Synaptic control of the shape of the motoneuron pool input-output function

Motoneuron activity is generally considered to reflect the level of excitatory drive. However, the activation of voltage-dependent intrinsic conductances can distort the relation between excitatory drive and the total output of a pool of motoneurons. Using a pool of realistic motoneuron models, we show that pool output can be a highly nonlinear function of synaptic input but linearity can be achieved through adjusting the time course of excitatory and inhibitory synaptic inputs.

Abstract

Although motoneurons have often been considered to be fairly linear transducers of synaptic input, recent evidence suggests that strong persistent inward currents (PICs) in motoneurons allow neuromodulatory and inhibitory synaptic inputs to induce large nonlinearities in the relation between the level of excitatory input and motor output. To try to estimate the possible extent of this nonlinearity, we developed a pool of model motoneurons designed to replicate the characteristics of motoneuron input-output properties measured in medial gastrocnemius motoneurons in the decerebrate cat with voltage-clamp and current-clamp techniques. We drove the model pool with a range of synaptic inputs consisting of various mixtures of excitation, inhibition, and neuromodulation. We then looked at the relation between excitatory drive and total pool output. Our results revealed that the PICs not only enhance gain but also induce a strong nonlinearity in the relation between the average firing rate of the motoneuron pool and the level of excitatory input. The relation between the total simulated force output and input was somewhat more linear because of higher force outputs in later-recruited units. We also found that the nonlinearity can be increased by increasing neuromodulatory input and/or balanced inhibitory input and minimized by a reciprocal, push-pull pattern of inhibition. We consider the possibility that a flexible input-output function may allow motor output to be tuned to match the widely varying demands of the normal motor repertoire.

NEW & NOTEWORTHY Motoneuron activity is generally considered to reflect the level of excitatory drive. However, the activation of voltage-dependent intrinsic conductances can distort the relation between excitatory drive and the total output of a pool of motoneurons. Using a pool of realistic motoneuron models, we show that pool output can be a highly nonlinear function of synaptic input but linearity can be achieved through adjusting the time course of excitatory and inhibitory synaptic inputs.

all motor commands pass through motoneurons and thus are necessarily transformed by motoneuronal input-output functions. A fundamental question is whether these functions remain consistent as motor commands vary to meet the demands of the wide range of motor behaviors. There are considerable data showing that, in contrast to the classical view of motoneurons, their input-output behaviors are highly dependent on differences in the synaptic organization of motor commands (Heckman and Enoka 2012; Powers and Binder 2001). The primary source of these differences is neuromodulatory inputs that act via G protein-coupled receptors, which are known to mediate changes in electrical properties in many types of neurons (Hille 1994). Probably the most potent neuromodulatory actions on motoneurons are mediated by axons that descend from the brain stem and release either serotonin (5-HT) or norepinephrine (NE) (Heckman and Enoka 2012; Hounsgaard et al. 1988; Hultborn et al. 2004; Powers and Binder 2001). The effects of 5-HT and NE on the electrical properties of motoneurons result in a remarkably strong increase in overall excitability. Recruitment thresholds are lowered by depolarization of the resting membrane potential and hyperpolarization of the spike threshold. Rate modulation is transformed by reduction in the duration of the spike afterhyperpolarization (AHP), by decreased input conductance, and by facilitation of persistent inward currents (PICs).

PICs are especially striking in their actions. PICs are primarily generated by voltage-sensitive Na⁺ and Ca²⁺ channels in the dendrites of motoneurons and consequently act as potent amplifiers of synaptic inputs (Bennett et al. 1998; Hultborn et al. 2003; Lee and Heckman 2000). Our studies show that this amplification can be as large as three- to fivefold, depending on the level of monoaminergic input (Lee and Heckman 2000). Thus the neuromodulatory input provided by brain stem 5-HT and NE systems has the potential to increase the overall gain of the motor pool by 300–500%—nearly 10 times as much as that estimated to be achievable by variations in ionotropic inputs (Heckman and Binder 1993). Moreover, PICs, by their very persistence, have an inherent tendency to prolong synaptic inputs. Recent studies, however, also show that PICs are highly sensitive to synaptic inhibition (Bui et al. 2008; Hultborn et al. 2003; Hyngstrom et al. 2007), which can reduce both their amplification and prolongation effects.

PICs induce strong nonlinearities in the input-output functions of motoneurons. Initially, PIC amplification induces a rapid increase in firing rate, but as the PIC becomes strongly active the rate of increase of firing decreases (Bennett et al. 1998; Lee et al. 2003). Moreover, as input decreases the PIC prolongs firing, inducing a tendency for derecruitment to occur at a lower input level than recruitment (e.g., Bennett et al. 1998, 2001; Hounsgaard et al. 1988; Lee and Heckman 1998a). Recordings of human motor units during linearly increasing and decreasing torque patterns clearly demonstrate these same nonlinear behaviors (e.g., De Luca and Contessa 2012; Fuglevand et al. 2015; Gorassini et al. 2002; Mottram et al. 2014), suggesting that PICs play a major role in normal human rate modulation (Heckman and Enoka 2012). However, it has recently been shown that a strong background of inhibition, which is known to reduce PIC amplitude (Kuo et al. 2003), can linearize these effects (Revill and Fuglevand 2017). In addition, since increasing drive will recruit additional motoneurons, it is possible that the net output of the motoneuron pool will vary linearly with the level of excitatory drive, even though individual motoneurons exhibit a nonlinear input-output relation.

The goal of the present study was to develop a pool of model motoneurons that matched as closely as possible the motoneuron properties measured in cat medial gastrocnemius (MG) motoneurons via voltage and current clamp (Bennett et al. 1998; Lee and Heckman 1998a, 1998b, 1999) and then to drive this pool with different patterns of synaptic excitation and inhibition and levels of neuromodulatory drive in order to quantify the relation between excitatory input and pool output. Our guiding hypothesis is that the gain and shape of this pool input-output function are greatly affected both by the level of neuromodulation and by the temporal pattern of inhibition, with the PICs in motoneurons playing a central role in both effects. Our results revealed that the PICs not only enhance gain but also induce a strong nonlinearity in the relation between the average motoneuron firing rate of the pool and the level of excitatory input. The relation between the total simulated force output and input was somewhat more linear because of higher force outputs and less hysteresis (prolongation of synaptic effects) in later-recruited units. We also found that the nonlinearity can be minimized by a reciprocal, push-pull pattern of inhibition. We consider the possibility that a flexible input-output function may allow motor output to be tuned to match the widely varying demands of the normal motor repertoire.

