Models of recruitment and rate coding organization in motor-unit pools

1. Isometric muscle force and the surface electromyogram (EMG) were simulated from a model that predicted recruitment and firing times in a pool of 120 motor units under different levels of excitatory drive. The EMG-force relationships that emerged from simulations using various schedules of recruitment and rate coding were compared with those observed experimentally to determine which of the modeled schemes were plausible representations of the actual organization in motor-unit pools. 2. The model was comprised of three elements: a motoneuron model, a motor-unit force model, and a model of the surface EMG. Input to the neuron model was an excitatory drive function representing the net synaptic input to motoneurons during voluntary muscle contractions. Recruitment thresholds were assigned such that many motoneurons had low thresholds and relatively few neurons had high thresholds. Motoneuron firing rate increased as a linear function of excitatory drive between recruitment threshold and peak firing rate levels. The sequence of discharge times for each motoneuron was simulated as a random renewal process. 3. Motor-unit twitch force was estimated as an impulse response of a critically damped, second-order system. Twitch amplitudes were assigned according to rank in the recruitment order, and twitch contraction times were inversely related to twitch amplitude. Nonlinear force-firing rate behavior was simulated by varying motor-unit force gain as a function of the instantaneous firing rate and the contraction time of the unit. The total force exerted by the muscle was computed as the sum of the motor-unit forces. 4. Motor-unit action potentials were simulated on the basis of estimates of the number and location of motor-unit muscle fibers and the propagation velocity of the fiber action potentials. The number of fibers innervated by each unit was assumed to be directly proportional to the twitch force. The area of muscle encompassing unit fibers was proportional to the number of fibers innervated, and the location of motor-unit territories were randomly assigned within the muscle cross section. Action-potential propagation velocities were estimated from an inverse function of contraction time. The train of discharge times predicted from the motoneuron model determined the occurrence of each motor- unit action potential. The surface EMG was synthesized as the sum of all motor-unit action-potential trains. 5. Two recruitment conditions were tested: narrow (limit of recruitment <50% maximum excitation) and broad recruitment range conditions (limit of recruitment >70% maximum excitation). Three rate coding conditions were tested: 1) low-threshold units attained greater firing rates than high-threshold units, 2) all units were assigned the same peak firing rate, and 3) peak firing rates were matched for each unit to the stimulus frequency required for maximum tetanic force. 6. The relation between EMG and force was linear when recruitment operated over a broad force range, and peak firing rates were not the same for all units. When recruitment was complete at low force levels (<57% maximum) the EMG- force relation, in all cases, was nonlinear and unlike that observed experimentally. 7. For the conditions that yielded linear EMG-force relationships, the relation between EMG and excitatory drive and between force and excitatory drive were both nonlinear. Because the shape of those nonlinear relationships were similar, when EMG was plotted as a function of force, a linear relation resulted. 8. When recruitment operated over a broad range and the peak firing rates were similar for all motor units, the EMG- force relation exhibited a slightly parabolic shape. As excitatory drive increased and the mean firing rates of the units converged toward the same value, rhythmic bursting was evident in the EMG. The bursting was associated with an augmentation of EMG amplitude, which induced a degree of concavity on an otherwise linear EMG-force relationship. 9. Unexpectedly, the maximum force capacity of the modeled muscle was not achieved in conditions where peak firing rates were set for each unit equivalent to the stimulus rate required for maximum tetanic force. The natural variability in interspike intervals combined with nonlinear force-firing rate curves for each unit diminished force from what would have been exerted had units discharged with constant interspike intervals. 10. The relation between the twitch force of a unit and the muscle force at which the unit was recruited was linear. However, the force added by the recruitment of a new unit was not a constant fraction of the muscle force. The force contributed by newly recruited units, relative to muscle force, declined hyperbolically as muscle force increased. This occurred because low-threshold units generated a larger proportion of their maximum force capacity when discharging at the threshold rate as compared with high-threshold units.