Asymptotic behavior for a non-autonomous model of neural fields with variable external stimulus

TL;DR

This paper analyzes the long-term behavior of a generalized nonlocal neural field model with variable external stimuli, establishing existence, uniqueness, and stability of solutions and attractors, with implications for understanding neuronal responses.

Contribution

It extends the classical Amari neural model by studying non-autonomous dynamics, proving existence, uniqueness, and stability of solutions, and analyzing how external stimuli influence neuronal activity.

Findings

Existence and uniqueness of solutions established.

Pullback attractors are shown to exist and depend continuously on stimuli.

Upper semicontinuity of attractors with respect to external stimulus functions.

Abstract

In this work we consider a class of nonlocal non-autonomous evolution equations, which generalizes the model of neuronal activity that arises in Amari (1979). Under suitable assumptions on the nonlinearity and on the parameters present in the equation, we study, in an appropriated Banach space, the assimptotic behavior of the evolution process generated by this equation. We prove results on existence, uniqueness and smoothness of the solutions and on the existence of pullback attracts for the evolution process associated to this equation. We also prove a continuous dependence of the evolution process with respect to external stimulus function present in the model. Furthermore, using the result of continuous dependence of the evolution process, we also prove the upper semicontinuity of pullback attracts with respect to stimulus function. We conclude with a small discussion about the model and about a biological interpretation of the result of continuous dependence of neuronal activity with respect to the external stimulus function.