# Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

Authors: Yuanhua Lin
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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]:
(1)v1′(t)=-v13(t)+av1(t)-w1(t)+c1tanh(v2(t-τ)),w1′(t)=v1(t)-b1w1(t),v2′(t)=-v23(t)+av2(t)-w2(t)+c2tanh(v1(t-τ)),w2′(t)=v2(t)-b2w2(t).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]:
(2)x1′(t)=-x13(t)+ax1(t)-x2(t)+c1tanh(x3(t-τ1)),x2′(t)=x1(t)-b1x2(t),x3′(t)=-x33(t)+ax3(t)-x4(t)+c2tanh(x1(t-τ2)),x4′(t)=x3(t)-b2x4(t).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]:
(3)u1′=-13u13+cu12+du1-u2+αu12+β[f(u3(t-τ))+f(u5(t-τ))],u2′=ε(u1-bu2),u3′=-13u33+cu32+du3-u4+αu32+β[f(u1(t-τ))+f(u5(t-τ))],u4′=ε(u3-bu4),u5′=-13u53+cu52+du5-u6+αu52+β[f(u1(t-τ))+f(u3(t-τ))],u6′=ε(u5-bu6),
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:
(4)u1′=-13u13+(c+α)u12+du1-u2+2βf(u1(t-τ)),u2′=ε(u1-bu2),
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:
(5)u1′=-13u13+c1u12+d1u1-u2+α1u12+β1[f(u3(t-τ3))+f(u5(t-τ5))],u2′=ε1(u1-b1u2),u3′=-13u33+c2u32+d2u3-u4+α2u32+β2[f(u1(t-τ1))+f(u5(t-τ5))],u4′=ε2(u3-b2u4),u5′=-13u53+c3u52+d3u5-u6+α3u52+β3[f(u1(t-τ1))+f(u3(t-τ3))],u6′=ε3(u5-b3u6),
wherebi,ci,di,αi,βi,εi (i=1,2,3) are constants.τj>0 (j=1,3,5) represent the time delays in signal transmission. System (5) can be rewritten as follows:
(6)u1′=[d1+(α1+c1)u1-13u12]u1-u2+β1[f(u3(t-τ3))+f(u5(t-τ5))],u2′=ε1u1-ε1b1u2,u3′=[d2+(α2+c2)u3-13u32]u3-u4+β2[f(u1(t-τ1))+f(u5(t-τ5))],u4′=ε2u3-ε2b2u4,u5′=[d3+(α3+c3)u5-13u52]u5-u6+β3[f(u1(t-τ1))+f(u3(t-τ3))],u6′=ε3u5-ε3b3u6.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(ui(t-τi)) (i=1,3,5) are continuous bounded monotone increasing functions, satisfying
(7)limui→0f(ui(t))ui(t)=γi(>0),i=1,3,5;f(0)=0.For example, activation functionstanh(ui(t)), arctan(ui(t)), and (1/2)(|ui(t)+1|-|ui(t)-1|) satisfy condition (7). From assumption (7), the linearization of system (6) about the zero point leads to the following:
(8)u1′=d1u1-u2+β1[γ3u3(t-τ3)+γ5u5(t-τ5)],u2′=ε1u1-ε1b1u2,u3′=d2u3-u4+β2[γ1u1(t-τ1)+γ5u5(t-τ5)],u4′=ε2u3-ε2b2u4,u5′=d3u5-u6+β3[γ1u1(t-τ1)+γ3u3(t-τ3)],u6′=ε3u5-ε3b3u6.The matrix form of system (8) is as follows:
(9)U′(t)=AU(t)+BU(t-τ),
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,(10)A=(d1ε10000-1-ε1b1000000d2ε20000-1-ε2b2000000d3ε30000-1-ε3b3),B=(00β2γ10β3γ10000000β1γ3000β3γ30000000β1γ50β2γ5000000000).


