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 multi-valued convexification and
obtain the corresponding Volterra Hammerstein integral inclusion. We
investigate the solvability of this inclusion using the properties of
upper semi-continuous multi-valued 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 multi-valued
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.