METHODS

Model Development

The goal of the simulations was to develop a pool of model motoneurons whose voltage-clamp and current-clamp behavior approximates the range measured in MG motoneurons in the decerebrate cat (without exogenous monoamines; Lee and Heckman 1999). For most of the simulations we used a pool of 20 motoneurons, since this sped up the simulations and we found that the essential features of pool behavior were adequately captured. In some of the simulations we used a larger number to confirm this. The following properties were approximated: 1) input conductance, 2) threshold (rheobase), 3) frequency-current (F-I) relations, and 4) current-voltage (I-V) relations. In addition, the relation between input conductance and properties 2–4 needs to be matched. In the initial simulations, we used the two-compartment model originally developed by Kim and colleagues that has spike- and AHP-generating conductances in the soma compartment and L-type (Ca_v1.3) calcium conductances in the dendritic compartment (Kim et al. 2009, 2014; Kim and Jones 2011, 2012). The approach was to first specify model parameters controlling the passive behavior of the model (i.e., without voltage- or calcium-activated conductances) to match the behavior of the motoneuron with the lowest input conductance and then that of the highest-conductance motoneuron. Model parameters were then interpolated between these two extremes to generate a pool of models that when endowed with active conductances (see below) exhibited an appropriate range of thresholds. The interpolation was nonlinear, so that there are more motoneurons with relatively low thresholds than there are high-threshold motoneurons (Gustafsson and Pinter 1984a). Next, a similar procedure was used to specify active conductances for the pool. The soma compartment has a transient and persistent Na conductance and a delayed-rectifier K conductance to generate action potentials and a high-threshold Ca conductance and a calcium-activated K conductance to generate AHPs. The dendritic compartment has a low-threshold (L-type) Ca current, and both compartments have a hyperpolarization-activated mixed-cation current.

Although the parameters of the two-compartment models could be tuned to provide a reasonable approximation of MG motoneuron behaviors, they tended to show less hysteresis in their F-I and I-V behavior than that observed experimentally, and the secondary range slopes of their F-I relations were too high. As previously described in relation to single-cable motoneuron models (Elbasiouny 2014), this reflects the fact that there is just a single plateau-generating dendritic compartment, whereas the branching dendrites in real motoneurons have many plateau-generating compartments with a range of activation thresholds. To overcome this problem we developed a model in which the single dendritic compartment was replaced with four dendritic compartments with the same total surface area. The axial resistance in the four compartments was then adjusted to replicate the passive attenuation factors in the two-compartment model. Variations in the plateau activation thresholds were achieved by varying the maximum Ca conductances of the different dendritic compartments.

Determination of Passive Parameters for Two-Compartment Models

The passive behavior of the two-compartment model is set by the values of six parameters: 1) somatic conductance (G_m,S), 2) somatic capacitance (C_m,S), 3) dendritic conductance (G_m,D), 4) dendritic capacitance (C_m,D), 5) coupling conductance between the two compartments (G_C), and 6) the proportion of total membrane area in the somatic compartment (p). Kim and colleagues (Kim et al. 2009; Kim and Jones 2012) showed that these values could be derived analytically based on the following properties of completely reconstructed motoneurons (from Cullheim et al. 1987; Fleshman et al. 1988): 1) input resistance (R_N), 2) membrane time constant (τ_m), 3) voltage attenuation of a DC input from the soma to the dendrites at a specified distance (${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{D}\mathrm{C}}$), 4) voltage attenuation of a DC input from the dendrites to the soma (${\mathrm{V}\mathrm{A}}_{\mathrm{D}\mathrm{S}}^{\mathrm{D}\mathrm{C}}$) (either from a single point or all points at the same path distance), 5) attenuation of an AC input from the soma to the dendrites (${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{A}\mathrm{C}}$), and 6) the proportion of the total surface area from the soma to the specified dendritic distance. Kim et al. (2014) showed that when spiking and AHP conductances were added to the soma compartment and L-type (Ca_v1.3) Ca conductances were added to the dendritic compartment the two-compartment model could closely approximate the both the F-I and I-V behavior of the fully reconstructed motoneuron from which it was derived (their Fig. 2).

For the implementation of the two-compartment model in NEURON (Hines 1989), the soma and dendrite were represented as two connected cylindrical compartments of equal diameter with the axial resistance calculated to achieve a specified coupling conductance between the midpoint of each compartment. The variation in total surface area was chosen to represent the range estimated for MG motoneurons: the smallest motoneuron has a total length of 5,800 μm and a diameter of 22 μm for a surface area of 400,867 μm², and the corresponding values for the largest motoneuron were 7,200 μm, 30 μm, and 678,584 μm². Values for R_N and τ_m were chosen to cover the range reported by Zengel et al. (1985): the smallest motoneuron had an R_N of 3.5 MΩ and a τ_m of 14 ms, and the largest motoneuron had an R_N of 0.6 MΩ and a τ_m of 5 ms. These R_N values are slightly higher than those reported by Zengel et al. (1985) but approximate them more closely when a hyperpolarization-activated mixed-cation conductance is added to the compartments (see below).

Although there is some evidence based on simulation work that dendritic Ca channels in larger motoneurons are located farther away from the soma than in smaller motoneurons (Grande et al. 2007), for simplicity, we chose voltage attenuation factors corresponding to a dendritic path length of 700 μm for all motoneurons, which for a typical distribution of surface area in reconstructed motoneurons (cf. Fig. 3 of Fleshman et al. 1988) represents about half of the surface area. The soma-to-dendrite AC attenuation factors appear to vary similarly with distance in the sample of reconstructed motoneurons analyzed in Kim et al. (2014), so we chose a value (at 700 μm) from a type S motoneuron for the smallest cell (${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{A}\mathrm{C}}$ = 0.19) and from a type FF motoneuron for the largest cell (${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{A}\mathrm{C}}$ = 0.20).

