Upper semi-continuous convergence of attractors for a Hopfield-type lattice model

Abstract

To investigate dynamical behavior of the Hopfield neural network model when its dimension becomes increasingly large, a Hopfield-type lattice system is developed as the infinite dimensional extension of the classical Hopfield model. The existence of global attractors is established for both the lattice system and its finite dimensional approximations. Moreover, the global attractors for the finite dimensional approximations are shown to converge to the attractor for the infinite dimensional lattice system upper semi-continuously.

Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

“…Throughout the rest of this paper, we always use D to denote the collection of all tempered families D = {D(τ, ω) ⊆ C H ρ : τ ∈ R, ω ∈ Ω} which satisfies (23), that is,…”

“…The key is to establish the convergence of solutions in C([−ρ, 0], H) and H as intensity of noise and time delay goes to zero, respectively. For more details about the convergence of attractors, we refer the readers to [23,41,45,46,49,52] for deterministic and stochastic differential equations without delays and to [25,28,29,50,51,56] for deterministic and stochastic ODEs as well as PDEs with delays.…”

Attractors of Hopfield-type lattice models with increasing neuronal input

“…Basic properties of solutions. In [20] system (2) was studied in the usual Hilbert space of bi-infinite sequences 2 . However, due to the global interconnection term j∈Z λ i,j f j (u j (t)) system (3) is not well-posed in 2 except under very restrictive assumptions.…”

“…Hopfield-type neural networks have been studied extensively during the past decades (see, e.g., [1,6,13,17,24,25,29,30,37]). Very recently Han, Usman & Kloeden studied Hopfield-type lattice dynamical system [20]:…”

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

“…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.…”