Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

Abstract

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.

1. Introduction

The FitzHugh-Nagumo (FHN) neuron [1, 2], a simplified model of Hodgkin-Huxley neuron [3], describes the generation and propagation of nerve impulse in planar autonomous systems. In recent years, to understand information processing in the brain, the FHN neuron model is commonly used to study neural spiking due to its simplicity. A complete qualitative investigation of the FHN neuron was done by Bautin [4]. Further, a hard oscillation, separatrix loops, and bifurcations of equilibria and limit cycles can occur under suitable values of the parameters in this system [5].

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 [6–9], in which many rich bifurcation behaviors for equilibrium point and limit cycle are observed. Based on the finite propagating speed in the signal transmission between the neurons [10, 11], a coupled FHN neural system with time delays has been established: 𝑢̇₁=−𝑢₁𝑢₁−1𝑢₁−𝑎−𝑢₂+𝑐tanh𝑢₃𝑡−𝜏₁,𝑢̇₂=𝑏𝑢₁−𝛾𝑢₂,𝑢̇₃=−𝑢₃𝑢₃−1𝑢₃−𝑎−𝑢₄+𝑐tanh𝑢₁𝑡−𝜏₂,𝑢̇₄=𝑏𝑢₃−𝛾𝑢₄, (1) where 𝑎, 𝑏, and 𝛾 are the positive constants, 𝑢_1,3 represent the membrane potentials of neurons 1 and 2, 𝑢_2,4 are the slow refractory variables, which model the time dependence of several physical quantities, 𝑐 measures the coupling weight, and 𝜏_1,2>0 represent the time delays in signal transmission between the neurons.

The research for coupled FHN neural systems with time delay has attracted many authors’ attentions. Burić and Todorović [12] studied the Hopf bifurcation (inverse and direct) and fold bifurcation of limit cycle in the delay-coupling FHN neurons. Using the ODE (ordinary differential equation) to approximate the dynamics of the coupled FHN neurons with small delays, the codimension-2 generalized Hopf bifurcation was investigated [13]. The different synchronization states were observed with the variation of the coupling strength and time delay [14]. Their results illustrated that the patterns of exactly synchronous oscillations were dependent on the type of excitability and coupling. Wang et al. [15] presented that time delay can control the transition from the original chaotic motions to periodic ones in two coupled nonidentical FHN models with synaptic connection. Regarding the sum of two delays as a parameter, Fan and Hong [16] investigated the stability and local Hopf bifurcation in the synaptically coupled nonidentical FHN model. The simple zero singularity (both pitchfork and transcritical bifurcations) was studied employing the center manifold reduction and normal form method [17]. Further, by applying the Bautin bifurcation theorem, the coexistence of rest state and periodic spiking was reported in the synchronous solution of a coupled FHN neural system with delay [18].

Recently, some criteria to determine the periodic oscillation were provided in the multiple delayed FHN neural system with three nonidentical cells [19]. In [20], the Hopf bifurcation and Bogdanov-Takens bifurcation were investigated in a coupled FHN neural system with gap junction. The spatiotemporal patterns of bifurcating periodic solutions were considered by using the symmetric bifurcation theory. The neuron behavior can transit between resting and spiking. By the proportional-spatial derivative control approach, Yang et al. [21] achieved the control problem of the FHN equation. All works mentioned above have promoted greatly a deep understanding for dynamic behavior of the coupled FHN systems with time delay. However, in reality, the neural dynamics may be influenced by multiple independent parameters, such as external inputs [22, 23], time delays, and coupling weight [24, 25]. Their combined effects on dynamic behavior may be an important topic in the coupled FHN neural systems with multiple delays. The purpose of the present paper is to consider the combined effects of coupling weight and multiple delays on the stability of equilibrium point and obtain the stability switches in the coupled FHN neural system with multiple delays (1).

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 (1) may have one/three equilibrium points for the varying of the coupled weight by employing the saddle-node bifurcation of nontrivial equilibrium point and transcritical bifurcation of trivial one. In Section 3, the stability of equilibrium point for system (1) is investigated by analyzing 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 involving the delay-independence stability and delay-dependence stability. Increasing time delay can induce the stability switches between resting state and periodic activity in some parameter regions of coupling weight. In Section 4, some numerical simulations are employed to support the theoretical results. Conclusions are given in Section 5.