Our choice for the DC voltage attenuation factors was a bit more complicated. Kim et al. (2014) found that the threshold for the activation of dendritic Ca channels and the amount of hysteresis in the F-I and I-V relations were sensitive to these DC voltage attenuation factors. Lower values of ${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{D}\mathrm{C}}$ (more attenuation) increase the threshold and decrease the amount of hysteresis, whereas changes in ${\mathrm{V}\mathrm{A}}_{\mathrm{D}\mathrm{S}}^{\mathrm{D}\mathrm{C}}$ have the opposite effects. These voltage attenuation factors in turn depend upon the distance between the soma and the dendritic Ca channels (more attenuation with increasing distance) and specific membrane resistivity (more attenuation with decreasing specific membrane resistivity). The variations in input conductance among MG motoneurons are thought to reflect differences both in surface area and specific membrane resistivity; high-threshold, high-conductance motoneurons are thought to be larger and to have a lower specific membrane resistivity (e.g., Burke et al. 1982; Gustafsson and Pinter 1984a; Kernell and Zwaagstra 1981). For a given motoneuron, voltage attenuation factors also depend upon the spatial variation in membrane resistivity; since this cannot be precisely determined from the available data (cf. Fleshman et al. 1988), we have chosen attenuation factors for low- and high-threshold motoneurons that allowed us to achieve a reasonable match to the variations in F-I and I-V behavior observed in the MG motoneuron pool: for the smallest motoneuron ${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{D}\mathrm{C}}$ = 0.77 and ${\mathrm{V}\mathrm{A}}_{\mathrm{D}\mathrm{S}}^{\mathrm{D}\mathrm{C}}$ (calculated for inputs to all points at a path distance of 700 μm) = 0.76, and the corresponding values for the largest motoneuron were 0.63 and 0.43. Although analysis by Kim et al. (2009, 2014) suggests a much wider variation in attenuation factors between low- and high-conductance motoneurons, we found that this range was hard to reconcile with the observation that I-V hysteresis is greater in low-conductance motoneurons (Lee and Heckman 1999).

Determination of Active Conductance Parameters for Two-Compartment Models

Somatic conductances.

The soma compartment contained active conductances responsible for spike and AHP generation. Spikes were generated by a transient Na conductance and a delayed-rectifier K conductance. The values of these conductances were set to produce narrow spikes (~1-ms duration) that exhibited positive overshoots throughout most of the firing rate range (cf. Fig. 3A). The densities for Na and K conductances were 10 mS/cm² and 15 mS/cm² for the lowest-threshold motoneuron and 22 mS/cm² and 20 mS/cm² for the highest-threshold motoneuron. It is important to note that in the two-compartment model the surface area of the somatic compartment is much larger than the actual soma size, so that the conductance densities that are chosen to produce action potentials of appropriate size and shape are not meant to reflect the actual densities in the soma, axon hillock, and initial segment of real motoneurons.

Fig. 3.

Example frequency-current behavior of a motoneuron model (same model as in Fig. 1). A: from top to bottom, instantaneous firing rate, the proportion of Ca channel activation in 2 different dendritic compartments, membrane voltage in the soma (black) and 2 different dendritic compartments (blue and red), and injected current (I_inj). B: F-I relation. Diagonal arrows indicate direction of current-clamp command.

The range of current thresholds (rheobase) of motoneurons is largely determined by the variations in R_N; if resting potentials and voltage thresholds for spike initiation are roughly similar, less current is required to reach threshold in high-resistance motoneurons. However, the range of variation in current thresholds is significantly larger than that of R_N, and this is thought to reflect the fact that the subthreshold activation of a voltage-sensitive inward current contributes to depolarization (Gustafsson and Pinter 1984b). The amount of inward current generated by this conductance is thought to be similar in different motoneurons and therefore proportionally greater in low-threshold motoneurons (Gustafsson and Pinter 1984b). We included a persistent sodium current in the soma compartment to provide this function, with a conductance density of 0.026 mS/cm² in the lowest-threshold motoneuron and 0.02 mS/cm² in the highest-threshold motoneuron. This conductance generated peak currents that ranged from 1.7 to 2.9 nA at voltages that were just below threshold. Together with variations in R_N this led to a range of rheobases of from 3.4 to 23.5 nA, which covers almost the entire range reported by Zengel et al. (1985).

The repetitive firing properties of motoneurons are strongly influenced by the characteristics of the medium-duration AHP that follows each spike (Kernell 2006; Powers and Binder 2001). Low-threshold motoneurons have longer and larger AHPs than high-threshold motoneurons. Zengel et al. (1985) found that peak AHP amplitudes were on average 4.9 mV in low-threshold type S motoneurons and 3.0 mV in high-threshold type FF motoneurons. However, these values were obtained in anesthetized cats, and it is thought that the activation of the channels mediating the AHP may be reduced under conditions of higher neuromodulatory drive (Hultborn et al. 2004). Accordingly, we set the conductance densities mediating the AHP to yield peak AHP amplitudes that were 3.5 mV in the lowest-threshold cell and 2.0 mV in the highest-threshold cell. The time constants of AHP decay were set to yield AHP durations of 162 ms in the lowest-threshold cell and 47 ms in the highest-threshold cell, which are slightly shorter than the values reported by Zengel et al. (1985). As repetitive discharge rate increases the mean AHP conductance also increases because of summation, and that rate of increase influences the slope of the F-I relation (Powers 1993). If the peak AHP conductance following a single spike is nearly maximal, then this increase during repetitive discharge is limited, leading to a very steep F-I relation. The AHP results from calcium influx mediated by a high-threshold Ca channel activated by the spike that in turn activates a Ca-activated K channel (Viana et al. 1993). We set the conductance density of the high-threshold Ca channel such that the activation of the K channel reached ~20% of the maximal activation at the AHP peak, which led to primary range F-I slopes that were similar to those observed experimentally (see below).

Dendritic conductances.

The PIC generated in motoneuron dendrites is thought to be largely due to the activation of a low-threshold Ca conductance (Heckman and Enoka 2012; Kernell 2006; Powers and Binder 2001). The net dendritic PIC is also thought to reflect the contribution of persistent Na currents (Lee and Heckman 2001) as well as outward currents generated by Ca-activated K channels (Li and Bennett 2007). The slow decline in PIC seen particularly in high-threshold motoneurons may result either from increasing activation of an outward K current or from slow inactivation of a persistent Na current (Lee and Heckman 1999). The calcium channels themselves may show both voltage- and calcium-dependent inactivation (Jones 2003). For simplicity, we decided to represent the net effect of these processes by just including a Ca channel that exhibited slow, voltage-dependent inactivation that was more prominent in high-threshold cells.

Hyperpolarization-activated mixed-cation conductance.

Hyperpolarization-activated, cyclic nucleotide-gated (HCN) channels are thought to be distributed on both the soma and dendrites of motoneurons (Manuel et al. 2007; Zhao et al. 2010). These channels mediate a mixed-cation current that is partly activated at the resting potential and acts to oppose both depolarization and hyperpolarization around the resting potential, thus increasing effective input conductance. An HCN conductance of equal density was included in both compartments. The contribution of this conductance is thought to be less in low-threshold motoneurons; accordingly, HCN conductance density was set to 0.03 mS/cm² in the lowest-threshold motoneuron and 0.23 mS/cm² in the highest-threshold motoneuron. Both the persistent Na conductance and the HCN conductance affect R_N; R_N values ranged from to 0.4 to 2.9 MΩ when these conductances were included (compared with passive values of 0.6–3.5 MΩ).

