A Single, continuously applied control policy for modeling reaching movements with and without perturbation

Zhe Li, Pietro Mazzoni, Sen Song, and Ning Qian, Neural Computation, 2018, 30: 397-427.
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It has been debated whether kinematic features, such as the number of peaks or decomposed sub-movements in a velocity profile, indicate the number of discrete motor impulses or result from a continuous control process. The debate is particularly relevant for tasks involving target perturbation which can alter movement kinematics. To simulate such tasks, finite-horizon models require two preset movement durations to compute two control policies before and after the perturbation. Another model employs infinite- and finite-horizon formulations to determine, respectively, movement duration and control policy at each time step, with the number of durations/policies equal to the number of time steps. We adopted an infinite-horizon optimal feedback control model that, unlike previous approaches, does not preset movement durations or use multiple control policies. It contains both control-dependent and independent noises in system dynamics, state-dependent and independent noises in sensory feedbacks, and different delays and noise levels for visual and proprioceptive feedbacks. We analytically derived an optimal solution which can be applied continuously to move an effector toward a target regardless of whether, when, or where the target jumps. This single policy produces different numbers of peaks and ``sub-movements" in velocity profiles for different conditions and for different trials of the same condition. Movements that are slower or perturbed later appear to have more sub-movements. The model is also consistent with the observation that subjects can perform the perturbation task even without detecting the target jump or seeing their hands during reaching. Finally, because the model incorporates Weber's law via a state representation relative to the target, it explains why initial and terminal visual feedback are, respectively, less and more effective in improving endpoint accuracy. Our work suggests that the number of peaks or sub-movements in a velocity profile does not necessarily reflect the number of motor impulses, and that the difference between initial and terminal feedback does not necessarily imply a transition between open- and closed-loop strategies.

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