Spatiotemporal Kernel of a Three-Component Differential Equation Model with Self-control Mechanism in Vision

This paper examines a three-component differential equation model with a self-control mechanism in vision as a slight extension of the lateral inhibition model proposed by Peskin (Partial differential equations in biology: Courant Institute of Mathematical Sciences Lecture Notes, New York, 1976). First, we derive a condition under which the exact solution of our differential equation model for time-dependent input I = I ( t ) is described by the convolution integral with a temporal biphasic function. Second, we analyze the model with the input signal I = I ( t , x ) depending on time t and position x ∈ R , and we prove that the solution can be represented in convolution integral form and that t 1 > 0 exists such that the spatiotemporal integral kernel K u ( t 1 , x ) is positive for x ∈ R and t ∈ ( 0 , t 1 ) . Moreover, we numerically demonstrate that there exists t 2 ( > t 1 ) such that K u ( t 2 , x ) includes the Mexican-hat function and a temporal biphasic function under certain parameter conditions. From these mathematical and numerical results, we find that there is a time lag before the Mexican-hat function appears in K u ( t , x ) , and the shape of K u ( t , x ) is similar to the receptive field structure observed in experiments in the field of neurophysiology. We conclude that the partial differential equations for visual information processing can be used to analytically determine the shape of the spatiotemporal kernel indicating the self-control mechanism.