Conversion from Models with One Dendritic Compartment to Those with Four Dendritic Compartments

As mentioned above, the presence of one dendritic plateau-generating site in the two-compartment models made it difficult to precisely reproduce certain aspects of the F-I and I-V behavior seen in full multicompartmental models and real motoneurons (cf. Elbasiouny 2014). To partially rectify these shortcomings we replaced the single dendritic compartment with four separate dendritic compartments, each at the same electrotonic distance from the soma but with one-fourth of the original surface area, so that the total surface area equaled that of the original single compartment. Two dendritic compartments were attached to one end of the soma cylinder and two to the opposite end. If axial resistance of the soma is significant, this means that the dendritic compartments at one end of the soma are more strongly connected to each other than to the compartments at the opposite end. To achieve equal coupling between the compartments we made the axial resistance of the soma very low (0.001 Ω-cm) and then adjusted the axial resistance of the dendritic compartments so that values of ${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{D}}^{\mathrm{D}\mathrm{C}}$ and ${\mathrm{V}\mathrm{A}}_{\mathrm{D}\mathrm{S}}^{\mathrm{D}\mathrm{C}}$ matched those of the original two-compartment models. If the density of dendritic Ca channels was equal to that of the two-compartment model and the same in all four dendritic compartments, the I-V and F-I behaviors of the five-compartment models were indistinguishable from those of the two-compartment models (data not shown). However, when Ca channel density varied across the four dendritic compartments, model behavior became more realistic, since staggered activation and deactivation of plateau in the four dendritic compartments could produce more I-V hysteresis (Fig. 1) and a shallower secondary range slope in the F-I relation (Fig. 3). The range of Ca channel densities was tuned by hand to produce reasonable I-V hysteresis and F-I slopes, without large “steps” in the input-output functions resulting from nonoverlapping activations and deactivations of plateau depolarizations in the four compartments. This was achieved by making the Ca channel density in the lowest-density compartment 70–80% of that of the highest-density compartment.

Fig. 1.

Example of the current-voltage behavior of a motoneuron model. A: from top to bottom, voltage-clamp current (estimated leak current indicated by the dotted line), the proportion of Ca channel activation in 2 different dendritic compartments (m_Ca), and membrane voltage in the soma (black) and 2 different dendritic compartments (blue and red). Arrow next to red trace indicates a slight sag in the plateau depolarization reflecting Ca channel inactivation. B: I-V relation. Diagonal arrows indicate direction of voltage-clamp command.

Simulation Protocols

We studied three types of model behavior: 1) the voltage-clamp current evoked by a slowly increasing then decreasing somatic voltage-clamp command, 2) the firing rate elicited in response to a slowly increasing then decreasing somatic injected current command, and 3) the firing rate elicited by a slowly increasing then decreasing noisy conductance command applied to the dendritic compartments, combined with either a steady level of inhibition or inhibition that varied in proportion to the amount of excitation (“balanced inhibition”) or inversely to the level of excitation (“push-pull inhibition”). The voltage-clamp current was measured during a slow (7 mV/s) somatic voltage clamp that varied from −70 to −28 mV and back over the course of 12 s. Somatic conductances were set to zero, so that the clamp current only reflected the contribution of dendritic Ca channels. F-I relations were obtained by applying an injected current command that increased and then decreased at a rate of 6 nA/s over the course of 12 s. For higher-threshold motoneurons, the duration of the ramp was increased up to 19 s in order to evoke secondary range firing (see results). The starting current varied according to motoneuron threshold, from −15 nA for the lowest-threshold motoneuron to 10 nA for the highest-threshold motoneuron.

The noisy conductance inputs were based on those described in Destexhe et al. (2001) and used in our previous simulation studies (Powers et al. 2012; Powers and Heckman 2015). Excitatory and inhibitory conductance noise were generated by two independent random Gaussian noise processes filtered by separate time constants for excitatory and inhibitory inputs. In our previous studies, those time constants were chosen to reflect the time course of individual excitatory and inhibitory inputs. This is appropriate if individual inputs fluctuate independently; however, there is evidence for significant correlation among inputs (e.g., Farina et al. 2014) that includes lower-frequency oscillations. To represent these effects we chose a filtering time constant of 20 ms for the synaptic conductances and set the standard deviation of the filtered noise at a value that produced a physiological level of discharge variability.

NEURON files that describe model parameters and produce model responses to current-clamp, voltage-clamp, and conductance commands will be made available at http://senselab.med.yale.edu/modeldb.

Data Analysis

We characterized the I-V behavior of the models based on a plot of voltage-clamp current vs. the voltage command. Figure 1B shows the I-V relation for a motoneuron model at the median of the recruitment range; the I-V relation depends upon the direction of change of the voltage-clamp command, indicated by diagonal arrows. The PIC can be characterized by a number of features of the I-V relation (Lee and Heckman 1998b, 1999). We focus on the following four: 1) initial PIC amplitude (PIC_i), the difference between the peak inward current on the ascending limb of the I-V relation and the estimated leak current (Fig. 1B, dashed line) at the same voltage, 2) the voltage onset (V_on), which is the voltage at the first zero slope point on the ascending portion of the I-V relation, 3) voltage offset (V_off), the voltage at the last zero slope point on the descending limb of the I-V relation, and 4) voltage hysteresis (V_hist), the difference between V_on and V_off.

We characterized the F-I behavior of the models by plotting instantaneous discharge rate as a function of the level of injected current (F-I relation). Figure 3B shows the F-I relation obtained for a motoneuron at the median of the recruitment range. As in the case of the I-V relation the F-I relation differs depending on the direction of change of the input. The arrows in the lower left portion of Fig. 3B show the direction of current change. On the ascending limb, frequency initially increases linearly with increasing injected current (primary range), followed by a steeper linear increase (secondary range). The slopes of these two linear ranges were estimated by computing the best bilinear fit over relevant portion of the F-I relation. The firing rate over much of the descending limb of the F-I relation was often greater than that on the ascending limb, reflecting the contribution of a dendritic plateau of depolarization; we used the terminology of Bennett et al. (1998) and calculated the amplitude of this firing rate plateau (FRP) as the difference in firing rate between the descending and ascending F-I relations at the current at which the secondary range begins (double arrow in Fig. 3B).

