Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise

The well-posedness and long term dynamics of a stochastic non-autonomous neural field lattice system on vector-valued indices $${\mathbb {Z}}^d$$ driven by state dependent nonlinear noise are investigated in a weighted space of infinite sequences. First the existence and uniqueness of a mean square solution to the lattice system is established under the assumptions that the nonlinear drift and diffusion terms are component-wise continuously differentiable with weighted equi-locally bounded derivatives. Then the existence and uniqueness of a tempered weak pullback mean random attractor associated with the solution is proved. Finally the existence of invariant measures for the neural field lattice system is obtained by uniform tail-estimates and Krylov–Bogolyubov’s method.