2. Equilibrium Point Analysis

It is obvious that (𝑢₁,𝑢₂,𝑢₃,𝑢₄)=(0,0,0,0) is the trivial equilibrium point of system (1). Linearizing system (1) near the point (0,0,0,0) produces the following linearized system: 𝑢̇₁=−𝑎𝑢₁−𝑢₂+𝑐𝑢₃𝑡−𝜏₁,𝑢̇₂=𝑏𝑢₁−𝑏𝛾𝑢₂,𝑢̇₃=−𝑎𝑢₃−𝑢₄+𝑐𝑢₁𝑡−𝜏₂,𝑢̇₄=𝑏𝑢₃−𝑏𝛾𝑢₄. (2) The corresponding characteristic equation is 𝐷𝜆,𝜏₁,𝜏₂,𝑐=𝜆⁴+𝑑₁𝜆³+𝑑₂𝜆²+𝑑₃𝜆+𝑑₄=0, (3) where 𝑑₁𝑑=2𝑎+2𝑏𝛾,₂=𝑎²+2𝑏−𝑐²e^{−𝜆𝜏1−𝜆𝜏2}+4𝑎𝑏𝛾+𝑏²𝛾²,𝑑₃=2𝑎𝑏+2𝑎²𝑏𝛾+2𝑏²𝛾−2𝑏𝑐²e^{−𝜆𝜏1−𝜆𝜏2}𝛾+2𝑎𝑏²𝛾²,𝑑₄=𝑏²+2𝑎𝑏²𝛾+𝑎²𝑏²𝛾²−𝑏²𝑐²e^{−𝜆𝜏1−𝜆𝜏2}𝛾². (4) Then, we have the following conclusion on the eigenvalues of system (1).

Theorem 1. System (1) has a zero eigenvalue if and only if 𝑐=±1+𝑎𝛾𝛾,𝜏₁≠2−𝑏𝛾𝜏₂+𝛾2+𝑎𝜏₂.𝑏𝛾1+𝑎𝛾 (5) Further, system (1) exhibits a double zero eigenvalue if and only if 𝑐=±1+𝑎𝛾𝛾,𝜏₁=2−𝑏𝛾𝜏₂+𝛾2+𝑎𝜏₂,𝑏𝛾1+𝑎𝛾𝑎≠−1−2𝑏𝛾²+𝑏²𝛾⁴.2𝛾 (6)

Proof. 𝜆=0 is a zero eigenvalue of system (1) if and only if 𝑏²+2𝑎𝑏²𝛾+𝑎²𝑏²𝛾²−𝑏²𝑐²𝛾²=0,𝜕𝐷(𝜆,𝑐)𝜕𝜆_𝜆=0=2𝑎𝑏+2𝑎²𝑏𝛾+2𝑏²𝛾−2𝑏𝑐²𝛾+2𝑎𝑏²𝛾²+𝑏²𝑐²𝛾²𝜏₁+𝜏₂≠0. (7) Further, 𝜆=0 is a double root of system (1) if and only if 𝑐=±1+𝑎𝛾𝛾,𝜏₁=2−𝑏𝛾𝜏₂+𝛾2+𝑎𝜏₂,𝜕𝑏𝛾1+𝑎𝛾₂𝐷(𝜆,𝑐)𝜕𝜆²_𝜆=0=2+4𝑎𝛾+4𝑏𝛾²−2𝑏²𝛾⁴𝛾²≠0. (8)

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 (1) are the solution of the following equations: 𝛾𝑢₂𝛾𝑢₂−1𝛾𝑢₂−𝑎+𝑢₂−𝑐tanh𝛾𝑢₄=0,𝛾𝑢₄𝛾𝑢₄−1𝛾𝑢₄−𝑎+𝑢₄−𝑐tanh𝛾𝑢₂=0. (9)

Solutions of (9), that is, equilibrium points of system (1), depend on the parameters 𝑎, 𝑐, and 𝛾 but are independent of the delays 𝜏₁ and 𝜏₂. Geometrically, equilibrium points are the intersection points of two curves given by (9), as shown in Figure 1. These two curves are called the nullclines of system (1). Thus, it is seen that system (1) exhibits either one or three equilibriums for the different coupling weight 𝑐. In fact, for the small value of the coupling weight 𝑐, the system has just the trivial equilibrium point (0,0), as shown in Figure 1(a). With the increasing of the coupling weight 𝑐, two nontrivial equilibrium points are emerged immediately at the right side of the trivial point (0,0) (Figure 1(b)), which is the saddle-node bifurcation of the nontrivial equilibrium point. Further, the trivial equilibrium point (0,0) evolves into somewhere between the nontrivial ones, as shown in Figure 1(c). It is called the transcritical bifurcation of the trivial point. Figure 2 shows the corresponding one-dimension bifurcation diagram in the (𝑐,𝑢₁)-plane (Figure 2(a)) and (𝑐,𝑢₂)-plane (Figure 2(b)).

