Qualitative and quantitative stability analysis of penta-rhythmic circuits

Abstract

Inhibitory circuits of relaxation oscillators are often-used models for dynamics of biological networks. We present a qualitative and quantitative stability analysis of such a circuit constituted by three generic oscillators (of a Fitzhugh–Nagumo type) as its nodes coupled reciprocally. 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 robustness of the rhythms 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 properties of polyrhythmicity do not always align.