Existence of physical measures in some excitation–inhibition networks* *MT acknowledges funding from the H2020 Marie Słodowska-Curie Actions, project: ‘Ergodic Theory of Complex Systems’ (Project No. 843880). LSY was supported in part by NSF Grant DMS 1901009.

Matteo Tanzi,Lai-Sang Young

doi:10.1088/1361-6544/ac3eb6

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.

Highlights

As dynamical systems come in too many flavors to be classified or even systematically described, when studying the subject one usually learns from paradigms
In the mathematical theory of chaotic systems, a great deal of intuition has been derived from classical examples such as expanding circle maps [KS69], geodesic flows on manifolds of negative curvature [Hop[39], Ano67], hyperbolic billiards [Sin[70], Bun74] [BSC90, CM06], the Lorenz attractor [Lor[62], GW79], logistic maps [Jak[81], CE80, BC85], Henon attractors and generalizations [BC91, YB93, WY08], and so on
Our notion of what a chaotic dynamical system looks like has been intimately tied to these examples

Summary

Introduction

As dynamical systems come in too many flavors to be classified or even systematically described, when studying the subject one usually learns from paradigms. In the belief that examples will contribute to our understanding of hyperbolicity in high dimensional systems, we present in this paper a new class of models — new in the sense that their ergodic theory has not been studied — and demonstrate how they can be analyzed using existing tools. These models are examples of ExcitationInhibition networks. The interaction is bi-directional and the model is more general than a skew-product: an excitatory subsystem excites inhibitory components which when activated send feedback inhibition to the excitatory subsystem These examples are inspired by biology, but we do not pretend that they are depictions of any specific biological system. We wish to demonstrate that dynamical systems theory has the language and the tools to shed light on natural phenomena, to offer insight on a conceptual level even when analysis of the system exactly as defined is out of reach

Model Description and Main Results

The full model

Statement of the main results and an illustrative example

Distribution of pushed-forward mass: a preliminary estimate

Proof of SRB property

Lyapunov charts and related results