The primary focus of this paper lies in exploring the limiting dynamics of a neural field lattice model with state dependent superlinear noise. First, we established the well-posedness of solutions to these stochastic systems and subsequently proved the existence of periodic measures for the system in the space of square-summable sequences using Krylov-Bogolyubov's method. The cutoff techniques of uniform estimates on tails of solutions was employed to establish the tightness of a family of probability distributions for the system's solutions.

In this article, we investigate the Wong-Zakai approximations of a class of second order non-autonomous stochastic lattice systems with additive white noise. We first prove the existence and uniqueness of tempered pullback random attractors for the original stochastic system and its Wong-Zakai approximation. Then, we establish the upper semicontinuity of these attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.

The focus of this paper lies in exploring the limiting dynamics of stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions and nonlinear noise in weighted space. To begin, we established the well-posedness of solutions to these stochastic delay lattice systems and subsequently proved the existence and uniqueness of invariant measures.