Large-Scale Structure of Chaotic Attractors in FitzHugh-Rinzel model
Mohammadreza Razvan, Sheida Shahidi
TL;DR
This work tackles the challenge of characterizing large-scale structure in chaotic attractors of the FitzHugh-Rinzel neural model during bursting transitions. It introduces a coarse-grained Markov-chain representation by partitioning the attractor into recognizable regions (the Leviathan's heads and wings) and modeling inter-region transitions as a directed graph. By comparing the Markov-chain entropy rate with the dynamical system's topological entropy and validating randomness with Lempel-Ziv complexity, the approach reveals when the partition omits essential dynamics and guides iterative graph refinement. The findings show how new attractor regions emerge with parameter changes and demonstrate that refining the graph yields a more faithful stochastic description of the chaotic transitions, offering a tool for detecting structural changes in neural chaos.
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.