The purpose of this study was to investigate the role of feedback gain in optimal feedback control (OFC) theory for musculoskeletal systems. We could not apply OFC to the musculoskeletal systems because of the nonlinearity. Thus, we modeled the systems as a linear plant with a constraint, and computed an optimal feedback gain using model predictive control (MPC) which was based on iterative, finite horizon optimization of plant model.
Neural studies have shown that directional tuning, known as the preferred direction (PD), is a basic functional property of cell activity in the primary motor cortex (M1). However, it is not clear which directions the M1 codes for, because neural activities can correlate with several directional parameters, such as joint torque and end-point motion. Besides, OFC has been proposed as a plausible model could predict various motor behaviors even though it has remained an open question how to realize OFC in the brain.
To examine the computational mechanism in the M1, we modeled the isometric motor task of a musculoskeletal system required to generate the desired joint torque. Then, we computed the optimal feedback gain according to MPC with a nonlinear constraint for muscle activities, which must not be negative. The optimal solution of MPC was provided as a state feedback form just like OFC, and the current state including joint torques and muscle activities was estimated by the Kalman filter. Thereby, the feedback controller could be divided into the predictive state feedback and output feedback components. The output feedback gain was given by the inner product of the state feedback and Kalman gains.
As a result, the state and output feedback gains indicated directional tunings of the joint torque and end-point motion in Cartesian space that were similar to the M1 neuron PDs observed in primate studies. Besides, the state feedback gain was also projected by muscle activities included in the predicted state. Then, the directions of the feedback gain were rotated from the mechanical PDs by existence of biarticulate muscles. Therefore, we suggest that an optimal feedback gain is represented in the M1, and indicated by the PD. Furthermore, the muscle and neural activities may be optimized for the musculoskeletal system having biarticulate muscles.