%)18.349.61*50.4020.6920.8810.30* High (25%)19.4712.92*59.1623.3123.318.12* * denotes the lowest norm of residual over the three fitting functions

Table 1The Norm of Residuals of Fitting Functions in SD of Continuous and Discrete Forces at Different Ranges of Recruitment Threshold in Excitatory Unit Threshold rangeContinuousDiscrete LinearQuadraticSquare-rootLinearQuadraticSquare-root Low (20)62.0762.1639.21*48.2948.3030.51* Moderate (40)13.10*13.6741.0122.9423.0117.84* High (70)33.9913.31*68.5320.6620.6910.30* * denotes the lowest norm of residual over the three fitting functions

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The scaling functions (Table 2) changed from quadratic (low PFR) to square-root (moderate and high PFR) with increasing PFR. The results in both continuous and discrete forces also revealed that the force level produced by the same excitatory drive increased with higher PFR.

Influence of Synchronized Firing on Force Variability.

Figure 3 illustrates the relation between force variability and force level in continuous and discrete force production as a function of motor unit synchrony. The synchronized firing had a stronger effect on variability in the continuous as opposed to the discrete force task. The variability in the continuous force showed a positive quadratic increase over force level regardless of the synchronization level, whereas the variability in the discrete force followed a square-root increase over the peak force. A comparison of the fitted residuals of the functions is displayed in Table 3.

The residual results showed that the quadratic function in the continuous force and the square root function in the discrete force had the best fits.

Time Constraints and Variability in Discrete Force.

A series of discrete forces with four peak force levels and three time-to-peak constraints were produced with adjusted excitatory drive. Figure 4 shows that both the SD and CV of force decreased with longer time-to-peak force at all of the four peak force levels. The mean difference of the SD between the 200 ms and 400 ms constraints was 12.62 au ± 6.81 au, and the mean difference of the CV between

Figure 3 — Variability in continuous and discrete forces at different synchronization levels.

Open symbols represent the SD of continuous force and filled symbols represent the discrete force. Triangles represent the SD of force without synchronization (0%) in ISIs, circles represent the SD of force with moderate synchrony (10%), and squares represent the SD of force with high synchrony (25%). The best fitting functions (quadratic functions in the continuous force and square-root functions in the discrete force) are plotted.

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the two time constraints was 0.14% ± 0.07%. Figure 4 also shows that the SD of force increased with respect to the force level regardless of the time constraints, and a reverse relation was found in the CV of force. Force level had a stronger effect than the time constraints on force variability, which is consistent with the experimental finding that the force variability was scaled up as a square-root function of force level and scaled down with a quarter power function of time-to-peak force (Carlton & Newell, 1993).

Figure 4 — Scaling down of variability with time-to-peak in discrete force. A: SD of peak force with three time-to-peak force constraints (approximately 200 ms, 300ms, and 400 ms) at four peak force levels (approximately 20%, 40%, 60%, and 80% MVC). B: CV of peak force with three time-to-peak force constraints at four peak force levels.

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Discussion

The focus of the study was to identify the mechanisms of neuromuscular properties (motor unit recruitment and rate coding) that mediate the different scaling relations between force and force variability in constant continuous and discrete pulse isometric force production. We used simulated forces produced by the motor-unit pool model (Fuglevand et al., 1993) to determine the motor unit properties that relate to the contrasting force variability scaling functions. The general outcome of the simulations was that the range of recruitment, the peak firing rate, and the discharge synchrony between motor units contributed to the different variability scaling functions in continuous and discrete contractions. The results suggest that the motor-unit pool model (Fuglevand et al., 1993) can produce the distinct force variability functions of continuous (Enoka et al., 1999; Hong et al., 2007; Slifkin

& Newell, 1999) and discrete (Carlton & Newell, 1993; Christou & Carlton, 2002;

Schmidt et al., 1985; Sherwood et al., 1988) isometric force production. The simulations also revealed that the model can produce outcomes that are consistent with properties of the force output other than the amount of variability (SD), such as time-to-peak force in the discrete pulse. Thus, the Fuglevand et al. (1993) model generates a much broader set of task relevant force variability findings than that has previously been shown.

