Bipedal locomotion was simulated to generate a pattern of activating muscles for walking using electrical stimulation in persons with spinal cord injury (SCI) or stroke. The simulation presented in this study starts from a model of the body determined with user-specific parameters, individualized with respect to the lengths, masses, inertia, muscle and joint properties. The trajectory used for simulation was recorded from an ablebodied subject while walking with ankle-foot orthoses. A discrete mathematical model and dynamic programming were used to determine the optimal control. A cost function was selected as the sum of the squares of the tracking errors from the desired trajectories, and the weighted sum of the squares of agonist and antagonist activations of the muscle groups acting around the hip and knee joints. The aim of the simulation was to study plausible trajectories keeping in mind the limitations imposed by the spinal cord injury or stroke (e.g., spasticity, decreased range of movements in some joints, limited strength of paralyzed, externally activated muscles). If the muscles were capable of generating the movements required and the trajectory was achieved, then the simulation provided two kinds of information: 1) timing of the onset and offset of muscle activations with respect to the various gait events and 2) patterns of activation with respect to the maximum activation. These results are important for synthesizing a rule-based controller.

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A model of isometric force production by skeletal muscle was developed in which the response to each stimulus in a train was described by a critically damped, linear second-order system. The parameters describing the system were constrained to be constant within an interstimulus interval, but were allowed to vary between interstimulus intervals. The ability of this model to match experimental data, and the time variation in the parameters (low-frequency gain and natural frequency) required to do so were examined in soleus and plantaris muscles of the cat stimulated by synchronous whole-nerve stimulation. The model produced good fits across firing rates from twitch to tetanus for slow and fast muscle, rested and fatigued muscle, and maximal submaximal stimulation. Both gain and natural frequency generally varied smoothly and predictably under all conditions. Gain increased at intermediate stimulation rates and in potentiated muscle, and decreased with fatigue and submaximal stimulation. Natural frequency was higher in fast muscle, and decreased with stimulation rate and fatigue. This modeling approach may provide a useful alternative to current models of skeletal muscle force, as its implementation is simple and it can describe force under conditions (fatigue, potentiation) where the muscle dynamics change with time.

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The kinetics relating Ca2+ transients and muscle force were examined using data obtained with the photoprotein aequorin in skeletal muscles of the rat, barnacle, and frog. These data were fitted by various models using nonlinear methods for minimizing the least mean square errors. Models in which Ca2+ binding to troponin was rate limiting for force production did not produce good agreement with the observed data, except for a small twitch of the barnacle muscle. Models in which cross-bridge kinetics were rate limiting also did not produce good agreement with the observed data, unless the detachment rate constant was allowed to increase sharply on the falling phase of tension production. Increasing the number of cross-bridge states did not dramatically improve the agreement between predicted and observed force. We conclude that the dynamic relationship between Ca2+ transients and force production in intact muscle fibers under physiological conditions can be approximated by a model in which (a) two Ca2+ ions bind rapidly to each troponin molecule, (b) force production is limited by the rate of formation of tightly bound cross-bridges, and (c) the rate of cross-bridge detachment increases rapidly once tension begins to decline and free Ca2+ levels have fallen to low values after the last stimulus. Such a model can account not only for the pattern of force production during a twitch and tetanus, but also the complex, nonlinear pattern of summation which is observed during an unfused tetanus at intermediate rates of stimulation.