Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

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.

Characterizing the dynamics of three FitzHugh-Nagumo neurons network

“…The case for n = 2 was recently published elsewhere [7]. In the bidirectional coupling the neurons interact mutually, all to all system, while for unidirectional coupling the neuron i interacts only with the neuron i + 1.…”

“…[7]). To observe hyperchaotic behavior in the Lyapunov diagrams, we plot the parameter-space with the second largest exponent of the LEs spectrum.…”

“…This procedure allows us to identify regions of periodic, chaotic, and hyperchaotic behavior, and recently it is applied in several models. A general feature is observed in these parameter-spaces, the existence of shrimp-shaped periodic structures embedded in chaotic domains [7,14]. These structures are stable periodic domains, i.e., inside them the system variables oscillate periodically with a well-defined period, and often bordered by chaotic regions.…”

“…More recently, a growing interest is in the study of the dynamics on the parameter-spaces of the neural system models [7,16]. The parameter-spaces are colorful diagrams when two parameters of the model are varied and the other are kept constant.…”

“…The colors codify some measure that can be computed on the model. Usually, this measure is the Lyapunov exponent [1,3,7,9,13,14], periods [7,14], or other invariant measure [3]. This procedure allows us to identify regions of periodic, chaotic, and hyperchaotic behavior, and recently it is applied in several models.…”