Desynchronization of large scale delayed neural networks

Abstract

Abstract. We consider a ring of identical neurons with delayed nearest neighborhood inhibitory interaction. Under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation with negative feedback. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, we show that the associated synchronous solution is unstable if the size of the network is large.

Existence, Uniqueness, and Stability of Slowly Oscillating Periodic Solutions for Delay Differential Equations with Nonnegativity Constraints

“…Consider a system of n ≥ 3 coupled DDEs arranged in a ring with nearest neighbor coupling; that is, the coupling matrix R n = (R jk n ) satisfies R jk n = 0 if j − k (mod n) > 1. Systems of DDEs arranged in a ring arise in biological applications [32,33,36,37] and have received attention in the mathematical literature [5,18]. Remark 9.5.…”

“…Of special interest for such a system is the synchronous state in which the elements of the system oscillate in unison. A natural problem in this context, which has received considerable attention in the literature, is to characterize conditions under which the synchronous state is stable; see, for example, [1,2,3,5,12,13,16,17,18,35,38,43].…”

“…Our focus is on linear stability of so-called synchronous slowly oscillating periodic solutions for the coupled DDE (1.1), in the asymptotic regime β → ∞. By focusing on this asymptotic regime, we are able to treat a broad class of nonlinearities f and coupling matrices G, which is in contrast to many related works that impose additional symmetry conditions on the system, such as restricting to oscillators arranged in a ring [3,5,16,17,18,43] or, more generally, symmetric circulant coupling matrices [2,40]. Before defining a synchronous slowly oscillating periodic solution for the coupled DDE (1.1), we first define a slowly oscillating periodic solution for the related scalar DDE (1.2) ẋ(t) = −αx(t) + βf (x(t − 1)), t > 0, where x is a real-valued continuous function on [−1, ∞) that is continuously differentiable on (0, ∞) and α, β and f are as in system (1.1).…”

“…The next class considers systems of DDEs arranged in a ring. In [5] Chen, Huang and Wu show that the synchronous SOPS is unstable when either n is even or n is odd and sufficiently large. The works [16,17] prove related results when allowing for a more general nonlinear feedback structure.…”

Global continuation of nonlinear waves in a ring of neurons

“…In the field of neural networks, rings are studied to gain insight into the mechanisms underlying the behaviours of recurrent networks [17,22]. Moreover, ring networks belong to the class of cyclic feedback systems whose asymptotic behaviour has been investigated in more detail [1,2,5,8,13,15,19,24,25,27,29,30]. These theoretical results help in better understanding the system's dynamics and are important complements to experimental and numerical investigations using analog circuits and digital computers.…”