Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

TL;DR

This study performs a detailed numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators, revealing complex dynamical behaviors, bifurcation structures, and multistability depending on coupling strength and parameters.

Contribution

It introduces a comprehensive numerical approach to analyze bifurcations and dynamical regimes in coupled FitzHugh-Nagumo neurons, including Arnold tongues and multistability.

Findings

Identification of Arnold tongues in unidirectionally coupled oscillators.

Connection between bifurcation curves and periodic structures.

Presence of multistability in the system's basins of attraction.

Abstract

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also present multistability shown in the planes of the basin of attractions.