On the stability of delay equation models of simple human stretch reflexes

Human stretch reflexes (SRs) are often too weak and ineffectual to provide adequate postural regulation or rhythmic movement boosting (e.g. in ankle pushoff at the end of stance phase in fast running). Recent improvements in methods of artificially enhancing skeletomotor responses, especially in therapeutic regimens, should not be widely employed until the clonus-resisting stability properties of SRs are better understood. We formulate an idealized linear servo model of a segmentally-mediated SR system which includes the often ignored electromechanical coupling delay. For typical closed-loop (delay/gain) ratios, the model is shown to be unstable for all values of loop gain when operating as a position servo, but maximally stable when operating as a velocity servo. We claim that the velocity servo or more of its nonlinear relatives is a better model for some well studied SRs than, e.g., Houk's stiff muscle hypothesis. We also present evidence that even feeble and quickly saturating monosynaptic postural servos are always unstable if operated as pure position regulators.