Long term behavior of a random Hopfield neural lattice model

Abstract

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable biinfinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.

Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

“…We now estimate the right-hand side of (46). For the first term, since e αz(θ−tω) v τ −t ∈ D(τ − t, θ −t ω) and D is tempered, which together with (23) implies that for all 0 < α ≤ α 0 , e − 5…”

“…where we have used (13). Since e αz(θ −t ω) vτ−t ∈ D(τ − t, θ−tω) and 0 < α ≤ α0, by (23), we obtain that for all t ≥ T1,…”

“…Up to now, long-range interaction has been studied for different physical systems. For examples, traveling wave solution for lattice model with infinite-range interaction can be found in [3,36], the continuum limit for discrete nonlinear Schrödinger equations with long-range lattice interactions are considered in [26,27], attractors of neural lattice model are studied in [23,24,34] for finite-range interaction and [22,39] for infinite-range interaction.…”

“…It is worth mentioning that the fractional discrete Laplacian (−∆ d ) s is nonlocal and hence deriving uniform estimates on the tails of solutions are much more involved than the discrete Laplacian −∆ d . In addition, lattice with fractional discrete Laplacian is different from the neural networks lattice with long-range interaction that are considered in [22,23,24,34,39].…”

Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

“…where ν 0 is the unit outward normal to ∂Q. Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4].…”