Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

Flank Bezerra,Antônio Pereira,Severino Da Silva

doi:10.14232/ejqtde.2017.1.41

Abstract

Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

Highlights

In this work we study the pullback dynamics of a class of nonlocal nonautonomous evolution equations for neural fields in a bounded smooth domain Ω
Under suitable assumptions on the nonlinearity f : R2 → R, we prove existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to this problem
In this paper we study the pullback dynamics for a class of nonlocal non-autonomous evolution equations generated as continuum limits of computational models of neural fields theory

Summary

Introduction

In this paper we study the pullback dynamics for a class of nonlocal non-autonomous evolution equations generated as continuum limits of computational models of neural fields theory. Neural field equations are tissue level models that describe the spatiotemporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons, see e.g. We analyze the following non-autonomous theoretical model for networks of nerve cells. We will assume that f : R2 → R is a sufficiently smooth function (some growth conditions about f are assumed, as presented along the Section 3). ∂tu(t, x) = −u(t, x) + J(x, y)( f ◦ u)(t, y)dy, where the strength of the connection depends only on the distance between cells, that is, J(x, y) = J(x − y) and the firing rate function is time-independent

Outline of the paper

Functional setting and background results

Well-posedness of the problem