To get an estimate of the average output of the pool of model motoneurons, each model spike train was convolved with a 1-s Hanning filter, resulting in a smoothed version of firing rate (Fig. 5B). The average pool output was simply the average of these smoothed waveforms (Fig. 5C). To estimate the relative force output of the pool, we used the Fuglevand et al. (1993) model and set the maximum twitch contraction time (time to peak tension) to 70 ms, resulting in a range of contraction times from 70 to 24 ms, which nearly covers the range reported by Zengel et al. (1985). To get a more accurate force output we increased the number of units to 120. When this pool was driven with a very large command (6× the standard level used) and dendritic calcium channel density was increased by 60%, the last-recruited unit reached a firing rate of 55.5 imp/s, which produced >95% of its peak force. We normalized the other simulated force outputs to the peak pool force output for this “maximum” contraction.

Fig. 5.

Responses of the models to noisy excitatory conductance command applied to the dendritic compartments. A: from top to bottom, raw (black) and smoothed (red) instantaneous firing rate, Ca channel activation in 2 of the dendritic compartments, membrane voltage of the somatic (black) and 2 of the dendritic compartments (blue and red), and conductance command (black; mean level indicated by red trace). B: time course of smoothed firing rates in 10 of the models compared with that of the mean conductance level (G_ex, black dashed lines). C: time course of average smoothed firing rate compared with that of the conductance command.

RESULTS

I-V Behavior

The voltage-dependent activation of dendritic Ca channels produces inward current that can be measured by voltage-clamping the soma at different membrane potentials and measuring the resultant voltage-clamp current. Figure 1A, bottom traces, show the somatic voltage (black) as well as the voltage of the dendritic compartments with the lowest (blue) and highest (red) density of Ca channels. The activation of Ca channels (Fig. 1A, middle traces) leads to plateaus of depolarization in all of the dendritic compartments that in turn produce inward current (Fig. 1A, top solid trace). The dendritic plateaus turn off at a more hyperpolarized somatic voltage than that associated with plateau initiation. Figure 1B shows the I-V relation and four different measures of I-V behavior (see methods).

The I-V behavior of MG motoneurons varies according to motoneuron input conductance (Lee and Heckman 1999); higher-conductance motoneurons have somewhat larger PIC amplitudes (PIC_i), more depolarized voltage values for PIC onset and offset, and less hysteresis than low-conductance motoneurons. The values of Ca channel activation threshold, density, and degree of inactivation were chosen to match the experimental data as closely as possible. Figure 2 shows four characteristics of I-V behavior vs. input conductance for our pool of 20 model motoneurons, compared with the best linear fits to the experimental data in the presence and absence of methoxamine (an α₁-adrenergic receptor agonist that enhances PICs; Lee and Heckman 1999). Figure 2A shows that the values for initial PIC amplitude in our models are between the mean relations for the experimental data with and without methoxamine. Although our goal was to match the relation without methoxamine, we chose somewhat higher values since in Lee and Heckman (1999) data were collected with the nerve to the antagonist intact, which would lead to somewhat lower PIC amplitudes (due to the effect of tonic inhibition on PIC; see Hyngstrom et al. 2007), whereas the F-I behavior reported by Bennett et al. (1998) was obtained with antagonist nerves cut. Figure 2, B and C, show that the voltage onset and offset values for the models match linear fits to the experimental data for low-input conductance motoneurons but are less depolarized than predicted by the linear fits for high-conductance motoneurons. Nonetheless, the model values are well within the experimental range. Figure 2D shows that the voltage hysteresis values for the models are quite close to best linear fits to the experimental data. These results illustrate the flexibility of the reduced modeling approach and its ability to represent a wide range of experimental behaviors; in contrast, simulations using fully reconstructed motoneurons are typically restricted to matching the behavior of a small sample of motoneurons (e.g., Elbasiouny 2014; Grande et al. 2007).

Fig. 2.

I-V characteristics as a function of model input conductance. A: initial PIC amplitude. B: voltage onset. C: voltage offset. D: voltage hysteresis. Lines in each panel represent the best linear fit to experimental voltage-clamp values as a function of input conductance in the presence (thick lines) and absence (thin lines) of methoxamine (Lee and Heckman 1999). Filled circles in each panel are the values obtained from the motoneuron models.

F-I Behavior

The voltage-dependent activation of Ca channels and the resulting dendritic depolarization that produces the PIC measured under voltage-clamp conditions produce firing rate acceleration under current-clamp conditions. Figure 3 shows the F-I behavior of the same model whose I-V response is shown in Fig. 1. Figure 3A shows the instantaneous firing rate of the model (top trace) elicited by an injected current input (bottom trace). The traces in Fig. 3A, third panel, show the membrane voltage of the soma compartment (black) as well as that of the dendritic compartments with the lowest (blue) and highest (red) Ca channel density. In the second half of the ascending limb of the current command dendritic voltage reaches the threshold for opening Ca channels (Fig. 3A, second panel), leading to plateau depolarizations in the dendritic compartments. The period of firing rate acceleration is indicated by the dashed lines in Fig. 3A and starts when the dendritic compartment with highest Ca channel density starts to depolarize rapidly and ends when the compartment with lowest Ca channel density reaches a relatively steady level of depolarization. Termination of the dendritic plateau potential on the descending limb of the command leads to the termination of repetitive firing. The F-I relation is plotted in Fig. 3B, which also shows the demarcation of primary and secondary ranges of firing as well as the “plateau” of firing rate (FRP), indicating the amount by which firing rate is boosted by the dendritic plateaus.

The F-I plot shown in Fig. 3B was obtained from a model in the middle of the recruitment range and was typical of models with lower thresholds: these showed both a primary and a secondary range, firing rates that were consistently higher on the descending limb of the F-I relation, and firing rate offsets at lower current levels than the firing rate onset. Higher-threshold models could show a variety of F-I behavior, as also reported experimentally (Bennett et al. 1998). Figure 4 illustrates how this variation relates to motoneuron threshold; the F-I relation on the left is typical of low-threshold motoneurons, whereas the two relations on the right show the variation seen in higher-threshold models. Somewhat higher-threshold motoneurons could show clear secondary ranges but exhibited firing rates on the descending limb of the F-I relation that were greater over just a portion of the current range. The highest-threshold motoneuron just showed a primary range. (In the higher-threshold motoneurons we extended the range of current injection to ensure that we did not miss secondary-range firing.) Although some activation of dendritic Ca channels did occur in the highest-threshold model, and did contribute to discharge (when dendritic Ca channels were removed the slope of the F-I relation was slightly decreased), its contribution was insufficient to produce secondary range firing. Overall, models exhibited primary range slopes of 1.9 ± 0.2 imp·s⁻¹·nA⁻¹ compared with the Bennett et al. (1998) values of 1.7 ± 0.7 imp·s⁻¹·nA⁻¹. For those motoneurons that exhibited a secondary range, the slope of this range was somewhat higher than those reported in Bennett et al. (1998): 12.2 ± 2.8 vs. 7.5 ± 0.7 imp·s⁻¹·nA⁻¹. This difference probably reflected the fact that there were only four plateau-generating regions in the models. FRPs were 23.6 ± 3.9 imp/s compared with 20.2 ± 9.6 imp/s reported by Bennett et al. 1998. (Since prolonged discharge in the secondary range led to Ca channel inactivation in higher-threshold motoneurons, the amplitude of the FRP depended upon the duration of secondary range firing—this was presumably true of the experimental results as well).

