Chaos and stability in a model of inhibitory neuronal network

TL;DR

This paper investigates the dynamics of inhibitory neuronal networks, demonstrating how discontinuities lead to chaos while continuous regions produce stable orbits, and classifies systems based on their stability and chaos characteristics.

Contribution

It introduces a classification of inhibitory neuronal network models into chaotic, stable, and combined types, based on the properties of their Poincare maps.

Findings

Discontinuities in the Poincare map generate chaotic sets.

Continuous parts of the map produce stable orbits.

Almost everywhere stable systems exhibit limit cycles.

Abstract

We analyze the dynamics of a deterministic model of inhibitory neuronal networks proving that the discontinuities of the Poincare map produce a never empty chaotic set, while its continuity pieces produce stable orbits. We classify the systems in three types: the almost everywhere (a.e.) chaotic, the a.e. stable, and the combined systems. The a.e. stable are periodic and chaos appears as bifurcations. We prove that a.e. stable systems exhibit limit cycles, attracting a.e. the orbits.