FitzHugh, R. J. Math. Biology (1977) 4: 203. doi:10.1007/BF00280973
In the absence of detailed knowledge of how the CNS controls a muscle through its motor fibers, a reasonable hypothesis is that of optimal control. This hypothesis is studied using a simplified mathematical model of a single muscle, based on A. V. Hill's equations, with series elastic element omitted, and with the motor signal represented by a single input variable.
Two cost functions were used. The first was total energy expended by the muscle (work plus heat). If the load is a constant force, with no inertia, Hill's optimal velocity of shortening results. If the load includes a mass, analysis by optimal control theory shows that the motor signal to the muscle consists of three phases: (1) maximal stimulation to accelerate the mass to the optimal velocity as quickly as possible, (2) an intermediate level of stimulation to hold the velocity at its optimal value, once reached, and (3) zero stimulation, to permit the mass to slow down, as quickly as possible, to zero velocity at the specified distance shortened. If the latter distance is too small, or the mass too large, the optimal velocity is not reached, and phase (2) is absent. For lengthening, there is no optimal velocity; there are only two phases, zero stimulation followed by maximal stimulation.
The second cost function was total time. The optimal control for shortening consists of only phases (1) and (3) above, and is identical to the minimal energy control whenever phase (2) is absent from the latter.
Generalization of this model to include viscous loads and a series elastic element are discussed.