Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

Abstract

The existence of periodic oscillation for a coupled FHN neural system with delays is investigated. Some criteria to determine the oscillations are given. Simple and practical criteria for selecting the parameters in this network are provided. Some examples are also presented to illustrate the result.

1. Introduction

Recently, several researchers have studied the dynamics of coupled FHN neural systems [1–5]. Wang et al. have investigated the following model [6]: 𝑣^′₁(𝑡)=−𝑣³₁(𝑡)+𝑎𝑣₁(𝑡)−𝑤₁(𝑡)+𝑐₁tanh𝑣₂(),𝑤𝑡−𝜏_′1(𝑡)=𝑣₁(𝑡)−𝑏₁𝑤₁(𝑡),𝑣^′₂(𝑡)=−𝑣³₂(𝑡)+𝑎𝑣₂(𝑡)−𝑤₂(𝑡)+𝑐₂tanh𝑣₁(),𝑤𝑡−𝜏_′2(𝑡)=𝑣₂(𝑡)−𝑏₂𝑤₂(𝑡). (1)

The effects of time delay on bifurcation and synchronization in the two synaptically coupled FHN neurons have been investigated. The authors found that time delay can control the occurrence of bifurcation in the coupled FHN neural model and synchronization is sometimes related to bifurcation transition. Fan and Hong introduced second time delay in model (1) as follows [7]: 𝑥^′₁(𝑡)=−𝑥³₁(𝑡)+𝑎𝑥₁(𝑡)−𝑥₂(𝑡)+𝑐₁tanh𝑥₃𝑡−𝜏₁,𝑥^′₂(𝑡)=𝑥₁(𝑡)−𝑏₁𝑥₂(𝑡),𝑥^′₃(𝑡)=−𝑥³₃(𝑡)+𝑎𝑥₃(𝑡)−𝑥₄(𝑡)+𝑐₂tanh𝑥₁𝑡−𝜏₂,𝑥^′₄(𝑡)=𝑥₃(𝑡)−𝑏₂𝑥₄(𝑡). (2)