The different firing properties of the motor-unit pool contributed differently to the variability in discrete and continuous force production, namely lower peak firing rate of the motoneurons and higher synchronization of the firing among motor units led to greater variability in the continuous than in the discrete force output. The range of recruitment changed the scaling functions in the continuous force production. The narrow range of recruitment also resulted in greater variability in the discrete force than in the continuous force output. These findings are now discussed in detail.

Influence of Recruitment Range on Force Variability

The range of recruitment of motor units differs for different types of contractions and muscles (Desmedt & Godaux, 1977; Kukulka & Clamann, 1981), and it also influenced the scaling functions of force variability as a function of force level (Figure 2A). These results are consistent with previous constant force studies showing that the variability under the narrow range of recruitment of the motor units in the first dorsal interosseus resulted in square-root like function over force levels (Barry et al., 2007; Yao et al., 2000), and the moderate range of recruitment of the motor units in the extensor pollicis longus led to linear variability scaling with the force levels (Jones et al., 2002). The high recruitment range resulted in quadratic variability scaling function (Figure 2A), which supported the experimental findings showing quadratic variability function over force levels in index finger flexors (Hong et al., 2007; Slifkin & Newell, 1999) and elbow flexors (Enoka et al., 1999). Overall, these results suggest that the range of recruitment of the motor-unit pool is one factor that contributes to the different variability scaling functions in continuous force for different muscles.

However, in discrete contractions, the range of recruitment had no effect on the scaling function, although it had significant impact on the level of peak force

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variability especially at the moderate and high force level (Figure 2B). In the narrow range of recruitment setting during discrete ballistic contractions, the increase of force at the low and moderate force level is contributed primarily by the recruitment of additional larger motor units, and the increase of force at the high force level is contributed primarily by increasing the firing rate because most of the motor units in the pool have already been recruited. This arrangement increases the force variability dramatically at the low and moderate force level, because the newly recruited motor units fire at low rate for a long period, as a result the motor unit forces fluctuate with large amplitude. In contrast, in the wide range of recruitment setting, recruitment of additional motor units and increasing firing rate share more or less the same role on force increment over a large range of force levels.

Because the motor unit recruitment is spread out, and as soon as the motor units are recruited, they increase the firing rate to maintain the required force level. This effectively reduces the rate of variability increment as the force level approaches to the maximum.

Influence of Peak Firing Rate on Force Variability

The simulation results showed that the peak firing rate of motoneurons contributed to the enhanced variability in the continuous force. These findings are consistent with previous studies showing that low peak firing rate leads to enhanced force variability in continuous isometric force output (Fuglevand et al., 1993; Westgaard

& De Luca, 2001). Earlier studies have also shown that the firing rate of the motoneurons in ballistic discrete contractions is higher than that in constant contractions (Desmedt & Godaux, 1977; Grimby & Hannerz, 1977). Our results revealed that a high peak firing rate in the discrete contraction resulted in a square-root variability function (Figure 2D) and low peak firing rate in the continuous contraction led to a quadratic variability function (Figure 2C). This is consistent with experimental results showing a square-root variability function in discrete force output (Carlton et al., 1993; Carlton & Newell, 1993; Christou & Carlton, 2001, 2002; Schmidt et al., 1985; Sherwood et al., 1988) and a quadratic variability function in continuous force output (Enoka et al., 1999; Hong et al., 2007; Slifkin & Newell, 1999). The different peak firing rates of the motor units also influenced the scaling function in both types of force production tasks (Figure 2C and 2D), and may mediate the different variability scaling functions in different types of contractions (Christou

& Carlton, 2001) and muscle groups (Enoka et al., 1999).

The finding that the low peak firing rate led to high force output variability was inconsistent with previous observations revealing similar force output variability at different peak firing rates for different age populations (Barry et al., 2007).

