Neural field equations with neuron‐dependent Heaviside‐type activation function and spatial‐dependent delay

Abstract

We introduce a neural field equation with a neuron‐dependent Heaviside‐type activation function and spatial‐dependent delay. The basic object of the study is represented by a Volterra–Hammerstein integral equation involving a discontinuous nonlinearity with respect to the state variable that is both time and space dependent. We replace the discontinuous nonlinearity by its multivalued convexification and obtain the corresponding Volterra–Hammerstein integral inclusion. We investigate the solvability of this inclusion using the properties of upper semicontinuous multivalued mappings with convex closed values. Based on these results, we study the solvability of an initial‐prehistory problem for the former neural field equation with the Heaviside‐type activation function. The application of multivalued analysis techniques allowed us to avoid some restrictive assumptions standardly used in the investigations of the solutions to neural field equations involving Heaviside‐type activation functions.