Let𝜏=𝜏₁+𝜏₂be a parameter. The authors have shown that there is a critical value of the parameter; the steady state of model (2) is stable when the parameter is less than the critical value and unstable when the parameter is greater than the critical value. Thus, the zero equilibrium loses its stability when the parameter passes through the critical value, and a Hopf bifurcation occurs and oscillations induced by the Hopf bifurcation appeared. Zhen and Xu generated models (1) and (2) to a three coupled FHN neurons network with time delay as follows [8]: 𝑢^′₁1=−3𝑢³₁+𝑐𝑢²₁+𝑑𝑢₁−𝑢₂+𝛼𝑢²₁+𝛽𝑓𝑢₃(𝑡−𝜏)+𝑓𝑢₅(),𝑢𝑡−𝜏_′2=𝜀𝑢₁−𝑏𝑢₂,𝑢^′₃1=−3𝑢³₃+𝑐𝑢²₃+𝑑𝑢₃−𝑢₄+𝛼𝑢²₃+𝛽𝑓𝑢₁(𝑡−𝜏)+𝑓𝑢₅(,𝑢𝑡−𝜏)_′4=𝜀𝑢₃−𝑏𝑢₄,𝑢^′₅1=−3𝑢³₅+𝑐𝑢²₅+𝑑𝑢₅−𝑢₆+𝛼𝑢²₅+𝛽𝑓𝑢₁(𝑡−𝜏)+𝑓𝑢₃(),𝑢𝑡−𝜏_′6=𝜀𝑢₅−𝑏𝑢₆, (3) where𝛼, 𝛽represent the synaptic strength of self-connection and neighborhood interaction, respectively, and 𝑓(𝑥)is a sufficiently smooth sigmoid amplification function such as tanh(𝑥) and arctan(𝑥). The method of Lyapunov functional is used to obtain the synchronization conditions of the neural system. Noting that, for each neuron of model (3), the synaptic strength of self-connection and neighborhoodinteraction are the same under the same restrictive condition, the dynamics of (3) are completely characterized by the following system: 𝑢^′₁1=−3𝑢³₁+(𝑐+𝛼)𝑢²₁+𝑑𝑢₁−𝑢₂+2𝛽𝑓𝑢₁(),𝑢𝑡−𝜏_′2=𝜀𝑢₁−𝑏𝑢₂, (4) where[𝑢₁,𝑢₂]^𝑇is a completely synchronous solution of system (4). The Bautin bifurcation of synchronous solution for this neural system (4) in which𝛼,𝛽are regarded as the bifurcating parameters is investigated. However, generally speaking, the synaptic strength of self-connection, neighborhoodinteraction for each neuron, and the time delays are different. Therefore, in this paper, we will discuss the following model: 𝑢^′₁1=−3𝑢³₁+𝑐₁𝑢²₁+𝑑₁𝑢₁−𝑢₂+𝛼₁𝑢²₁+𝛽₁𝑓𝑢₃𝑡−𝜏₃+𝑓𝑢₅𝑡−𝜏₅,𝑢^′₂=𝜀₁𝑢₁−𝑏₁𝑢₂,𝑢^′₃1=−3𝑢³₃+𝑐₂𝑢²₃+𝑑₂𝑢₃−𝑢₄+𝛼₂𝑢²₃+𝛽₂𝑓𝑢₁𝑡−𝜏₁+𝑓𝑢₅𝑡−𝜏₅,𝑢^′₄=𝜀₂𝑢₃−𝑏₂𝑢₄,𝑢^′₅1=−3𝑢³₅+𝑐₃𝑢²₅+𝑑₃𝑢₅−𝑢₆+𝛼₃𝑢²₅+𝛽₃𝑓𝑢₁𝑡−𝜏₁+𝑓𝑢₃𝑡−𝜏₃,𝑢^′₆=𝜀₃𝑢₅−𝑏₃𝑢₆, (5) where𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝛼_𝑖,𝛽_𝑖,𝜀_𝑖 (𝑖=1,2,3) are constants.𝜏_𝑗>0 (𝑗=1,3,5) represent the time delays in signal transmission. System (5) can be rewritten as follows: 𝑢^′₁=𝑑₁+𝛼₁+𝑐₁𝑢₁−13𝑢²₁𝑢₁−𝑢₂+𝛽₁𝑓𝑢₃𝑡−𝜏₃+𝑓𝑢₅𝑡−𝜏₅,𝑢^′₂=𝜀₁𝑢₁−𝜀₁𝑏₁𝑢₂,𝑢^′₃=𝑑₂+𝛼₂+𝑐₂𝑢₃−13𝑢²₃𝑢₃−𝑢₄+𝛽₂𝑓𝑢₁𝑡−𝜏₁+𝑓𝑢₅𝑡−𝜏₅,𝑢^′₄=𝜀₂𝑢₃−𝜀₂𝑏₂𝑢₄,𝑢^′₅=𝑑₃+𝛼₃+𝑐₃𝑢₅−13𝑢²₅𝑢₅−𝑢₆+𝛽₃𝑓𝑢₁𝑡−𝜏₁+𝑓𝑢₃𝑡−𝜏₃,𝑢^′₆=𝜀₃𝑢₅−𝜀₃𝑏₃𝑢₆. (6)

It is known that if all solutions of system (6) are bounded and there exists a unique unstable equilibrium point of system (6), then this particular instability will force system (6) to generate a limit cycle, namely, a periodic oscillation [9]. We will provide some restrictive conditions which are easy to check to ensure the existence of periodic oscillation. It was pointed out that bifurcating method to determine the periodic solution of system (6) is very difficult.