In the Barry et al. (2007) study, the peak firing rate increased with recruitment threshold, whereas in our study, an inverse relation was used based on the “onion skin” phenomenon (i.e., later recruited motor units discharge at lower peak firing rate) (De Luca et al., 1982). This departure may also be due to the fact that other properties (which were held constant across peak firing conditions in the current study) such as the range of motor unit forces (Barry et al., 2007). Therefore, the interactive effect of these properties may result in similar variability over a range of peak firing rates.

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Influence of Synchronized Firing on Force Variability

The findings also showed that synchronized firing between motor units increased the amount of force variability, which is consistent with earlier observations that the synchronization between motor units led to high force variability in continuous isometric contractions (Semmler & Nordstrom, 1998; Yao et al., 2000), especially in the elderly population (Semmler, Kornatz, & Enoka, 2003). Although the scaling functions were invariant at different levels of synchronization in both the continuous and the discrete force output, the synchronization had stronger enhancement of force variability at higher force level, especially in the continuous force output.

Therefore, synchronized firing also contributes to the accelerated increase of variability with increments of force level in the continuous force output.

The effects of synchronized firing of motor units and the peak firing rate (discussed earlier) on force variability were stronger in the continuous than in the discrete forces. Since the variability in discrete force was calculated at instantaneous peaks from trial to trial while the variability in continuous force was calculated at successive samples in a single trial, it is likely that the effect of synchronized firing accumulated over time had a stronger influence on the enhancement of force variability. However, empirical evidence from experimental motor unit studies on discrete contractions is needed to test this inference.

Other Factors Contributing to Force Variability

The time-to-peak constraint of the discrete force was also found to relate to the higher variability in the peak force. This is consistent with the experimental studies showing that the force variability is correlated with the time constraints in discrete force production (Carlton & Newell, 1993; Newell & Carlton, 1988). The findings are due to the fact that the maximum peak force decreases with shorter time-to-peak force and results in the prescribed force level being located at a higher percentage of maximum peak force. Since peak force variability is scaled with the percentage of maximum peak force, a shorter time-to-peak force leads to higher force variability.

A previous simulation study (Taylor et al., 2003) has shown that the low frequency oscillations in the excitatory drive contribute to variability in the continuous force output. Since the discrete peak force corresponds to an instantaneous point of the excitatory drive, it is possible that the fluctuations in the excitatory drive also contribute to the different variability functions in these two types of tasks.

Furthermore, during the simulations we manipulated one variable of the motor- unit pool at a time to determine its influence on force variability; however, the interactive effect of multiple properties of motor-unit pool may also contribute to the distinct scaling functions.

In the current simulations, the force variability in continuous and discrete forces, especially the CV in discrete force, was smaller than the values in experimental findings (Carlton & Newell, 1993; Enoka et al., 1999). This may be due to the parameter settings in the model. In the current model, the CV of ISI was set at 20%. Although different ISI variability does not change the force variability scaling function, it does influence the absolute values of force variability. With higher ISI variability, higher variability in the force output is observed (Barry et al., 2007;

Jones et al., 2002). Therefore, a higher value of CV of ISI may lead to a better fit to

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the experimental data. A recent study has also shown that varying CV of ISI with force level results in a better fit to the experimental data (Moritz et al., 2005). A second parameter to be considered on this issue is the minimum firing rate. Lower minimum firing rate leads to higher variability in the force output. In the model, the minimum firing rate was set at 8 Hz. Lower values of minimum firing rate have been observed (e.g., 5 Hz in Westgaard & De Luca, 2001). Varying minimum firing rate with force output has also been found (Moritz et al., 2005). Therefore, a lower or varying minimum firing rate may also result in a better fit to the experimental data.

In summary, the simulations showed that the recruitment and rate coding (the peak firing rate and synchronization) of the motor-unit pool contributed to the different variability scaling functions in the continuous and discrete force output.

The results also revealed that the motor-unit pool model (Fuglevand et al., 1993) can capture the qualitative and quantitative distinct force variability functions in continuous and discrete force production, even though the model did not include feedback processes that are typically available in human experimental studies.

Clearly, a further challenge is to include in the model the different time scales of the sensory receptors that have been shown to influence isometric force variability (Vaillancourt & Russell, 2002).

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