Effects of memristor-based coupling in the ensemble of FitzHugh–Nagumo elements

Abstract

In this paper, we study the impact of electrical and memristor-based couplings on some neuron-like spiking regimes, previously observed in the ensemble of two identical FitzHugh-Nagumo elements with chemical excitatory coupling. We demonstrate how increasing strength of these couplings affects on such stable periodic regimes as spiking in-phase, antiphase and sequential activity. We show that the presence of electrical and memristor-based coupling does not essentially affect regimes of in-phase activity. Such regimes do not changes remaining stable ones. However, it is not the case for regimes of anti-phase and sequential activity. All such regimes can transform into periodic or chaotic ones which are very similar to the regimes of in-phase activity. Concerning the regimes of sequential activity, this transformation depends continuously on the coupling parameters, whereas some anti-phase regimes can disappear via a saddle-node bifurcation and nearby orbits tend to regimes of in-phase activity. Also, we show that new interesting neuron-like phenomena can appear from the regimes of sequential activity when increasing the strength of electrical and/or memristor-based coupling. The corresponding regimes can be associated with the appearance of spiral attractors containing a saddle-focus equilibrium with homoclinic orbit and, thus, they correspond to chaotic motions near the invariant manifold of synchronization, which contains all in-phase limit cycles. Such new regimes can lead to the emergence of extreme events in the system of coupled neurons. In particular, the interspike intervals can become arbitrarily large when orbits (corresponding to this regime) pass very close to the saddle-focus. Finally, we show that the 1 arXiv:1903.05472v1 [nlin.CD] further increase in the strength of electrical coupling and/or memristorbased coupling leads to decreasing interspike intervals and, thus, it helps to avoid such extreme behavior.

Scenarios for the creation of hyperchaotic attractors in 3D maps

“…A similar scenario (but for a three-dimensional system of differential equations) was observed e.g. in [51].…”

Hopf Bifurcation Analysis for the Fitzhugh-Nagumo Model of a Spiking Neuron

“…together, we obtain again condition (14). Therefore, if there is Hopf bifurcation at E − and E + , it is a supercritical Hopf bifurcation.…”

Synchronization of Memristive FitzHugh-Nagumo Neural Networks

“…The researches on memristive FitzHugh-Nagumo and Hindmarsh-Rose neural networks in ordinary differential equations have been expanding in the recent decade, cf. [1,8,16,24,40] and many references therein. Various synchronization results with memristive effect of these models are achieved [10,13,19,22,31,32,35] mainly by the methods of generalized Hamiltonian functions, Lyapunov exponents, and the computational algebra with numerical simulations.…”