Fig. 4.

Different types of F-I relations. From left to right, F-I relations for motoneurons that are 5th, 15th, and 20th (of 20 models) in recruitment order. F-I relations for lower-threshold models show clear secondary ranges, unlike the highest-threshold motoneuron, which only exhibits primary-range firing.

Motoneuron Responses to Conductance Commands

Excitatory conductance inputs applied to the dendritic compartment were particularly effective at activating dendritic calcium channels, in keeping with previous experimental (Bennett et al. 1998; Lee et al. 2003) and simulation (Bui et al. 2006; Elbasiouny 2014; ElBasiouny et al. 2006; Powers et al. 2012) results. Figure 5A shows the response of the medium-threshold motoneuron model to a noisy excitatory conductance command (bottom panel); rapid calcium channel activation (second panel) and rapid dendritic depolarization (third panel) begin right at the onset of repetitive discharge, leading to a rapid increase in firing rate (top panel). (The red trace in the top panel of Fig. 5A shows the firing rate smoothed by a 1-s Hanning filter.) Further increases in excitatory drive produce little change in calcium channel activation, dendritic depolarization, or firing rate. This same pattern of rapid firing rate acceleration followed by saturation was seen in all models in the pool as illustrated in Fig. 5B, which shows the smoothed firing rate responses of 10 of the 20 motoneurons in the pool; the time course of the average excitatory conductance is superimposed for comparison (thick black dashed trace) to demonstrate the extent to which the firing rate profiles represent a very distorted picture of the time course of the excitatory command. Similar firing rate profiles have been reported in a number of studies of human motor unit discharge patterns during slowing increasing and decreasing isometric contractions (e.g., De Luca and Contessa 2012; Fuglevand et al. 2015; Heckman et al. 2008b; Mottram et al. 2014). Thus our model reproduces the pattern of activation of motor units that is characteristic of the pattern in human subjects, including orderly recruitment and nonlinear rate modulation.

Comparison of Single-Cell to System Motor Outputs

We hypothesized (see introduction) that the system output of the motor pool might accurately and linearly track the triangular input command. This hypothesis was based on the possibility that continued recruitment of higher-threshold units would compensate for the rate saturation seen in lower-threshold units, so that the system output signal would have a time course similar to that of the excitatory input. Figure 5C shows that this is clearly not the case; the time course of the smoothed firing rate averaged across all units (thin solid line) is somewhat more linear than the firing patterns of individual units (Fig. 5B) but still much different from that of the excitatory command (thick dashed trace).

We also looked at the response of the models, both individually and as a pool, to the same excitatory input combined with an inhibitory conductance that varied in proportion to the excitatory input (balanced inhibition) or varied inversely to excitatory input (push-pull inhibition). Figure 6A shows the time course of the three types of inhibition used; balanced, push-pull, and constant along with the time course of the excitatory input. In the simulations presented in Fig. 5, we included a small amount of constant inhibition to ensure the derecruitment of the lowest threshold motoneurons. Figure 6, B and C, show examples of model firing rates for push-pull and balanced inhibition, respectively. As in our previous study (Powers et al. 2012), in individual motoneurons push-pull inhibition leads to firing rate profiles that rise and fall more smoothly with changes in excitatory input, whereas balanced inhibition leads to more pronounced firing rate saturation. These profound changes in firing patterns clearly persist in the behavior of the pools as a system, as Fig. 6D shows that the same features are present in the averaged firing rate profiles. Overall these results show that the strong nonlinearities in input-output behaviors of individual motoneurons induce a highly nonlinear system behavior.

Fig. 6.

Responses of the models to noisy excitatory conductance command combined with different forms of inhibition. A: mean excitatory command (black dashed line) and mean inhibitory commands for constant (black), push-pull (blue), and balanced (red) inhibition. B: time course of smoothed firing rates in 10 of the models for push-pull inhibition. C: same as B for balanced inhibition. D: time course of average smoothed firing rate for constant (black), push-pull (blue), and balanced (red) inhibition compared with that of the conductance command.

Effects of Changing Neuromodulatory Drive

We simulated changes in neuromodulatory drive by increasing or decreasing the density of dendritic calcium channels. Figure 7 shows the effects of increasing and decreasing calcium channel density on the average firing rate (Fig. 7, top) and gain (Fig. 7, bottom) of the whole motoneuron pool for the three patterns of inhibition. Increases in density produce higher firing rates, but increasing rate saturation, particularly for balanced inhibition. Increases in density also produce more hysteresis, leading to discharge that outlasts the excitatory command, particularly for constant and balanced inhibition. Note the marked changes in excitability of the pool input-output function: the peak of the averaged firing rate for the pool increases by ~8 imp/s from across the −40% to +40% change in Ca channel density for each of the organizations. Note, however, that peak average firing rates for the push-pull are slightly higher than for constant inhibition (35.1 vs. 34.5 imps/s) and much higher than for the balanced inhibition (~35.1 vs 25.8 imp/s). Figure 7, bottom, shows the slopes of the average firing rate functions, i.e., their gains. Gain peaks at low levels of input, particularly for high Ca channel densities. This reflects the activation of the PIC. Since this activation is smoother for the push-pull condition, its gain is more modest than for the other two conditions, but in all cases peak gain increased markedly with increased neuromodulatory level (~5-fold for constant and balanced and ~3-fold for push-pull).

Fig. 7.

Effects of changing neuromodulatory drive on the average firing rate response (top) and firing rate gain (bottom). Results are shown for excitation (with a small amount of constant inhibition), excitation with push-pull inhibition, and excitation with balanced inhibition. Thick line in each panel represents responses with the standard density of dendritic calcium channels, whereas responses above and below these indicate the effects of increasing (red) or decreasing density (blue) by 20% (thin lines) and 40% (thick lines) of the standard value.

