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

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 Bogdanov–Takens bifurcation indicates that the neuron signals can transit between resting and spiking.