Topological-numerical analysis of a two-dimensional discrete neuron model

TL;DR

This paper presents a rigorous topological and numerical analysis of a 2D neuron model, introducing new algorithms to identify parameter regions with potential chaotic dynamics, applicable to various dynamical systems.

Contribution

The paper introduces a novel algorithm for analyzing return times in chain recurrent sets and a new method to identify parameters leading to chaos in neuron models.

Findings

Identification of parameter sets with chaotic dynamics

Development of a new return time analysis algorithm

Application of topological methods to neuron models

Abstract

We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at http://www.pawelpilarczyk.com/neuron/.