## 2. Preliminaries

Lemma 1.
Suppose thatbi>0, 0<εi≪1, di<0, (αi+ci)2+(4/3)di<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 existNj>0 such that|f(uj(t-τj))|≤Nj (j=1,3,5). Sincedi<0, (αi+ci)2+(4/3)di<0 (i=1,2,3), this implies that there exist constantski>0 such that for any valuesuiwe have di+(αi+ci)ui-(1/3)ui2≤-ki<0 (i=1,2,3).   From (6) we get
(11)d|u1(t)|dt≤-k1|u1|+|u2|+|β1|(N3+N5),d|u2(t)|dt≤ε1|u1|-ε1b1|u2|,d|u3(t)|dt≤-k2|u3|+|u4|+|β2|(N1+N5),d|u4(t)|dt≤ε2|u3|-ε2b2|u4|,d|u5(t)|dt≤-k3|u5|+|u6|+|β3|(N1+N3),d|u6(t)|dt≤ε3|u5|-ε3b3|u6|.
Noting that system (11) is the first-order linear system of equations with constant coefficients, the eigenvalues of system (11) areλi1,i2=(-(ki+εibi)±(ki+εibi)2-4εi(kibi+1))/2 (i=1,2,3). Sinceki>0, εi>0, bi>0 (i=1,2,3), λi1,i2<0 if (ki+εibi)2-4εi(kibi+1)>0 or λi1,i2 are complex numbers with Reλi1,i2<0if(ki+εibi)2-4εi(kibi+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 matrixC (=A+B) is a nonsingular matrix. Then, system (9) has a unique equilibrium point.Proof.
An equilibrium pointu*=[u1*,u2*,…,u6*]Tis the solution of the following algebraic equation:
(12)AU*+BU*=(A+B)U*=0.
Assume thatU* and V* are two equilibrium points of system (9); then we have
(13)(A+B)(U*-V*)=C(U*-V*)=0.
SinceC 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 thatbi>0,0<εi≪1, di<0, (αi+ci)2+(4/3)di<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 ρi>0, i∈(1,2,…,6), and the following inequalities hold:
(14)|ωi|τ*e-ρiτ*>1,|ωi|τ*e-ρiτ*>1.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:
(15)det(λI-A-Be-λτ*)=0.
Equation (15) is equal to
(16)∏k=16(λ-ρk-ωke-λτ*)=0.
Therefore, we are led to an investigation of the nature of the roots for the following equations:(17)λ-ρk-ωke-λτ*=0,k=1,2,…,6.
For someρi>0, consider equation
(18)λ-ρi-ωie-λτ*=0.
Ifλ<0is a solution of (18), then|λ|=-λ; from (18) we have
(19)|λ|≥|ωi|e|λ|τ*-ρi.
Using the formulaex≤ex(x>0), leads  to the fact that
(20)1≥|ωi|e|λ|τ*|λ|+ρi=|ωi|τ*e-ρiτ*·e(|λ|+ρi)τ*(|λ|+ρi)τ*≥|ωi|τ*e-ρiτ*.
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-ρi-ωie-λτ*=0, we also have a contradiction with the second inequality of assumption (14). Since τ*≤τi≤τ* (i=1,3,5), one can conclude that there exists a positive real part of the eigenvalue of system (8) for anyτi (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 thatbi>0,0<εi≪1, di<0, (αi+ci)2+(4/3)di<0 (i=1,2,3), and C is a nonsingular matrix. Letρk=ρk1+iρk2 (ρk2 may equal zero) and ωk=ωk1+iωk2 (ωk2 may equal zero) (k=1,2,…,6) denote the eigenvalues of matricesAandB, respectively. If, for someρi, |ρi1|<ωi1 as ρi1<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 ρi, let λ=λ1+iλ2; from (18) we have
(21)λ1-ρi1-ωi1e-λ1τ*cos(λ2τ*)=0,λ2-ρi2+ωi2e-λ1τ*sin(λ2τ*)=0.
We will show thatλ1>0 and there is an eigenvalue which has positive real part of system (18). Let f(λ1)=λ1-ρi1-ωi1e-λ1τ*cos(λ2τ*); then f(λ1)is a continuous function of λ1. Ifρi1>0, then select suitable delayτ*such thatωi1cos(λ2τ*)>-ρi1. Therefore,f(0)=-ρi1-ωi1cos(λ2τ*)<0. Noting thate-λ1τ*→0asλ1→+∞, obviously, there exists a suitably large λ1(>0) such thatf(λ1)=λ1-ρi1-ωi1e-λ1τ*cos(λ2τ*)>0. By the continuity off(λ1), there exists a positive λ1*∈(0,λ1)such thatf(λ1*)=0. Ifρi1<0, since |ρi1|<ωi1 (ωi1≠0), then there exists a suitable delayτ*and a positiveλ-1such thatωi1cos(λ2τ*)<-ρi1 and λ-1-ωi1e-λ-1τ*cos(λ2τ*)<0 both hold. Then, f(0)=-ρi1-ωi1cos(λ2τ*)>0 and f(λ-1)=λ-1-ωi1e-λ-1τ*cos(λ2τ*)<0. Again, from the continuity of f(λ1), there exists a positive λ1**∈(0,λ-1)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 anyτi≥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.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)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)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)Periodic oscillatory behavior, activation function:tanh(u),  and delays: (9, 10, 12).
(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)


## 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.



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