The nonlinear nature of the input-output functions can be illustrated more clearly by normalizing both the input and the outputs by their maximum values and plotting these normalized input-output relations for different conditions. Figure 8, A and B, show input-output relations for the excitatory command with different levels of neuromodulation for the ascending (Fig. 8A) and descending (Fig. 8B) portions of the command. The degree of nonlinearity clearly increases with increasing calcium channel density for the ascending portion of the command. For the descending portion, the output decreases in a fairly linear fashion for the highest channel densities, but with a considerable offset reflecting the increased discharge hysteresis. Figure 8, C and D, compare the normalized input-output relations at the standard calcium channel density for the three forms of inhibition for ascending (Fig. 8C) and descending (Fig. 8D) input. Push-pull inhibition clearly leads to a more linear input-output relation than either constant or balanced inhibition, particularly for decreasing excitation.

Fig. 8.

Normalized input-output relations for average pool firing rate vs. the level of excitatory input for the ascending (A and C) and descending (B and D) portions of the excitatory command. A and B: effects of different levels of neuromodulation on input-output relations for an excitatory command (with a small constant amount of inhibition). C and D: input-output relations for the standard level of neuromodulation and different forms of inhibition: small, constant (black), balanced (red), and push-pull (blue). The dashed line in each panel is the line of identity.

Simulated Pool Force Output

We used the force-generating portion of the Fuglevand et al. (1993) model to calculate the total force produced by different-sized commands (expressed as % of the maximum force, see methods). The average firing rate puts equal weight on the contribution of each motoneuron, whereas the force output of later-recruited units is much greater than that of earlier-recruited units (Fuglevand et al. 1993). The greater force contribution of the later-recruited units together with the fact that they exhibited less saturation and hysteresis than earlier-recruited units led to a somewhat more linear force output. Figure 9 shows the normalized average firing rate and force output for the ascending (Fig. 9A) and descending (Fig. 9B) portions of the contraction for the same level of excitatory input conductance used in the previous simulations, which produced a total force output of ~58% of the maximum level. The input-output relation is somewhat more linear for the force output than for the average firing rate, particularly on the descending phase of the contraction. Nonetheless, the transformation from synaptic conductance input to force output remains nonlinear during PIC activation on the rising phase.

Fig. 9.

Comparison of normalized input-output relations for average firing rate (black) and simulated force output (gray) for the ascending (A) and descending (B) portions of an excitatory command. Thick diagonal line is line of identity.

Deriving a Motor Command to Produce a Linear Motoneuron Pool Output

The nonlinear input-output relations shown in Figs. 8 and 9 suggest that producing a linear rise and fall in average pool output (or total force output) requires an input command with a nonlinear rise and fall. Figure 10A shows how this command can be derived for the case of excitation with a small amount of constant inhibition and the standard level of neuromodulation. In comparison to the linearly rising and following excitatory command (gray), the force response (black) initially lags the command and then rises more rapidly than the command. On subsequent simulations we modified the command based on the force “error” until the force rose and fell in a fairly linear fashion (red). This linear output was achieved by first increasing drive rapidly to advance force development, then reducing the rate of increase of drive as the input is being boosted by PICs, and finally increasing drive steeply to compensate for rate saturation. Figure 10B shows a sample of motor unit firing in response to this nonlinear command. The initial rapid increase in the command produces rapid recruitment of a number of low-threshold motor units, and the rapid increase in the command toward the end of the ascending phase produces a slight upward curvature in most of the firing rate profiles, but in other respects the firing rate profiles are similar to those produced by a linearly rising command (see Fig. 5B). These results show that moderate changes in motor command can produce a nicely linear output if this linearity is required by the motor task.

Fig. 10.

Nonlinear command can produce a linear total force output. A: linear and nonlinear conductance commands and the resultant force outputs. B: sample of motor unit firing rates produced by the nonlinear conductance command.

DISCUSSION

In this study, we employed biologically realistic computer simulations to explore the variations in motoneuron input-output functions induced by different levels of neuromodulatory input and different patterns of inhibitory input. These variations proved to be remarkably strong at the level of the single neuron. Summing the effects of recruitment and rate modulation together produced a system input-output function that remained highly sensitive to both neuromodulation and inhibition. Overall, a push-pull pattern of inhibition generated the most linearity and the least sensitivity to changes in neuromodulation. These results imply that the input-output function of the motor pool has a wide range of forms and, moreover, that these different forms are controllable by descending motor commands.

Limitations

As in all modeling studies, the accuracy of the recreation of the biology is imperfect. Fortunately in the case of spinal motoneurons, decades of study have provided a wealth of data to constrain model behavior. Of particular importance for the present effort are the systematic current- and voltage-clamp studies of cat motoneurons in vivo (e.g., Bennett et al. 1998; Lee and Heckman 1998a, 1998b, 1999). As detailed in methods and results, our model replicates these data very well, including reasonably accurate representation of the systematic differences in low-, medium-, and high-threshold motoneurons. There remain uncertainties about the distribution of voltage-sensitive ion channels along the extensive dendritic trees of motoneurons, and our models should not be taken as accurate estimations of these distributions. Rather, we sought to accurately capture the net input-output behavior of motoneurons, and in this respect our model performs reasonably well. Several components of overall pool behavior, however, were not systematically investigated in the present work. Previous studies (Heckman and Binder 1991, 1993) have demonstrated that the distribution of excitatory ionotropic synaptic input among motoneurons with different thresholds has a modest effect on overall gain and as well as on rate modulation (Heckman and Binder 1993). Effects on recruitment order can be substantial (Heckman and Binder 1993). Differences in the relative spacing of the intrinsic thresholds of motoneurons also impact these behaviors (Heckman and Binder 1991). Further work will be required to fully quantify the interactions of these variations with neuromodulation and inhibition.

