# Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

Authors: Changjin Xu; Peiluan Li
Journal: Advances in Artificial Neural Systems
 (2012)

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## Abstract

A four-dimensional neural network model with delay is investigated. With the help
of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium
undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the
normal form theory and the center manifold argument, we derive the explicit formulae for
determining the properties of the bifurcating periodic solutions. Numerical simulations are
performed to illustrate the analytical results.


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## Body

## 1. Introduction

The interest in the periodic orbits of a delay neural networks has increased strongly in recent years and substantial efforts have been made in neural network models, for example, Wei and Zhang [1] studied the stability and bifurcation of a class of n-dimensional neural networks with delays, Guo and Huang [2] investigated the Hopf bifurcation behavior of a ring of neurons with delays, Yan [3] discussed the stability and bifurcation of a delayed trineuron network model, Hajihosseini et al. [4] made a discussion on the Hopf bifurcation of a delayed recurrent neural network in the frequency domain, and Liao et al. [5] did a theoretical and empirical investigation of a two-neuron system with distributed delays in the frequency domain. For more information, one can see [6–23]. In 1986 and 1987, Babcock and Westervelt [24, 25] had analyzed the stability and dynamics of the following simple neural network model of two neurons with inertial coupling:(1)dx1dt=x3,dx2dt=x4,dx3dt=-2ξx3-x1+A2tanh(x2),dx4dt=-2ξx4-x2+A1tanh(x1),
where xi(i=1,2) denotes the input voltage of the ith neuron, xj(j=3,4) is the output of the jth neuron, ξ>0 is the damping factor and Ai(i=1,2) is the overall gain of the neuron which determines the strength of the nonlinearity. For a more detailed interpretation of the parameters, one can see [24, 25]. In 1997, Lin and Li [26] made a detailed investigation on the bifurcation direction of periodic solution for system (1).Considering that there exists a time delay (we assume that it isτ) in the response of the output voltages to changes in the input, then system (1) can be revised as follows:(2)dx1dt=x3,dx2dt=x4,dx3dt=-2ξx3-x1+A2tanh(x2(t-τ)),dx4dt=-2ξx4-x2+A1tanh(x1(t-τ)).
As is known to us that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation is very critical. When delays are incorporated into the network models, stability, and Hopf bifurcation analysis become much complex. To obtain a deep and clear understanding of dynamics of neural network model with delays, we will make a investigation on system (2), that is, we study the stability, the local Hopf bifurcation for system (2).The remainder of the paper is organized as follows. In Section2, local stability for the equilibrium state of system (2) is discussed. We investigate the existence of the Hopf bifurcations for system (2) choosing time delay as the bifurcation parameter. In Section 3, the direction and stability of the local Hopf bifurcation are analyzed by using the normal form theory and the center manifold theorem by Hassard et al. [27]. In Section 4, numerical simulations for justifying the theoretical results are illustrated.