In the following, we first assume that𝑓(𝑢_𝑖(𝑡−𝜏_𝑖)) (𝑖=1,3,5) are continuous bounded monotone increasing functions, satisfying lim_𝑢𝑖→0𝑓𝑢_𝑖(𝑡)𝑢_𝑖(𝑡)=𝛾_𝑖(>0),𝑖=1,3,5;𝑓(0)=0. (7)

For example, activation functions tanh(𝑢_𝑖(𝑡)), arctan(𝑢_𝑖(𝑡)), and (1/2)(|𝑢_𝑖(𝑡)+1|−|𝑢_𝑖(𝑡)−1|) satisfy condition (7). From assumption (7), the linearization of system (6) about the zero point leads to the following: 𝑢^′₁=𝑑₁𝑢₁−𝑢₂+𝛽₁𝛾₃𝑢₃𝑡−𝜏₃+𝛾₅𝑢₅𝑡−𝜏₅,𝑢^′₂=𝜀₁𝑢₁−𝜀₁𝑏₁𝑢₂,𝑢^′₃=𝑑₂𝑢₃−𝑢₄+𝛽₂𝛾₁𝑢₁𝑡−𝜏₁+𝛾₅𝑢₅𝑡−𝜏₅,𝑢^′₄=𝜀₂𝑢₃−𝜀₂𝑏₂𝑢₄,𝑢^′₅=𝑑₃𝑢₅−𝑢₆+𝛽₃𝛾₁𝑢₁𝑡−𝜏₁+𝛾₃𝑢₃𝑡−𝜏₃,𝑢^′₆=𝜀₃𝑢₅−𝜀₃𝑏₃𝑢₆. (8)

The matrix form of system (8) is as follows: 𝑈^′(𝑡)=𝐴𝑈(𝑡)+𝐵𝑈(𝑡−𝜏), (9) where𝑈(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢₆(𝑡))^𝑇, 𝑈(𝑡−𝜏)=(𝑢₁(𝑡−𝜏₁),0,𝑢₃(𝑡−𝜏₃),0,𝑢₅(𝑡−𝜏₅),0)^𝑇,𝑑𝐴=¹𝜀₁0000−1−𝜀₁𝑏₁000000𝑑₂𝜀₂0000−1−𝜀₂𝑏₂000000𝑑₃𝜀₃0000−1−𝜀₃𝑏₃,00𝛽𝐵=²𝛾₁0𝛽₃𝛾₁0000000𝛽₁𝛾₃000𝛽₃𝛾₃0000000𝛽₁𝛾₅0𝛽₂𝛾₅000000000. (10)

2. Preliminaries

Lemma 1. Suppose that𝑏_𝑖>0, 0<𝜀_𝑖≪1, 𝑑_𝑖<0, (𝛼_𝑖+𝑐_𝑖)²+(4/3)𝑑_𝑖<0 (𝑖=1,2,3); then each solution of system (6) is bounded.

