Existence of physical measures in some Excitation-Inhibition Networks

TL;DR

This paper rigorously analyzes large finite excitation-inhibition networks inspired by biological systems, demonstrating that their long-term dynamics are governed by physical measures with SRB properties using hyperbolic theory.

Contribution

It provides a mathematical proof that certain coupled biological-inspired networks have physical measures with SRB properties, extending understanding of their long-term behavior.

Findings

Large networks exhibit complex dynamics influenced by coupling.

Existence of physical measures with SRB properties is established.

Lyapunov exponents and entropy characterize the network dynamics.

Abstract

In this paper we present a rigorous analysis of a class of coupled dynamical systems in which two distinct types of components, one excitatory and the other inhibitory, interact with one another. These network models are finite in size but can be arbitrarily large. They are inspired by real biological networks, and possess features that are idealizations of those in biological systems. Individual components of the network are represented by simple, much studied dynamical systems. Complex dynamical patterns on the network level emerge as a result of the coupling among its constituent subsystems. Appealing to existing techniques in (nonuniform) hyperbolic theory, we study their Lyapunov exponents and entropy, and prove that large time network dynamics are governed by physical measures with the SRB property.