(a)

(b)

(c)

Figure 1 

Intersection points of two curves given by (9), showing the number of equilibriums: (a) unique equilibrium for 𝑐=2.0, (b) three equilibriums for 𝑐=2.3, and (c) three equilibriums for 𝑐=3.0, where the system parameter values are fixed as 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

(a)

(b)

Figure 2 

The one-dimensional bifurcation diagrams in the (a) (𝑐,𝑢1)-plane and (b) (𝑐,𝑢2)-plane for the fixed parameters 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

3. Stability Switches and Hopf Bifurcation

It is well known that the equilibrium point is locally asymptotically stable if and only if each eigenvalue of the characteristic equation (3) has the negative real part. In the following, we investigate the stability of the equilibrium point by obtaining the eigenvalues with the maximum real part. The FHN neural system exhibits the parameter regions involving the delay-independence stability and delay-dependence stability. The multiple delays can induce the stability switches between resting state and periodic activity in some parameter regions of coupling weight. To simplify, the investigation begins with the case 𝜏₁=𝜏₂=0 in (3) as follows: 𝐷(𝜆,0,0,𝑐)=𝜆⁴+𝑚₃𝜆³+𝑚₂𝜆²+𝑚₁𝜆+𝑚₀=0, (10) where 𝑚₃𝑚=2𝑎+2𝑏𝛾,₂=𝑎²+2𝑏−𝑐²+4𝑎𝑏𝛾+𝑏²𝛾²,𝑚₁=2𝑎𝑏+2𝑎²𝑏𝛾+2𝑏²𝛾−2𝑏𝑐²𝛾+2𝑎𝑏²𝛾²,𝑚₀=𝑏²+2𝑎𝑏²𝛾+𝑎²𝑏²𝛾²−𝑏²𝑐²𝛾². (11) It follows from the Routh-Hurwitz criterion that the necessary and sufficient conditions for all roots of (10) having negative real parts are given by 𝑚₃>0,𝑚₀>0,𝑚₂𝑚₃−𝑚₁𝑚>0,₁𝑚₂𝑚₃−𝑚²₁−𝑚₀𝑚²₃>0. (12) That is (i) 1+𝑎𝛾>𝑐𝛾 and 1+𝑏𝛾>𝑐,

or(ii)1+𝑎𝛾>𝑐𝛾, 𝑎²+𝑎³𝛾+2𝑎𝑏𝛾+2𝑎²𝑏𝛾+𝑏²𝛾²+𝑎𝑏²𝛾³<𝑎𝑐²𝛾, and 𝑎³+𝑎𝑏+4𝑎²𝑏𝛾+𝑏²𝛾²+4𝑎𝑏²𝛾³+𝑏³𝛾³>𝑎𝑐².

Based on the dynamical theory, we have the following.

Theorem 2. 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 𝜏₁, the trivial equilibrium point of system (1) will lose its stability. The neural system may exhibit the periodic activity. To obtain such critical values, supposing 𝜆=±𝑖𝑣(𝑣>0) is a pair of purely imaginary roots of the characteristic equation (3) with the case 𝜏₂=0, one has 𝐷𝑖𝑣,𝜏₁,0,𝑐=𝑣⁴+𝑑₃𝑣³+𝑑₂𝑣²+𝑑₁𝑣+𝑑₀=0, (13) where 𝑑₃𝑑=−2𝑖𝑎−2𝑖𝑏𝛾,₂=𝑐²e^{−𝑖𝑣𝜏1}−𝑎²−2𝑏−4𝑎𝑏𝛾−𝑏²𝛾²,𝑑₁=2𝑖𝑎𝑏−2𝑖𝑏𝑐²𝛾e^{−𝑖𝑣𝜏1}+2𝑖𝑎²𝑏𝛾+2𝑖𝑏²𝛾+2𝑖𝑎𝑏²𝛾²,𝑑₀=𝑏²+2𝑎𝑏²𝛾−𝑏²𝑐²𝛾²e^{−𝑖𝑣𝜏1}+𝑎²𝑏²𝛾². (14) That is 𝑣⁴−2𝑖𝑎+𝑏𝛾𝑣³−4𝑎𝑏𝛾+𝑏²𝛾²+𝑎²+2𝑏𝑣²+2𝑖𝑏𝑎+𝑎²𝛾+𝑎𝑏𝛾²𝑣+𝑎+𝑏𝛾₂𝛾²+1+2𝑎𝛾𝑏²+𝑣²−2𝑖𝑏𝑣𝛾−𝑏²𝛾²𝑐²cos𝑣𝜏₁−𝑖𝑣²+2𝑏𝑣𝛾−𝑖𝑏²𝛾²𝑐²sin𝑣𝜏₁=0. (15) Separating (15) into the real and imaginary parts yields 𝑏²−𝑎²𝑣²−2𝑏𝑣²+𝑣⁴+2𝑎𝑏²𝛾−4𝑎𝑏𝑣²𝛾+𝑎²𝑏²𝛾²−𝑏²𝑣²𝛾²+𝑐²𝑣²−𝑏²𝑐²𝛾²cos𝑣𝜏₁−2𝑏𝑐²𝑣𝛾sin𝑣𝜏₁=0,2𝑎𝑏𝑣−2𝑎𝑣³+2𝑎²𝑏𝑣𝛾+2𝑏²𝑣𝛾−2𝑏𝑣³𝛾+2𝑎𝑏²𝑣𝛾²−2𝑏𝑐²𝑣𝛾cos𝑣𝜏₁+𝑏²𝑐²𝛾²−𝑐²𝑣²sin𝑣𝜏₁=0. (16) Eliminating 𝜏₁ from (16) using sin²𝑣𝜏₁+cos²𝑣𝜏₁=1, we have 𝐿(𝑣)=𝑣⁸+𝑛₁𝑣⁶+𝑛₂𝑣⁴+𝑛₃𝑣²+𝑛₄=0, (17) where 𝑛₁=2𝑎²−4𝑏+2𝑏²𝛾²,𝑛₂=𝑎⁴−4𝑎²𝑏+6𝑏²−𝑐²−𝑐⁴+4𝑎𝑏²𝛾+4𝑎²𝑏²𝛾²−4𝑏³𝛾²+𝑏⁴𝛾⁴,𝑛₃=2𝑎²𝑏²−4𝑏³+4𝑎³𝑏²𝛾−8𝑎𝑏³𝛾+2𝑎⁴𝑏²𝛾²−4𝑎²𝑏³𝛾²+2𝑏⁴𝛾²−2𝑏²𝑐²𝛾²−2𝑏²𝑐⁴𝛾²+4𝑎𝑏⁴𝛾³+2𝑎²𝑏⁴𝛾⁴,𝑛₄=𝑏⁴+4𝑎𝑏⁴𝛾+6𝑎²𝑏⁴𝛾²+4𝑎³𝑏⁴𝛾³+𝑎⁴𝑏⁴𝛾⁴−𝑏⁴𝑐²𝛾⁴−𝑏⁴𝑐⁴𝛾⁴. (18) Based on the dynamical theory, the equilibrium point is locally asymptotically stable if and only if each of eigenvalues has the negative real part. Therefore, one can obtain the effect of delay 𝜏₁ on the eigenvalue of characteristic equation (13).

Theorem 3. If the parameters values of system (1) are satisfied with one of the conditions (i) and (ii), the following statements are true. (a)When the polynomial 𝐿(𝑣) has no positive root, all eigenvalues of the characteristic equation (13) have the negative real parts for the arbitrary 𝜏₁. The trivial equilibrium point exhibits the delay-independent stability.(b)When the polynomial 𝐿(𝑣) has only one positive and simple root ν, there exists the critical delayed value 𝜏^𝑐₁>0 such that all the eigenvalues of the characteristic equation (13) have negative real parts for 𝜏₁∈(0,𝜏^𝑐₁) and at least one root has a positive real part for 𝜏₁>𝜏^𝑐₁. The FHN neural system exhibits the periodic activity when delay 𝜏₁ increases and passes through the critical delayed value 𝜏^𝑐₁.(c)When the polynomial 𝐿(𝑣) has at least two positive and simple roots 0<𝑣₁<𝑣₂<⋯, there exist a finite number of the delayed 𝜏₁ intervals in which all eigenvalues of the characteristic equation (13) have negative real parts. The system exhibits the stability switches between resting state and periodic activity.

