Pair of excitable FitzHugh-Nagumo elements: Synchronization, multistability, and chaos

Abstract

We analyze a pair of excitable FitzHugh-Nagumo elements, each of which is coupled repulsively. While the rest state for each element is globally stable for a phase-attractive coupling, various firing patterns, including cyclic and chaotic firing patterns, exist in an phase-repulsive coupling region. Although the rest state becomes linearly unstable via a Hopf bifurcation, periodic solutions associated to the firing patterns is not connected to the Hopf bifurcation. This means that the solution branch emanating from the Hopf bifurcation is subcritical and unstable for any coupling strength. Various types of cyclic firing patterns emerge suddenly through saddle-node bifurcations. The parameter region in which different periodic solutions coexist is also found.

Spike synchronization of chaotic oscillators as a phase transition

“…Here a novel feature emerges: while each individual system is chaotic and slightly perturbed by the input, the global array behaves as an excitable system, with a sharp transition from a steady high value of entropy to a low one. At variance with previous models which consist of arrays of excitable individuals, usually taken as FitzHugh-Nagumo systems (Fujii and Tsuda 2004;Yanagita et al 2005), here the excitable behaviour emerges as a collective property of the coupled chaotic array. To test the excitable property of an array, we show in Fig.…”

Stochastic resonance in FitzHugh–Nagumo system with time-delayed feedback

“…The FHN displays a rich phase diagram that include periodic oscillations, stable fixed points, excitability and chaotic behaviors when the parameters of the system are varied [43][44][45][46]. Different from the deterministic bifurcation, a bifurcation in stochastic systems refers mainly to qualitative changes in the stationary distribution of the variable x(t) [47,48].…”

Control of transient synchronization with external stimuli

“…The use of a column as a node is due to a recent results on chaotic spiking dynamics which can arise through an interaction of simple FHN units. 18 Moreover, it has been proposed in Ref. 19 that the network of coupled FHN neurons can lead to an appearance of heteroclinic connections between saddle regions ͑fixed points or limit cycles͒, giving a network the possibility for efficient encoding of inputs.…”