Feedback parameters for a closed-loop multiple-input multiple-output model of the upper limb

Both closed-loop models and multi-input multi-output (MIMO) models of the neuromusculoskeletal system of the upper limb are important for simulating and understanding motor control. Yet no large-scale linear neuromusculoskeletal models of the upper limb that are both closed-loop and MIMO have been developed. The primary difficulty in creating such models is choosing appropriate feedback parameters (such as feedback gains and delays), as such a collection of parameters is not available in the literature. The purpose of this work is to 1) present a method for developing MIMO models of short-loop afferent feedback and 2) offer estimates of average feedback parameter values and ranges based on the currently available literature. To this end, we combined measurements of feedback-related parameters available in 26 prior studies with known properties of system stability and behavior. As a result, we present estimated feedback gains and delays for a linear model of the upper limb with inputs into the 13 major superficial muscles and outputs to the 7 main joint degrees of freedom from the shoulder to the wrist. This model includes homonymous feedback mediated by Golgi tendon organs and both homonymous and heteronymous feedback mediated by muscle spindles. As a partial validation of muscle-spindle feedback gains, we compared the sign of the estimated gains to known differences in excess central delay between excitatory and inhibitory connections. The comparison proved correct in all 39 muscle pairs for which we had both estimated a feedback gain and found a measured excess central delay value in the literature. Furthermore, as a partial validation of delay times, we compared estimated delay times to measured innervation lengths. We found a strong fit for efferent delays (R = 0.88) and a moderate fit for afferent delays (R = 0.65). In addition, we demonstrate the effect of feedback on model behavior and present brief comparisons between this behavior and experimentally observed behaviors of the human upper limb with and without feedback.