Desynchronization of delayed neural networks
The subject of the paper is a model of a network of identical neurons of the following form \[ \dot u_j(t) = - \mu u_j(t) + \sum_{l=1}^{n} J_{jl} f(u_l(t-\tau)), \quad j=1,\dots,n, \] where \(\mu\) and \(\tau\) are nonnegative constants, \(f:\mathbb{R}\to\mathbb{R}\) is bounded and \(C^1\)-smooth with \(f(0)=0\), \(f'(x)>0\) for all \(x\in\mathbb{R}\). It has been shown by \textit{K. P. Hadeler} and \textit{J. Tomiuk} [Arch. Ration. Mech. Anal. 65, 87-95 (1977; Zbl 0426.34058)] that the reduced system for the synchronized solutions \(u_1=u_2=\cdots=u_n\), i.e., the system for the single neuron \(\dot u(t) = -\mu u(t) -a f(u(t-\tau))\) has a stable slowly oscillatory periodic solution \(p\) under some conditions. The main result of this paper states that, if the connection matrix \(J=(J_{jl})\) satisfies \(-a=\sum_{l=1}^n J_{lj}\) and has a real eigenvalue greater or equal to \(a\), then the corresponding synchronized slowly oscillatory solution \(p^s=(p,\dots,p)^T\) is unstable under asynchronous perturbations. The present paper extends results by \textit{Y. Chen, Y. S. Huang} and \textit{J. Wu} [Proc. Am. Math. Soc. 128, 2365-2371 (2000; Zbl 0945.34056)] for an inhibitory ring of neurons to a network with arbitrary connection topology.
scientific article; zbMATH DE number 1882986