Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation
Tomás Caraballo, Zhang Chen, Lingyu Li
Abstract
This paper is mainly concerned with limiting behaviors of invariant measures for neural field latticemodels in a random environment. First of all, we consider the convergence relation of invariantmeasures between the stochastic neural field lattice model and the corresponding deterministic modelin weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice modelmust be an invariant measure of its limiting system as the noise intensity tends to zero. Then we aredevoted to studying the numerical approximation of invariant measures of such a stochastic neurallattice model. To this end, we first consider convergence of invariant measures between such a neurallattice model and the system with neurons only interacting with its n-neighborhood; then we furtherprove the convergence relation of invariant measures between the system with an n-neighborhood andits finite dimensional truncated system. By this procedure, the invariant measure of the stochasticneural lattice models can be approximated by the numerical invariant measure of a finite dimensionaltruncated system based on the backward Euler--Maruyama (BEM) scheme. Therefore, the invariantmeasure of a deterministic neural field lattice model can be observed by the invariant measure ofthe BEM scheme when the noise is not negligible.