Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 358-382, March 2024.
Abstract. This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler–Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible.