Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Abstract

In this paper, Hopf bifurcation and Bogdanov-Takens bifurcation with codimension 2 in a coupled FitzHugh-Nagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov-Takens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov-Takens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of BogdanovTakens bifurcation indicates that the neuron signals can transit between resting and spiking.

Bifurcation structure of two coupled FHN neurons with delay

“…Thus the periodic solutions are due to coupling of the neurons, or time delay. Synchronized solutions in many works in the literature are caused by the symmetry of the proposed systems, like the study of Zhen and Xu [39], and also Li and Jiang [43]. In this paper, we show the existence of both almost synchronized and almost anti-phase solutions for non-identical neurons in different range of parameters.…”

“…Motivated by Lemma 2.1 of the work of Li and Jiang [43], we have the following conclusions on the eigenvalues of the system (1).…”

“…Zhen and Xu [42] studied Bautin bi-furcation of completely synchronous FHN neurons. Li and Jiang [43], investigated Hopf and Bogdanov-Takens bifurcations in a coupled FHN neural system with gap junction for identical neurons. In [44] it was shown that in a simple motif of two coupled FHN systems with two different delay times and with self-feedback, complex dynamics arise.…”

“…In this paper, we investigate the effect of the coupling strength and time delay on the stability and bifurcations of the system of two coupled FHN neurons. Note that neurons in the work of Zhen and Xu [39], and also Li and Jiang [43] are assumed identical which leads to a symmetry for their system of equations, that can not be found in our system. Instead the symmetry in our solutions is due to the inherent symmetry of the FHN neurons, namely our system of equations is equivariant under antipodal map.…”

Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

“…Different types of neural network systems with time delays have been proposed and developed. In these models, various types of dynamical behaviors including stability [8], chaos [9,10], Hopf bifurcation [11][12][13][14][15], global Hopf bifurcation [16,17], fold-Hopf bifurcation [18][19][20][21][22][23], Bogdanov-Takens bifurcation [24], and Hopf-Hopf bifurcation [25] were investigated. However, most of the above mentioned work focused only on the analysis of discrete time delays or distributed delays [26].…”