## 2. Stability of the Equilibrium and Local Hopf Bifurcations

The object of this section is to investigate the stability of the equilibrium and the existence of local Hopf bifurcations for system (2). It is easy to see that if the following condition:(H1)
A1A2<1holds, then (2) has a unique equilibrium E(0,0,0,0). To investigate the local stability of the equilibrium state we linearize system (2). We expand it in a Taylor series around the orgin and neglect the terms of higher order than the first order. The linearization of (2) near E(0,0,0,0) can be expressed as:(3)dx1dt=x3,dx2dt=x4,dx3dt=-x1-2ξx3+A2x2(t-τ),dx4dt=-x2-2ξx4+A1x1(t-τ),
whose characteristic equation has the form(4)det⁡(λ0-100λ0-11-A2e-λτλ+2ξ0-A1e-λτ10λ+2ξ)=0,
namely,(5)λ4+4ξλ3+(4ξ2+2)λ2+4ξλ+1-A1A2e-2λτ=0.
In order to investigate the distribution of roots of the transcendental equation (5), the following Lemma is necessary.Lemma 1 (see [28]).
For the transcendental equation:(6)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+[p1(1)λn-1+⋯+pn-1(1)λ+pn(1)]e-λτ1+⋯+[p1(m)λn-1+⋯+pn-1(m)λ+pn(m)]e-λτm=0,
as (τ1,τ2,τ3,…,τm) vary, the sum of orders of the zeros of P(λ,e-λτ1,…,e-λτm) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.Forτ=0, (5) becomes(7)λ4+4ξλ3+(4ξ2+2)λ2+4ξλ+1-A1A2=0.
In view of the Routh-Hurwitz criteria, we know that all roots of (7) have a negative real part if the following condition:(H2)
4ξ2+A1A2>0is satisfied.Forω>0,iω is a root of (5) if and only if(8)ω4-4ξω2i-(4ξ2+2)ω2+4ξωi+1-A1A2(cos⁡⁡2ωτ-isin⁡⁡2ωτ)=0.
Separating the real and imaginary parts gives(9)A1A2cos⁡2ωτ=ω4-(4ξ2+2)ω2+1,A1A2sin⁡2ωτ=4ξω2-4ξω.
It follows from (9) that(10)[ω4-(4ξ2+2)ω2+1]2+[4ξω2-4ξω]2=A12A22,
which is equivalent to(11)ω8+m1ω6+m2ω4+m3ω3+m4ω2+m5=0,
where(12)m1=-4(2ξ2+1),m2=16ξ4+32ξ2+6,m3=-32ξ2,m4=8ξ2-4,m5=1-A12A22.
Without loss of generality, we assume that (11) has eight positive roots, denoted by ωk(k=1,2,3,…,8). Then by (9), we derive(13)τk(j)=12ωk{arccos[ω4-(4ξ2+2)ω2+1A1A2]+2jπ},
where k=1,2,3,…,8;j=0,1,…, then ±iωk are a pair of purely imaginary roots of (5) when τ=τk(j). Define(14)τ0=τk0(0)=min⁡k∈{1,2,3,…,8}{τk(0)}.
The above analysis leads to the Lemma as follows.Lemma 2.
If (H1) and (H2) hold, then all roots of (5) have a negative real part when τ∈[0,τ0) and (5) admits a pair of purely imaginary roots ±ωk when τ=τk(j)(k=1,2,3,…,8;j=0,1,2,…).Letλ(τ)=α(τ)+iω(τ) be a root of (5) near τ=τk(j), and α(τk(j))=0, and ω(τk(j))=ωk. Due to functional differential equation theory, for every τk(j),k=1,2,…,8;j=0,1,2,…, there exists ɛ>0 such that λ(τ) is continuously differentiable in τ for |τ-τk(j)|<ɛ. Substituting λ(τ) into the left hand side of (5) and taking derivative with respect to τ, we have(15)[dλdτ]-1=2λ3+6ξλ2+2(2ξ2+1)λ+2ξλA1A2e-2λτ-τλ.
Then(16)[d(Reλ(τ))dτ]τ=τk(j)-1=Re{2λ3+6ξλ2+2(2ξ2+1)λ+2ξλA1A2e-2λτ}τ=τk(j)=Re{2ξ-6ξωk2+(4ξ2ωk+2ωk-2ωk3)iA1A2ωksin⁡2ωkτk(j)+iA1A2ωkcos⁡2ωkτk(j)}=1A12A22ωk2{(2ξ-6ξωk2)A1A2ωksin⁡2ωkτk(j)-(4ξ2ωk+2ωk-2ωk3)A1A2ωkcos⁡2ωkτk(j)}=1A12A22ωk2{(2ξ-6ξωk2)ωk(4ξωk2-4ξωk)-(4ξ2ωk+2ωk-2ωk3)ωk×[ωk4-(4ξ2+2)ωk2+1]}=2ΛA12A22,
where(17)Λ=4ωk6-3(2ξ2+1)ωk4-12ξωk3+(8ξ4+8ξ2+12ξ+3)ωk2+4ξ2ωk-(6ξ2+1).
We assume that the following condition holds:(H3)
Λ≠0.According to above analysis and the results of Kuang [29] and Hale [30], we have the following theoremTheorem 3.
If (H1) and (H2) hold, then the equilibriumE(0,0,0,0) of system (2) is asymptotically stable for τ∈[0,τ0). Under the conditions (H1) and (H2), if the condition (H3) holds, then system (2) undergoes a Hopf bifurcation at the equilibrium E(0,0,0,0) when τ=τk(j),k=1,2,3,…,8;j=0,1,2,….


