On the integral equations of the linear theory of recurrent lateral interaction in vision

Abstract

In a spatially homogeneous visual system with linear, recurrent, lateral interactions that fall off with distance at large distances, the response field is related to the excitation field through a functional equation involving convolution of the response with an “interaction kernel” which characterizes the system. We discuss here the general theory of solutions of such equations in Bochner's F k-spaces, and we present explicit solutions for some special cases in which the interaction kernel is “bimodal,” i.e., achieves its maximum absolute value at a non-zero distance of separation of sensory points. Bimodal kernels are of interest because they appear to be in qualitative accord with physiological data, and, as is known, a linear recurrent system with such a kernel can yield a transfer function whose maximum value is finite and occurs at a finite, non-zero, spatial frequency. Such a recurrent system with a bimodal kernel and a regular transfer function can, however, also show “oscillations,” i.e., multiple turning points, in the response to a simple step-function excitation field. To help study this last phenomenon, we derive resolvent functions for integral equations in a class which generalizes the classical Lalesco-Picard equation.