Many Attractors, Long Chaotic Transients, and Failure in Small-World Networks of Excitable Neurons

Abstract

We study the dynamical states that emerge in a small-world network of recurrently coupled excitable neurons through both numerical and analytical methods. These dynamics depend in large part on the fraction of long-range connections or `short-cuts' and the delay in the neuronal interactions. Persistent activity arises for a small fraction of `short-cuts', while a transition to failure occurs at a critical value of the `short-cut' density. The persistent activity consists of multi-stable periodic attractors, the number of which is at least on the order of the number of neurons in the network. For long enough delays, network activity at high `short-cut' densities is shown to exhibit exceedingly long chaotic transients whose failure-times averaged over many network configurations follow a stretched exponential. We show how this functional form arises in the ensemble-averaged activity if each network realization has a characteristic failure-time which is exponentially distributed.