Proof. Note that the activation functions are bounded continuous nonlinear functions. Therefore, there exist 𝑁_𝑗>0 such that|𝑓(𝑢_𝑗(𝑡−𝜏_𝑗))|≤𝑁_𝑗 (𝑗=1,3,5). Since𝑑_𝑖<0, (𝛼_𝑖+𝑐_𝑖)²+(4/3)𝑑_𝑖<0 (𝑖=1,2,3), this implies that there exist constants𝑘_𝑖>0 such that for any values𝑢_𝑖we have 𝑑_𝑖+(𝛼_𝑖+𝑐_𝑖)𝑢_𝑖−(1/3)𝑢²_𝑖≤−𝑘_𝑖<0 (𝑖=1,2,3). From (6) we get 𝑑𝑢₁(𝑡)𝑑𝑡≤−𝑘₁𝑢₁+𝑢₂+𝛽₁𝑁₃+𝑁₅,𝑑𝑢₂(𝑡)𝑑𝑡≤𝜀₁𝑢₁−𝜀₁𝑏₁𝑢₂,𝑑𝑢₃(𝑡)𝑑𝑡≤−𝑘₂𝑢₃+𝑢₄+𝛽₂𝑁₁+𝑁₅,𝑑𝑢₄(𝑡)𝑑𝑡≤𝜀₂𝑢₃−𝜀₂𝑏₂𝑢₄,𝑑𝑢₅(𝑡)𝑑𝑡≤−𝑘₃𝑢₅+𝑢₆+𝛽₃𝑁₁+𝑁₃,𝑑𝑢₆(𝑡)𝑑𝑡≤𝜀₃𝑢₅−𝜀₃𝑏₃𝑢₆. (11)
Noting that system (11) is the first-order linear system of equations with constant coefficients, the eigenvalues of system (11) are𝜆_𝑖1,𝑖2=(−(𝑘_𝑖+𝜀_𝑖𝑏_𝑖)±(𝑘_𝑖+𝜀_𝑖𝑏_𝑖)²−4𝜀_𝑖(𝑘_𝑖𝑏_𝑖+1))/2 (𝑖=1,2,3). Since𝑘_𝑖>0, 𝜀_𝑖>0, 𝑏_𝑖>0 (𝑖=1,2,3), 𝜆_𝑖1,𝑖2<0 if (𝑘_𝑖+𝜀_𝑖𝑏_𝑖)²−4𝜀_𝑖(𝑘_𝑖𝑏_𝑖+1)>0 or 𝜆_𝑖1,𝑖2 are complex numbers with Re𝜆_𝑖1,𝑖2<0if(𝑘_𝑖+𝜀_𝑖𝑏_𝑖)²−4𝜀_𝑖(𝑘_𝑖𝑏_𝑖+1)<0 (𝑖=1,2,3). This implies that all solutions of system (11), as well as the system, (6) are bounded according to the theory of the first-order linear system of equations with constant coefficients.
According to [10], there is the same oscillatory behavior for systems (8) and (6). So, in order to investigate the periodic oscillatory behavior of system (6), we only need to deal with system (8).

Lemma 2. Suppose that matrix 𝐶 (=𝐴+𝐵) is a nonsingular matrix. Then, system (9) has a unique equilibrium point.

Proof. An equilibrium point𝑢^∗=[𝑢^∗₁,𝑢^∗₂,…,𝑢^∗₆]^𝑇is the solution of the following algebraic equation: 𝐴𝑈^∗+𝐵𝑈^∗=(𝐴+𝐵)𝑈^∗=0. (12)
Assume that 𝑈^∗ and 𝑉^∗ are two equilibrium points of system (9); then we have (𝐴+𝐵)𝑈^∗−𝑉^∗=𝐶𝑈^∗−𝑉^∗=0. (13)
Since 𝐶 is a nonsingular matrix, implying that𝑈^∗−𝑉^∗=0 and𝑈^∗=𝑉^∗ system (9) has a unique equilibrium point. Obviously, this equilibrium point is exactly the zero point.

3. Periodic Oscillation

Theorem 3. Suppose that𝑏_𝑖>0,0<𝜀_𝑖≪1, 𝑑_𝑖<0, (𝛼_𝑖+𝑐_𝑖)²+(4/3)𝑑_𝑖<0 (𝑖=1,2,3), and 𝐶 is a nonsingular matrix. Let𝜏_∗=min{𝜏₁,𝜏₃,𝜏₅},𝜏^∗=max{𝜏₁,𝜏₃,𝜏₅}, 𝜌₁,𝜌₂,…,𝜌₆, and 𝜔₁,𝜔₂,…,𝜔₆ denote the eigenvalues of matrices𝐴and𝐵, respectively. Assume that there is at least one 𝜌_𝑖>0, 𝑖∈(1,2,…,6), and the following inequalities hold: 𝜔_𝑖𝜏_∗𝑒^{−𝜌𝑖𝜏∗}>1,𝜔_𝑖𝜏^∗𝑒^{−𝜌𝑖𝜏∗}>1. (14)

Then, the trivial solution of system (8) is unstable, implying that there is a periodic oscillatory solution of system (6).

