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A FitzHugh-Nagumo (FHN) neural system with multiple delays has been proposed. The number of equilibrium point is analyzed. It implies that the neural system exhibits a unique equilibrium and three ones for the different values of coupling weight by employing the saddle-node bifurcation of nontrivial equilibrium point and transcritical bifurcation of trivial one. Further, the stability of equilibrium point is studied by analyzing the corresponding characteristic equation. Some stability criteria involving the multiple delays and coupling weight are obtained. The results show that the neural system exhibits the delay-independence and delay-dependence stability. Increasing delay induces the stability switching between resting state and periodic activity in some parameter regions of coupling weight. Finally, numerical simulations are taken to support the theoretical results.

The FitzHugh-Nagumo (FHN) neuron [

To understand the coupling effect and information transmission between neuron systems, the analysis of the dynamic behavior in coupled FHN neural systems has been the subject of many papers [

The research for coupled FHN neural systems with time delay has attracted many authors’ attentions. Burić and Todorović [

Recently, some criteria to determine the periodic oscillation were provided in the multiple delayed FHN neural system with three nonidentical cells [

The paper is organized as follows. In the next section, we study the number of equilibrium points in the coupled FHN neural system employing the static bifurcation. The neural system (

It is obvious that

System (

From the dynamic theory, when an eigenvalue passes through the imaginary axis along the real axis with the variation of system parameter, a static bifurcation will be exhibited, which results in the variation of the number of equilibrium points. In fact, all equilibrium points of system (

Solutions of (

Intersection points of two curves given by (

The one-dimensional bifurcation diagrams in the (a)

It is well known that the equilibrium point is locally asymptotically stable if and only if each eigenvalue of the characteristic equation (

or

Based on the dynamical theory, we have the following.

If the system parameters are satisfied with one of the conditions (i) and (ii), the trivial equilibrium point is locally asymptotically stable for the FHN neural system model without any time delays.

With the variation of delay

If the parameters values of system (

When the polynomial

When the polynomial

When the polynomial

In order to investigate the combined effects of multiple delays on the local stability of neural system (

The following assertions are true if all roots of (

If

If

If

In this section, some numerical results of system (

Firstly, we fix the time delay

Roofs of function

Time histories with the varying delay

However, when the coupling weight is fixed as

Time histories with the varying delay

The one-dimensional bifurcation diagram in (a)

Time histories of the transient chaos for (a) overall view, (b) transient behavior, and (c) long-term behavior for the fixed delay

The partial eigenvalues are exhibited in Figure

Distribution of partial eigenvalues for the fixed coupling weight

Furthermore, we fix the time delay

Function

Additionally, it follows from Figure

Eigenvalues of the maximum real parts show that the delay

Time delay is an inevitable factor in the signal transmission between neurons. The neural system with time delay exhibits the rich dynamical behaviors. In this paper, a coupled FHN neural system with two delays has been proposed. The analyses of the number of equilibrium points illustrate that the neural system has a unique equilibrium and three equilibria for the different values of coupling weights. It exhibits the multiple equilibrium points employing the saddle-node bifurcation of nontrivial equilibrium point and the transcritical bifurcation of the trivial point. Further, the stability of equilibrium point is analyzed employing the corresponding characteristic equation. Some stability criteria involving the multiple delays and coupling weight are obtained. The results show that the FHN neural system exhibits the parameter regions involved the delay-independence stability and delay-dependence stability. Time delay increasing can induce the stability switches between resting state and periodic activity. Finally, numerical simulations are taken to support the theoretical results.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments.