Reliability and robustness of oscillations in some slow-fast chaotic systems

TL;DR

This paper explores how chaotic systems with multiple timescales can exhibit regular behavior at macro levels, reconciling chaos with biological robustness, and identifies bifurcations as key transition points.

Contribution

It introduces a refined notion of chaos that accounts for homeostasis in slow-fast systems and demonstrates this through analysis of various models including the Rulkov map.

Findings

Chaotic dynamics can appear regular at macroscopic timescales in slow-fast systems.

Global bifurcations like crises can disrupt this regularity, leading to erratic activity.

The proposed mechanism is validated across multiple models, including biological and mathematical systems.

Abstract

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are thus consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.