Proof. From the assumptions, we know that system (8) has a unique equilibrium point and all solutions are bounded. We will prove that the unique equilibrium point is unstable. We first discuss the case that 𝜏₁=𝜏₃=𝜏₅=𝜏_∗ in system (8). The characteristic equation of system (8) is as follows: det𝜆𝐼−𝐴−𝐵𝑒^{−𝜆𝜏∗}=0. (15)
Equation (15) is equal to ⁶_𝑘=1𝜆−𝜌_𝑘−𝜔_𝑘𝑒^{−𝜆𝜏∗}=0. (16)
Therefore, we are led to an investigation of the nature of the roots for the following equations: 𝜆−𝜌_𝑘−𝜔_𝑘𝑒^{−𝜆𝜏∗}=0,𝑘=1,2,…,6. (17)
For some𝜌_𝑖>0, consider equation 𝜆−𝜌_𝑖−𝜔_𝑖𝑒^{−𝜆𝜏∗}=0. (18)
If 𝜆<0is a solution of (18), then|𝜆|=−𝜆; from (18) we have |𝜆|≥𝜔_𝑖𝑒^{|𝜆|𝜏∗}−𝜌_𝑖. (19)
Using the formula 𝑒𝑥≤𝑒^𝑥(𝑥>0), leads to the fact that 𝜔1≥^𝑖𝑒^{|𝜆|𝜏∗}|𝜆|+𝜌_𝑖=𝜔_𝑖𝜏_∗𝑒^{−𝜌𝑖𝜏∗}⋅𝑒^{(|𝜆|+𝜌𝑖)𝜏∗}|𝜆|+𝜌_𝑖𝜏_∗≥𝜔_𝑖𝜏_∗𝑒^{−𝜌𝑖𝜏∗}. (20)
Equation (20) contradicts the first inequality of assumption (14). Then, we discuss the case that𝜏₁=𝜏₃=𝜏₅=𝜏^∗in system (8). Similarly, if 𝜆<0is a solution of the equation𝜆−𝜌_𝑖−𝜔_𝑖𝑒^{−𝜆𝜏∗}=0, we also have a contradiction with the second inequality of assumption (14). Since 𝜏_∗≤𝜏_𝑖≤𝜏^∗ (𝑖=1,3,5), one can conclude that there exists a positive real part of the eigenvalue of system (8) for any𝜏_𝑖 (𝑖=1,3,5) under the assumptions. This means that the trivial solution of system (8) is unstable, implying that there is a periodic oscillatory solution of system (6) based on Chafee’s criterion.

Theorem 4. Suppose that𝑏_𝑖>0,0<𝜀_𝑖≪1, 𝑑_𝑖<0, (𝛼_𝑖+𝑐_𝑖)²+(4/3)𝑑_𝑖<0 (𝑖=1,2,3), and 𝐶 is a nonsingular matrix. Let𝜌_𝑘=𝜌_𝑘1+𝑖𝜌_𝑘2 (𝜌_𝑘2 may equal zero) and 𝜔_𝑘=𝜔_𝑘1+𝑖𝜔_𝑘2 (𝜔_𝑘2 may equal zero) (𝑘=1,2,…,6) denote the eigenvalues of matrices𝐴and𝐵, respectively. If, for some𝜌_𝑖, |𝜌_𝑖1|<𝜔_𝑖1 as 𝜌_𝑖1<0, then the trivial solution of system (8) is unstable, implying that system (6) has a periodic oscillatory solution.

