Fuzzy neuronal model of motor control inspired by cerebellar pathways to online and gradually learn inverse biomechanical functions in the presence of delay

Abstract

Contrary to forward biomechanical functions, which are deterministic, inverse biomechanical functions are generally not. Calculating an inverse biomechanical function is an ill-posed problem, which has no unique solution for a manipulator with several degrees of freedom. Studies of the command and control of biological movements suggest that the cerebellum takes part in the computation of approximate inverse functions, and this ability can control fast movements by predicting the consequence of current motor command. Limb movements toward a goal are defined as fast if they last less than the total duration of the processing and transmission delays in the motor and sensory pathways. Because of these delays, fast movements cannot be continuously controlled in a closed loop by use of sensory signals. Thus, fast movements must be controlled by some open loop controller, of which cerebellar pathways constitute an important part. This article presents a system-level fuzzy neuronal motor control circuit, inspired by the cerebellar pathways. The cerebellar cortex (CC) is assumed to embed internal models of the biomechanical functions of the limb segments. Such neural models are able to predict the consequences of motor commands and issue predictive signals encoding movement variables, which are sent to the controller via internal feedback loops. Differences between desired and expected values of variables of movements are calculated in the deep cerebellar nuclei (DCN). After motor learning, the whole circuit can approximate the inverse function of the biomechanical function of a limb and acts as a controller. In this research, internal models of direct biomechanical functions are learned and embedded in the connectivity of the cerebellar pathways. Two fuzzy neural networks represent the two parts of the cerebellum, and an online gradual learning drives the acquisition of the internal models in CC and the controlling rules in DCN. As during real learning, exercise and repetition increase skill and speed. The learning procedure is started by a simple and slow movement, controlled in the presence of delays by a simple closed loop controller comparable to the spinal reflexes. The speed of the movements is then increased gradually, and output error signals are used to compute teaching signals and drive learning. Repetition of movements at each speed level allows to properly set the two neural networks, and progressively learn the movement. Finally, conditions of stability of the proposed model as an inverter are identified. Next, the control of a single segment arm, moved by two muscles, is simulated. After proper setting by motor learning, the circuit is able to reject perturbations.

Keywords: Approximate inverse function; Cerebellum; Deep cerebellar nuclei (DCN); Fast movements; Forward model; LTD/LTP; Online gradual learning; Reflex pathways.