In order to investigate the combined effects of multiple delays on the local stability of neural system (1), we regard 𝜏₂ as the varying parameter for any fixed delay 𝜏₁=𝜏^∗₁. Letting 𝜆=𝑖𝜔 be the simple roots of the characteristic equation (3), one obtains 𝐷𝑖𝜔,𝜏^∗₁,𝜏₂,𝑐=𝜔⁴+𝑘₃𝜔³+𝑘₂𝜔²+𝑘₁𝜔+𝑘₀=0, (19) where 𝑘₃𝑘=−2𝑖𝑎−2𝑖𝑏𝛾,₂=−𝑎²−2𝑏+𝑐²e^{−𝑖𝜔𝜏∗1}e^{−𝑖𝜔𝜏2}−4𝑎𝑏𝛾−𝑏²𝛾²,𝑘₁=2𝑖𝑎𝑏+2𝑖𝑎²𝑏𝛾+2𝑖𝑏²𝛾−2𝑖𝑏𝑐²e^{−𝑖𝜔𝜏∗1}e^{−𝑖𝜔𝜏2}𝛾+2𝑖𝑎𝑏²𝛾²,𝑘₀=𝑏²+2𝑎𝑏²𝛾+𝑎²𝑏²𝛾²−𝑏²𝑐²e^{−𝑖𝜔𝜏∗1}e^{−𝑖𝜔𝜏2}𝛾². (20) That is 𝑏²+2𝑎𝑏²𝛾+𝑎²𝑏²𝛾²−𝑎²+2𝑏+4𝑎𝑏𝛾+𝑏²𝛾²𝜔²+𝜔⁴+𝑐²𝜔²−𝑏²𝑐²𝛾²cos𝜔𝜏₂cos𝜔𝜏^∗₁−2𝑏𝑐²𝛾𝜔cos𝜔𝜏^∗₁sin𝜔𝜏₂−2𝑏𝑐²𝛾𝜔cos𝜔𝜏₂sin𝜔𝜏^∗₁+𝑏²𝑐²𝛾²−𝑐²𝜔²sin𝜔𝜏₂sin𝜔𝜏^∗₁=0,2𝑎𝑏+2𝑎²𝑏𝛾+2𝑏²𝛾+2𝑎𝑏²𝛾²𝜔−2𝑎+𝑏𝛾𝜔³−2𝑏𝑐²𝛾𝜔cos𝜔𝜏₂cos𝜔𝜏^∗₁+𝑏²𝑐²𝛾²−𝑐²𝜔²cos𝜔𝜏^∗₁sin𝜔𝜏₂+𝑏²𝑐²𝛾²−𝑐²𝜔²cos𝜏₂𝜔sin𝜔𝜏^∗₁+2𝑏𝑐²𝛾𝜔sin𝜏₂𝜔sin𝜔𝜏^∗₁=0. (21) Eliminating 𝜏₂ from (21), one has cos𝜔𝜏₂=𝑓₁𝜏^∗₁,𝜔𝑓₃𝜏^∗₁,,𝜔sin𝜔𝜏₂=𝑓₂𝜏^∗₁,𝜔𝑓₃𝜏^∗₁,,𝜔 (22) where 𝑓₁𝜏^∗₁=()𝜔,𝜔𝑏𝜔+𝑎−𝜔²+𝑏²𝛾×1+𝑎𝛾−𝛾𝜔−𝑏𝜔+𝜔²(𝑎+𝜔)+𝑏²𝛾1+𝑎𝛾+𝜔𝛾×cos𝜔𝜏^∗₁−2𝜔𝑏−1+𝑏𝛾²+𝜔²×𝑏²𝛾1+𝑎𝛾+𝑎𝜔²sin𝜔𝜏^∗₁,𝑓₂𝜏^∗₁𝑏,𝜔=−2𝜔−1+𝑏𝛾²+𝜔²×𝑏²𝛾1+𝑎𝛾+𝑎𝜔²cos𝜔𝜏^∗₁+𝑏𝜔+(𝑎−𝜔)𝜔²+𝑏²𝛾×1+𝑎𝛾−𝛾𝜔−𝑏𝜔+𝜔²(𝑎+𝜔)+𝑏²𝛾1+𝑎𝛾+𝜔𝛾sin𝜔𝜏^∗₁,𝑓₃𝜏^∗₁,𝜔=𝑐²𝑏²𝛾²+𝜔²². (23) This implies that 𝐺𝜏^∗₁,𝜔=𝑓₁𝜏^∗₁,𝜔²+𝑓₂𝜏^∗₁,𝜔²−𝑓₃𝜏^∗₁,𝜔²=0. (24) If (24) has a number of positive and simple roots 𝜔_𝑖,𝑖=1,2,…, (3) has the following critical delays determined by (24): 𝜏_𝑖,𝑗=𝜑_𝑖+2𝑗𝜋𝜔_𝑖,𝑖=1,2,…;𝑗=0,1,2,…, (25) where 𝜑_𝑖∈[0,2𝜋) and is satisfied with cos𝜑_𝑖=𝑓₁𝜏^∗₁,𝜔_𝑖𝑓₃𝜏^∗₁,𝜔_𝑖,sin𝜑_𝑖=𝑓₂𝜏^∗₁,𝜔_𝑖𝑓₃𝜏^∗₁,𝜔_𝑖. (26) We determine whether the eigenvalue real part increases or decreases as delay 𝜏₂ crosses the critical values 𝜏_𝑖,𝑗,𝑖=1,2,…;𝑗=0,1,2,…. Differentiating 𝜆 with respect to 𝜏₂ in (3) reaches 𝜆^′𝜏₂=𝜆²e^{2𝜆𝜏∗1}𝑐²𝑏𝛾+𝜆²𝑔𝜏₂,,𝜆 (27) where 𝑔𝜏₂,𝜆=2e^{𝜆(𝜏∗1+𝜏2)}×𝜆()𝑎+𝑏𝛾+2𝜆𝑎+𝜆+𝑏1+𝑎𝛾+𝜆𝛾+𝑐²×𝜏𝑏𝛾+𝜆−2+𝑏𝛾+𝜆^∗₁+𝜏₂. (28) Based on the transversality condition (27) and the Hopf bifurcation theorem of delay differential equations, one has the following theorem.

Theorem 4. The following assertions are true if all roots of (13) have negative real parts. (a)If 𝐺(𝜏^∗₁,𝜔)=0 has no positive root, the trivial equilibrium point of system (1) is asymptotically stable for an arbitrary time delay 𝜏₂, which is called the delay 𝜏₂-independent stability.(b)If 𝐺(𝜏^∗₁,𝜔)=0 has at least one positive and simple root 𝜔_𝑐, there exists exactly a critical delay 𝜏^𝑐₂>0 such that the trivial equilibrium of system (1) is asymptotically stable for 𝜏₂∈[0,𝜏^𝑐₂). Furthermore, if the transversality condition Re(𝜆^′(𝜏₂))≠0 holds, system (1) undergoes a Hopf bifurcation for 𝜏₂=𝜏^𝑐₂. That is, it exhibits a periodic activity bifurcating from the trivial equilibrium near 𝜏^𝑐₂.(c)If 𝐺(𝜏^∗₁,𝜔)=0 has at least two positive and simple roots 0<𝜔₁<𝜔₂<⋯ and the transversality condition Re(𝜆^′(𝜏₂))≠0 holds, there exist a finite number of intervals. If time delay is fixed into these intervals, the trivial equilibrium point is locally asymptotically stable, while unstable if delay does not belong to ones. The rest sate of neural system switches from stable to unstable, and then back to stable when time delay increases and crosses the critical values in sequence.

4. Numerical Simulation

In this section, some numerical results of system (1) are presented for the different parameter values. Using the method of numerical simulation, we find that the theoretically predicted values are in excellent agreement with the numerical behaviors. For simplicity, the system parameters are fixed as 𝑎=0.33, 𝑏=1, and 𝛾=0.47. The coupling weight 𝑐 and time delays (𝜏₁,𝜏₂) are considered as the variable parameters.

Firstly, we fix the time delay 𝜏₂=0 and illustrate the dynamic behavior of system (1) for the different values of the coupling weight 𝑐 and time delay 𝜏₁. It follows from (17) that the polynomial 𝐿(𝑣) has no positive root (see Figure 3(a)) for the fixed coupling weight 𝑐=0.5. The trivial equilibrium point is locally asymptotically stable for the arbitrary delay 𝜏₁, which is called the delay 𝜏₁-independent stability. The corresponding eigenvalue has the negative real parts for the delay 𝜏₁ variation, as shown in Figure 3(b). The time histories with the fixed delay are shown in Figure 4(a) for 𝜏₁=1.0 and Figure 4(b) for 𝜏₁=15.0; both converge to the equilibrium point (0,0,0,0) spirally.

(a)

(b)

(c)

(d)

Figure 3 

Roofs of function 𝐿 ((a) and (c)) and eigenvalue real parts with 𝜏1 varying ((b) and (d)) for the different coupling weights (a)-(b) 𝑐=0.5 and (c)-(d) 𝑐=0.8. The other parameters are chosen as 𝑎=0.33, 𝑏=1, 𝛾=0.47, and 𝜏2=0.

(a)

(b)

Figure 4 

Time histories with the varying delay 𝜏1 (a) 𝜏1=1.0 and (b) 𝜏1=15.0 when the time delay and coupling weight are fixed as 𝜏2=0, 𝑐=0.5, where the other system parameters have the fixed values 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

However, when the coupling weight is fixed as 𝑐=0.8, the equation 𝐿(𝑣)=0 has two positive and simple roots 𝑣₁=0.8826, 𝑣₂=1.2372, as shown in Figure 3(c). The corresponding eigenvalue with 𝜏₁ varying is shown in Figure 3(d). There exist some critical delays determined by (25). From Theorem 3, system (1) exhibits the stability switches with the delay 𝜏₁ increasing. In fact, the trivial equilibrium point is asymptotically stable when 𝜏₁∈(0,3.9045). Figure 5(a) illustrates the time history for the fixed delay 𝜏₁=1.0. When 𝜏₁∈(3.9045,7.1184), the trivial equilibrium point loses its stability and evolves into the instability. The time history for 𝜏₁=5.0 is shown in Figure 5(b). Further, system (1) obtains the resting state when delay increases and crosses the critical value 𝜏₁=7.1184. The time history for the fixed delay 𝜏₁=8.0 is illustrated in Figure 5(c). Finally, the equilibrium point loses its stability when the delay 𝜏₁ continues to increase, which is shown in Figure 5(d) for the fixed delay 𝜏₁=10.0. Figure 6 shows the corresponding one-dimension bifurcation diagrams for the varying delay 𝜏₁. It supports the above theoretical results. Neural system (1) exhibits the multiple switches between resting state and periodic spiking. Further, it is worth mentioning that the neural system (1) exhibits the transient chaos (see Figure 7(a)), which implies that the system trajectory exhibits a seemingly chaotic solution for a longer time (see Figure 7(b)) but evolves into a final nonchaotic state, such as a periodic activity (see Figure 7(c)) or equilibrium eventually.

(a)

(b)

(c)

(d)

Figure 5 

Time histories with the varying delay 𝜏1 (a) 𝜏1=1.0, (b) 𝜏1=5.0, (c) 𝜏1=8.0, and (d) 𝜏1=10.0 when the delay 𝜏2 and coupled weight are fixed as 𝜏2=0, 𝑐=0.8, where the other system parameters have the fixed values 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

(a)

(b)

(c)

(d)

Figure 6 

The one-dimensional bifurcation diagram in (a) (𝜏1,𝑢1)-plane, (b) (𝜏1,𝑢2)-plane, (c) (𝜏1,𝑢3)-plane, and (d) (𝜏1,𝑢4)-plane for the fixed delay 𝜏2=0 and coupling weights 𝑐=0.8.

(a)

(b)

(c)

Figure 7 

Time histories of the transient chaos for (a) overall view, (b) transient behavior, and (c) long-term behavior for the fixed delay 𝜏1=14.15 when 𝜏2=0 and 𝑐=0.8.

The partial eigenvalues are exhibited in Figure 8 for the different delay 𝜏₁ to illustrate the stability switches between resting state and periodic spiking. Figure 8(a) shows that the maximum eigenvalues are a conjugate pair with negative part given by −0.0888±0.7515𝑖 for the fixed delay 𝜏₁=1.0. The neural system exhibits the resting state. Increasing delay 𝜏₁ yields that the conjugate pair passes through the imaginary axis and go into the right-half plane. For 𝜏₁=5.0, the pair becomes to be 0.0187±1.0846𝑖 shown in Figure 8(b). The trivial equilibrium point loses its stability. The FHN neural system exhibits the periodic spiking. Continuing to increase delay 𝜏₁, the conjugate pair returns and passes through the imaginary axis again. The maximum eigenvalues with negative real parts occurs in the left-hand plane. Figure 8(c) shows the eigenvalues are −0.0133±0.8189𝑖 for the fixed delay 𝜏₁=8.0. The neural system obtains again the resting state. Further, if delay 𝜏₁ is fixed as 𝜏₁=10.0, the maximum eigenvalues are the conjugate pair with positive part 9.0902×10⁻³±1.1452𝑖, as shown in Figure 8(d). The system (1) exhibits the periodic spiking. In a word, the increase of time delay can induce the stability switches between resting state and periodic activity on the neural system (1).

(a)

(b)

(c)

(d)

Figure 8 

Distribution of partial eigenvalues for the fixed coupling weight 𝑐=0.8 and time delay 𝜏2=0, where the asterisk in green represents the eigenvalues with negative real part and one with positive real part is in red color. The other parameters are fixed as 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

Furthermore, we fix the time delay 𝜏₁=𝜏^∗₁=0.5 and find the combined effects of delay 𝜏₂ and coupling weight 𝑐 on the dynamic behavior of neural system (1). The figures of function 𝐺 and eigenvalues real parts are displayed in Figure 9 for the different coupling weights 𝑐. Figure 9(a) shows that the curve 𝐺 with 𝑐=0.2. There is no intersection point when 𝜔>0 holds. It implies that the function 𝐺 has no positive root. The real parts of all eigenvalues have the negative part, as shown in Figure 9(b). The equilibrium point of system (1) is locally asymptotically stable for the arbitrary delay 𝜏₂. It is called the delay 𝜏₂-independent stability. With the coupling weight 𝑐 increasing to 0.8, the curve determined by function 𝐺 has two positive roots shown in Figure 9(c). It follows from Figure 9(d) that there exist the delay 𝜏₂ intervals in which the eigenvalues have the negative parts. The trivial equilibrium point switches from stable to unstable, and then back to stable state with delay 𝜏₂ increasing. The neural system exhibits the stability switches between resting state and periodic spiking. When coupling weight 𝑐 is increased to 1.8, function 𝐺 has two pairs of positive roots, as shown in Figure 9(e). However, at this time, the maximum eigenvalues of the characteristic equation (13) for the fixed delay 𝜏₂=0 have the positive real part Re(𝜆)=0.2787. The equilibrium point of system (1) is instable for any delay 𝜏₂, as shown in Figure 9(f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9 

Function 𝐺 curve ((a), (c), and (e)) and eigenvalues real parts with 𝜏2 varying ((b), (d), and (f)) for the different coupling weights (a)-(b) 𝑐=0.2, (c)-(d) 𝑐=0.8, and (e)-(f) 𝑐=1.8. The other parameters are fixed as 𝑎=0.33, 𝑏=1, 𝛾=0.47, and 𝜏1=0.5.

Additionally, it follows from Figure 10 that the time delay has the different effect on the periodic activity in the FHN neural system (1). When the coupling weight and time delay are chosen as 𝑐=1.0, 𝜏₁=1.0, the eigenvalues of the maximum real parts are changed from negative value to positive ones, as shown in Figure 10(a). It implies that the system behavior will evolve from the resting state into periodic spiking. That is to say, time delay promotes the occurrence of the periodic activity in the delay-coupling FHN neural system. However, for the parameter values of 𝑐=1.2 and 𝜏₁=0.1, the system has the periodic activity with the nondelay 𝜏₂=0. Increasing delay 𝜏₂, the eigenvalue of the maximum real parts exhibits the negative value, as shown in Figure 10(b). It implies that the trivial equilibrium point obtains the stability by the increasing delay 𝜏₂. Time delay suppresses the system dynamic from periodic activity into resting state, which can be used to control the neural system.

(a)

(b)

Figure 10 

Eigenvalues of the maximum real parts show that the delay 𝜏2 increasing promotes the system dynamic from resting state into periodic spiking for 𝑐=1.0, 𝜏1=1.0 (a) and suppresses ones from periodic activity into resting state for 𝑐=1.2, 𝜏1=0.1 (b), where the other parameters are fixed as 𝑎=0.33, 𝑏=1, and 𝛾=0.47.

5. Conclusion

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.

Conflict of Interests

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

Acknowledgments

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