Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

Abstract

Abstract. Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with ascillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the mean square norm for all times.

Reaction-diffusion equations on metric graphs with edge noise

“…In the literature stochastic (reaction-)diffusion equations on networks are treated e.g. in [5], [6], [8], [9] and [15]. In the first four papers the semigroup approach is utilized in a Hilbert space setting.…”

“…(5) There exist constants a , b , k, K > 0 with K ≥ k such that the function (6) For some constant κ F ≥ 0, the map F : [0, T ] × Ω × Z → E −κ F is globally Lipschitz continuous in the third variable, uniformly with respect to the first and second variables. Moreover, for all u ∈ Z the map (t, ω) → F (t, ω, u) is strongly measurable and adapted.…”

Stochastic reaction–diffusion equations on networks

“…In [11], additive Lévy noise is considered that is square integrable with drift being a cubic polynomial. In [14], multiplicative square integrable Lévy noise is considered but with globally Lipschitz drift and diffusion coefficients and with a small time dependent perturbation of the linear operator. Paper [10] treats the case when the noise is an additive fractional Brownian motion and the drift is zero.…”