Chaos in neural networks with a nonmonotonic transfer function

TL;DR

This paper analytically investigates the complex dynamics, including chaos, in diluted neural networks with nonmonotonic transfer functions, revealing the structure of strange attractors and the fragility of chaos.

Contribution

It provides a detailed analytical description of the macroscopic and microscopic dynamics of such neural networks, highlighting the nature and robustness of chaos.

Findings

Rich variety of behaviors including fixed points, periodicity, and chaos

Chaotic regions are densely intercalated with periodicity windows

Microscopic damage spreading shows persistent differences in initial states

Abstract

Time evolution of diluted neural networks with a nonmonotonic transfer function is analitically described by flow equations for macroscopic variables. The macroscopic dynamics shows a rich variety of behaviours: fixed-point, periodicity and chaos. We examine in detail the structure of the strange attractor and in particular we study the main features of the stable and unstable manifolds, the hyperbolicity of the attractor and the existence of homoclinic intersections. We also discuss the problem of the robustness of the chaos and we prove that in the present model chaotic behaviour is fragile (chaotic regions are densely intercalated with periodicity windows), according to a recently discussed conjecture. Finally we perform an analysis of the microscopic behaviour and in particular we examine the occurrence of damage spreading by studying the time evolution of two almost identical initial configurations. We show that for any choice of the parameters the two initial states remain microscopically distinct.