Proof. The assumptions guarantee that system (8) has a unique equilibrium point and all solutions are bounded. In this case, we first consider 𝜏₁=𝜏₃=𝜏₅=𝜏_∗ in system (8). Then, for some 𝜌_𝑖, let 𝜆=𝜆₁+𝑖𝜆₂; from (18) we have 𝜆₁−𝜌_𝑖1−𝜔_𝑖1𝑒^{−𝜆1𝜏∗}cos𝜆₂𝜏_∗𝜆=0,₂−𝜌_𝑖2+𝜔_𝑖2𝑒^{−𝜆1𝜏∗}sin𝜆₂𝜏_∗=0. (21)
We will show that𝜆₁>0 and there is an eigenvalue which has positive real part of system (18). Let 𝑓(𝜆₁)=𝜆₁−𝜌_𝑖1−𝜔_𝑖1𝑒^{−𝜆1𝜏∗}cos(𝜆₂𝜏_∗); then 𝑓(𝜆₁)is a continuous function of 𝜆₁. If𝜌_𝑖1>0, then select suitable delay𝜏_∗such that𝜔_𝑖1cos(𝜆₂𝜏_∗)>−𝜌_𝑖1. Therefore,𝑓(0)=−𝜌_𝑖1−𝜔_𝑖1cos(𝜆₂𝜏_∗)<0. Noting that𝑒^{−𝜆1𝜏∗}→0as𝜆₁→+∞, obviously, there exists a suitably large 𝜆₁(>0) such that𝑓(𝜆₁)=𝜆₁−𝜌_𝑖1−𝜔_𝑖1𝑒^{−𝜆1𝜏∗}cos(𝜆₂𝜏_∗)>0. By the continuity of𝑓(𝜆₁), there exists a positive 𝜆₁^∗∈(0,𝜆₁)such that𝑓(𝜆₁^∗)=0. If𝜌_𝑖1<0, since |𝜌_𝑖1|<𝜔_𝑖1 (𝜔_𝑖1≠0), then there exists a suitable delay𝜏_∗and a positive𝜆₁such that𝜔_𝑖1cos(𝜆₂𝜏_∗)<−𝜌_𝑖1 and 𝜆₁−𝜔_𝑖1𝑒^{−𝜆1𝜏∗}cos(𝜆₂𝜏_∗)<0 both hold. Then, 𝑓(0)=−𝜌_𝑖1−𝜔_𝑖1cos(𝜆₂𝜏_∗)>0 and 𝑓(𝜆₁)=𝜆₁−𝜔_𝑖1𝑒^{−𝜆1𝜏∗}cos(𝜆₂𝜏_∗)<0. Again, from the continuity of 𝑓(𝜆₁), there exists a positive 𝜆₁^∗∗∈(0,𝜆₁)such that𝑓(𝜆₁^∗∗)=0. Thus, there is an eigenvalue of system (18) that has positive real part. Implying that the trivial solution of system (8) is unstable. Thus, the trivial solution of system (6) is also unstable. Based on the theory of delay differential equation, the oscillatory behavior of the solution will maintain as time delay increasing. Therefore, for any𝜏_𝑖≥𝜏_∗ (𝑖=1,3,5), system (8), as well as system, (6) generates a periodic oscillatory solution. We select a suitable delay 𝜏_∗ such that system (6) has a periodic oscillatory solution. This oscillation is said to be induced by time delay.

