Large-Scale Structure of Chaotic Attractors in FitzHugh-Rinzel model

TL;DR

This paper investigates the structure of chaotic attractors in the FitzHugh-Rinzel neural model using a stochastic approach, Markov chains, and entropy measures to understand transition mechanisms and shape changes in chaotic bursting behavior.

Contribution

It introduces a novel stochastic method with Markov chains and entropy analysis to study the structure and transitions of chaotic attractors in the FitzHugh-Rinzel model.

Findings

Partitioned attractor regions reveal transition mechanisms.

Entropy comparisons identify shape changes in attractors.

Lempel-Ziv entropy estimates confirm reliability of results.

Abstract

Chaotic bursting behaviors have been observed by many authors in neural dynamics mainly in the transition between different kinds of bursting behavior. As a well-known three-dimensional ODEs model with various bursting solutions, the FitzHugh-Rinzel model has been considered in this paper. The structure of the strange attractor that appears in chaotic transitions of this model was investigated by introducing a stochastic approach to uncover the transition mechanism. To portray this idea the attractor of the dynamical system can be partitioned into some regions and a discrete evolution that is inspired by the flow between them is sketched. A suitable Markov chain has been associated with the strange attractor based on partition selected by the recognizable regions of the attractor. Then the entropy rate of the Markov chain and the topological entropy of dynamical systems are compared to decide if the associated Markov chain should be modified or not. Furthermore, the differences between entropies guide us to uncover some changes in the shape of the attractor including some new regions which play important roles in the chaotic behavior of our system. It can be also ensured with the help of Lempel-Ziv quantity that the estimated entropy rate is reliable.