Relationship of calculating the Jacobian matrices of nonlinear systems and population coding algorithms in neurobiology

Abstract

In this paper we examine how the calculation of Jacobian matrices in nonlinear systems is related to population coding algorithms in neurobiology. Population coding algorithms have been designed to retrodict sensory stimuli or predict motor behavior from neuronal responses. We simulate the visuomotor hand movement task of reaching in a plane. Adaptive feedback control updating of joint angles of a three-joint arm was used in the model. The control signal is the dot product between the visual error signal and the Jacobian matrix of the direct kinematic equation of hand movement. The x and y hand trajectories can follow the x and y time series of Lorenz and Ro¨ssler systems of coupled nonlinear equations. In this particular example, the elements of the Jacobian matrix can be estimated from observed changes in joint angles and Cartesian coordinate values of the hand. We also point out how the Jacobian matrices in nonlinear biological systems can be estimated in general from observed time series; e.g. joint angles, Cartesian coordinate values of hand movement, neural or muscle activity. Estimating Jacobian matrices of nonlinear functions from experimental observations can provide not only data analysis but also signal synthesis and can lead to a more realistic modeling of sensorimotor transformations.