Comparison to Previous Studies of Input-Output Behavior of Motor Pool

One intriguing aspect of the present simulation results is that a strong nonlinearity occurs in the output of the motor pool at relatively low input levels (see Figs. 5–10). This inflection, from a steep initial slope to a lower slope, occurs because of the activation of PICs in the motoneurons. In previous simulation studies of the pool, saturation does not occur until high input levels. In those simulations, it was primarily due to the attainment of maximal force outputs of the muscle fibers, especially in high-threshold, high-force motor units (Fuglevand et al. 1993; Heckman and Binder 1991). These earlier studies lacked accurate representation of the effects of neuromodulation on PICs. A steep slope at low input levels due to PIC activation imparts a high gain to the pool input-output function, but this rapid increase in gain at low levels has not been seen in studies of reflex gain in human subjects. Typically these studies show a gradual increase in reflex gain spanning at least the initial 50% of maximum activation (e.g., Burne et al. 2005; Cathers et al. 2004). One reason for this difference may be that studies of reflex gains use transient perturbations while background input to the motor pool is held constant, conditions that are very different from the slow rate of rise conditions for our simulations. We have not evaluated the validity of our simulations for transients and so have not investigated rate of rise issues. Instead we have targeted slowly rising inputs to match the conditions of a number of recent (e.g., De Luca and Contessa 2012; Fuglevand et al. 2015; Gorassini et al. 2002; Mottram et al. 2014; Stephenson and Maluf 2011; Wilson et al. 2015) and classic (De Luca et al. 1982) studies of rate modulation in humans subjects (see next section). A second reason is that neuromodulatory levels may change with effort, thus expanding the range of the increase in gain with effort. This issue is largely unexplored, although in the studies of Jacobs and Fornal and colleagues using chronic recording techniques in cats, the raphespinal neurons that are the source of 5-HT in the spinal cord exhibit clear increases in firing rate with speed of locomotion (Jacobs and Fornal 1997).

Implications

As noted in results, the sharp saturation in firing rate simulated here is a feature of firing patterns of motor units in a wide range of studies (e.g., De Luca and Contessa 2012; Fuglevand et al. 2015; Heckman et al. 2008b; Hu et al. 2014a; Mottram et al. 2014). It is important to realize, however, that this degree of saturation [or rate limiting as it is sometimes called (Heckman and Enoka 2012)] may well vary from muscle to muscle. For example, in the study of De Luca et al. (1982), saturation appeared to be much stronger in the proximal deltoid than the distal first dorsal interosseus in the hand. Nonetheless, our simulations clearly imply that in muscles where saturation is strong the input-output relation of the motor pool is highly nonlinear. We show in Fig. 10 that central motor systems could utilize a nonlinearity of the opposite form to compensate for this pool behavior. In this case, low-threshold units still showed an initial rapid rise in firing rate but less subsequent rate saturation. The presence of strong saturation in human motor unit firing patterns implies that this compensation may not always occur.

Another feature of motor unit firing patterns that is reproduced by our simulations is the onion-skin patterns of rate modulation noted early on by De Luca and colleagues (De Luca et al. 1982) and also reported in a number of other studies (see references in De Luca and Contessa 2015; Hu et al. 2014b). The onion-skin pattern refers to the appearance of superimposed firing rate profiles of simultaneously active motor units where the firing rates of later-recruited units are consistently lower than those of early-recruited units. In our simulations, this pattern emerges primarily from the differences in current threshold for repetitive firing. Essentially, higher-threshold motoneurons fire only in response to the input once it reaches a larger amplitude and thus have less suprathreshold current to drive rate modulation. Because these threshold differences are the foundation of Henneman’s size principle (Heckman and Enoka 2012; Henneman and Mendell 1981), onion skinning is tightly coupled to orderly recruitment. The higher voltage threshold of PICs in higher-threshold motoneurons (see Fig. 4) may also contribute to onion skinning. The advantage of this scheme is that later-recruited, fatigable motor units are driven at lower firing rates, thus delaying the onset of fatigue. At the highest levels of force output, rates of high-threshold units likely exceed those of low-threshold units (Oya et al. 2009; Moritz et al. 2005). The mechanism of this increase is not clear but may involve high levels of neuromodulatory input.

A final important implication of our simulations is that the form of both motoneuron firing patterns and the pool input-output function is strongly dependent on the synaptic organization of motor commands, because of the high sensitivity of these behaviors to the level of neuromodulation and the form of inhibition. One reason for motor commands to produce variations in motoneuron behaviors might be to control the gain of the input-output function. Tasks that require a high degree of precision might benefit from a low gain to minimize the impact of errors in the motor commands, while those that require high forces or speeds might benefit from high gains. Moreover, this general concept might also partly explain the different patterns of rate modulation in distal vs. proximal muscles. The latter are associated with stabilization and high forces and thus may inherently tend to have high gains, large PICs, and consequently a high degree of saturation in firing rates. In contrast, a distal hand muscle may generally operate in a lower-gain regime for fine motor control. This consideration suggests that neuromodulatory input is stronger on proximal than distal muscles. This issue has not been directly studied, but differences in neuromodulatory drive have recently been demonstrated in human subjects (Wilson et al. 2015), using the estimations of PIC amplitude from motor unit firing patterns developed by Gorassini and colleagues (Gorassini et al. 2002). Recent quantitative reconstructions of monoaminergic contacts on animal motoneurons suggest that part of this variation could arise from differences in the number of monoaminergic contacts on motoneurons innervating different muscles (Maratta et al. 2015; Montague et al. 2013). Another possibility is that differences in firing patterns between muscles may reflect differences in inhibitory control, with distal muscle perhaps operating more consistently in a push-pull fashion, which tends to reduce motor unit firing saturation (see Fig. 6) and lower gain at low input levels (Fig. 7). Consistent with this possibility, Revill and Fuglevand (2017) recently showed that a background of inhibition generated by electrical stimulation of a cutaneous nerve tends to linearize motor unit firing patterns. Note that cutaneous inhibition tends to fade during steady electrical stimulation (Heckman et al. 1994) and thus the background inhibition in Revill and Fuglevand (2017) study may have approximated the gradual decrease in inhibition that occurs during the rising phase of the push-pull organization simulated here. Alternatively, the firing rate saturation in human motor units often observed during slowly increasing contractions (e.g., DeLuca and Contessa 2012; Fuglevand et al. 2015) may result from balanced inhibition. Thus the flexibility in the form of the input-output function of motoneurons and the motor pool may provide motor commands with the capacity for matching these behaviors to the demands of different types of motor tasks, as we have previously suggested (Johnson et al. 2012; Heckman et al. 2008a, 2008b).

GRANTS

This work was supported by National Institute of Neurological Disorders and Stroke Grants R01 NS-062200, R01 NS-071951, and R01 NS-08931.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

R.K.P. and C.J.H. conceived and designed research; R.K.P. performed experiments; R.K.P. analyzed data; R.K.P. and C.J.H. interpreted results of experiments; R.K.P. prepared figures; R.K.P. drafted manuscript; R.K.P. and C.J.H. edited and revised manuscript; R.K.P. and C.J.H. approved final version of manuscript.