On Inclusions Arising in Neural Field Modeling

We study existence of solutions to the Hammerstein integral inclusion with delay generated by a neural field equation with discontinuities and time lags. Particular attention is given to the problem of continuous dependence of the solution set on a parameter. The investigation of this issue has important applications in mathematical neuroscience allowing to establish relations between continuous and discontinuous modeling approaches that are both widely used in mathematical theory of neural fields. In order to obtain the main results for the Hammerstein integral inclusion we develop a framework of abstract Volterra operator inclusions and prove statements on local solvability, on extendability of the solutions, and on parametric dependence of the solution set.