Qualitative and Quantitative Stability Analysis of Penta-rhythmic Circuits

TL;DR

This paper analyzes the stability of inhibitory circuits of relaxation oscillators, revealing how parameters influence polyrhythmicity and robustness of biological network models through bifurcation and phase reduction methods.

Contribution

It provides a combined qualitative and quantitative stability analysis of a penta-rhythmic circuit with Fitzhugh-Nagumo oscillators, highlighting the relationship between node dynamics and circuit stability.

Findings

Circuit exhibits up to five stable rhythms depending on parameters.

Qualitative and quantitative stability of rhythms can diverge.

Robustness of rhythms assessed via phase diffusion under perturbations.

Abstract

Inhibitory circuits of relaxation oscillators are often-used models for the dynamics of biological networks. We present a qualitative and quantitative stability analysis of such a circuit constituted by three reciprocally coupled oscillators of a Fitzhugh-Nagumo type as nodes. Depending on inhibitory strengths, and parameters of individual oscillators, the circuit exhibits polyrhythmicity of up to five simultaneously stable rhythms. With methods of bifurcation analysis and phase reduction, we investigate qualitative changes in stability of these circuit rhythms for a wide range of parameters. Furthermore, we quantify how robustly rhythms are maintained under random perturbations by monitoring phase diffusion in the circuit. Our findings allow us to describe how circuit dynamics relate to dynamics of individual nodes. We also find that quantitative and qualitative stability of polyrhythmicity do not always align.