4. Simulation Result

The parameter values are selected as𝛼₁=−1.5, 𝛼₂=−1.5, 𝛼₃=−1.2; 𝑏₁=0.16, 𝑏₂=0.25, 𝑏₃=0.12; 𝑐₁=1.3, 𝑐₂=1.302, 𝑐₃=1.305; 𝑑₁=−0.705, 𝑑₂=−0.706, 𝑑₃=−0.707; 𝛽₁=1.5, 𝛽₂=1.5, 𝛽₃=0.15; 𝜀₁=0.05, 𝜀₂=0.025, 𝜀₃=0.085, respectively. It is easy to check that the conditions of Lemmas 1 and 2 hold. The activation functions are selected as arctan(𝑢) and tanh(𝑢), respectively.In this case,𝛾₁=𝛾₃=𝛾₅=1, and eigenvalues of matrices 𝐴 and 𝐵 are𝜌₁=−0.6238,𝜌₂=−0.0892, 𝜌₃=−0.6682, 𝜌₄=−0.0440,𝜌₅=−0.5493, and 𝜌₆=−0.1679, and𝜔₁=1.7562,𝜔₂=−1.5000, 𝜔₃=−0.2562, 𝜔₄=0, and 𝜔₅=0, 𝜔₆=0, respectively. Since|𝜌₁|=0.6238<𝜔₁, there is a periodic oscillatory solution based on Theorem 4. Both in Figures 1 and 2, the time delays are selected as 𝜏₁=10, 𝜏₂=8, and 𝜏₃=4. Then, we change delays as𝜏₁=1, 𝜏₂=2, 𝜏₃=3; activation function is kept as tanh(𝑢); periodic oscillatory solution also occurred (Figure 3). In Figure 4, the parameter values are selected as𝛼₁=−0.95, 𝛼₂=−1.2, 𝛼₃=−1.25; 𝑏₁=0.18, 𝑏₂=0.2, 𝑏₃=0.16; 𝑐₁=1.4, 𝑐₂=1.42, 𝑐₃=1.45; 𝑑₁=−0.7, 𝑑₂=−0.72, 𝑑₃=−0.75; 𝛽₁=1.25, 𝛽₂=1.2, 𝛽₃=1.15; 𝜀₁=0.05, 𝜀₂=0.045, and 𝜀₃=0.065, respectively. The activation function is tanh(𝑢). The eigenvalues of matrices𝐴and𝐵are 𝜌₁=−0.6179, 𝜌₂=−0.0911, 𝜌₃=−0.6498, 𝜌₄=−0.0792, 𝜌₅=−0.6481, 𝜌₆=−0.1123 and 𝜔₁=2.3654, 𝜔₂=−1.2154, 𝜔₃=−1.1500, 𝜔₄=0, 𝜔₅=0, and 𝜔₆=0, respectively. We see that periodic oscillatory solution appeared.

(a)  Solid line: 𝑢1(𝑡), dashed line: 𝑢2(𝑡), dotted line: 𝑢3(𝑡)

(b) Solid line: 𝑢4(𝑡), dashed line: 𝑢5(𝑡), dotted line: 𝑢6(𝑡)

Figure 1 

Periodic oscillatory behavior, activation function: arctan(𝑢), and delays: (10, 8, 4).

(a) Solid line: 𝑢1(𝑡), dashed line: 𝑢2(𝑡), dotted line: 𝑢3(𝑡)

(b) Solid line: 𝑢4(𝑡), dashed line: 𝑢5(𝑡), dotted line: 𝑢6(𝑡)

Figure 2 

Periodic oscillatory behavior, activation function: tanh(𝑢), and delays: (10, 8, 4).

(a)  Solid line: 𝑢1(𝑡), dashed line: 𝑢2(𝑡), dotted line: 𝑢3(𝑡)

(b) Solid line: 𝑢4(𝑡), dashed line: 𝑢5(𝑡), dotted line: 𝑢6(𝑡)

Figure 3 

Periodic oscillatory behavior, activation function: tanh(𝑢), and delays: (1, 2, 3).

(a)  Solid line: 𝑢1(𝑡), dashed line: 𝑢2(𝑡), dotted line: 𝑢3(𝑡)

(b) Solid line: 𝑢4(𝑡), dashed line: 𝑢5(𝑡), dotted line: 𝑢6(𝑡)

Figure 4 

Periodic oscillatory behavior, activation function: tanh(𝑢), and delays: (9, 10, 12).

5. Conclusion

This paper discusses a three coupled FHN neurons model in which the synaptic strength of self-connection, neighborhoodinteraction for each neuron, and the time delays are different. Two theorems are provided to determine the periodic oscillatory behavior of the solutions based on Chafee’s criterion of limit cycle. Computer simulation suggested that those theorems only are sufficient conditions.

Acknowledgments

This research was supported by NNSF of China (11361010) and Scientific Research Foundation of the Education Department of Guangxi Province (201010LX463).