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we discuss the direction, stability and the period of the bifurcating periodic solutions. The used methods are based on the normal form theory and the center manifold theorem introduced by Hassard et al. [27]. From the previous section, we know that if τ=τk(j),k=1,2,3…,8;j=0,1,2,…. any root of (5) of the form λ(τ)=α(τ)+iω(τ) satisfies α(τk(j))=0,ω(τk(j))=ωk and dα(τ)/dτ∣τ=τk(j)≠0.For convenience, letx¯i(t)=xi(τt)(i=1,2,3,4) and τ=τk(j)+μ, where τk(j) is defined by (13) and μ∈R, drop the bar for the simplification of notations, then system (3) can be written as an FDE in C=C([-1,0],R4) as(18)u̇(t)=Lμ(ut)+F(μ,ut),
where u(t)=(x1(t),x2(t),x3(t),x4(t))T∈C and ut(θ)=u(t+θ)=(x1(t+θ),  x2(t+θ),x3(t+θ),  x4(t+θ))T∈C, and Lμ:C→R,F:R×C→R are given by(19)Lμϕ=(τk(j)+μ)(00100001-10-2ξ000-1-2ξ)(ϕ1(0)ϕ2(0)ϕ3(0)ϕ4(0))+(τk(j)+μ)(000000000A200A1000)(ϕ1(-1)ϕ2(-1)ϕ3(-1)ϕ4(-1)),f(μ,ϕ)=(τk(j)+μ)(00A2ϕ23(-1)+h.o.t.A1ϕ13(-1)+h.o.t.),
respectively, where ϕ(θ)=(ϕ1(θ),ϕ2(θ),ϕ3(θ),ϕ4(θ))T∈C.From the discussion in Section2, we know that if μ=0, then system (18) undergoes a Hopf bifurcation at the equilibrium E(0,0,0,0) and the associated characteristic equation of system (18) has a pair of simple imaginary roots ±ωkτk(j).By the representation theorem, there is a matrix function with bounded variation componentsη(θ,μ),θ∈[-1,0] such that(20)Lμϕ=∫-10dη(θ,μ)ϕ(θ)forϕ∈C.
In fact, we can choose(21)η(θ,μ)=(τk(j)+μ)(00100001-10-2ξ00-10-2ξ)δ(θ)-(τk(j)+μ)(000000000A200A1000)δ(θ+1),
where δ is the Dirac delta function.Forϕ∈C([-1,0],R4), define(22)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10dη(s,μ)ϕ(s),θ=0,Rϕ={0,-1≤θ<0,f(μ,ϕ),θ=0.
Then (18) is equivalent to the abstract differential equation(23)u̇t=A(μ)ut+R(μ)ut,
where ut(θ)=u(t+θ),θ∈[-1,0]. For ψ∈C([0,1],(R4)*), define(24)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10dηT(t,0)ψ(-t),s=0.
For ϕ∈C([-1,0],R4) and ψ∈C([0,1],(R4)*), define the bilinear form(25)〈ψ,ϕ〉=ψ¯(0)ϕ(0)-∫-10∫ξ=0θψT(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0), the A=A(0) and A* are adjoint operators. By the discussions in Section 2, we know that ±iωkτk(j) are eigenvalues of A(0), and they are also eigenvalues of A* corresponding to iωkτk(j) and -iωkτk(j) respectively. By direct computation, we can obtain(26)q(θ)=(1,α,β,γ)Teiωkτk(j)θ,q*(s)=D(1,α*,β*,γ*)eiωkτk(j)s,
where(27)α=1-ωk2+2ξωkiA2e-iωkτk(j),β=iωk,γ=iωk(1-ωk2+2ξωki)A2e-iωkτk(j),α*=1-ωk2-2ξωkiA1e-iωkτk(j),β*=12ξ-iωk,γ*=1-ωk2-2ξωkiA1e-iωkτk(j)(2ξ-iωk),D=11+α¯α*+β¯β*+γ¯γ*+τk(j)(A1γ*+A2β*α¯)eiωkτk(j).
Furthermore, 〈q*(s),q(θ)〉=1 and 〈q*(s),q¯(θ)〉=0.Next, we use the same notations as those in Hassard et al. [27] and we first compute the coordinates to describe the center manifold C0 at μ=0. Let ut be the solution of (18) when μ=0.Define(28)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)},
on the center manifold C0, and we have(29)W(t,θ)=W(z(t),z¯(t),θ),
where(30)W(z(t),z¯(t),θ)=W(z,z¯)=W20z22+W11zz¯+W02z¯22+⋯,
and z and z¯ are local coordinates for center manifold C0 in the direction of q* and q¯*. Noting that W is also real if ut is real, we consider only real solutions. For solutions ut∈C0 of (18),(31)ż(t)=iωkτk(j)z+q¯*(θ)f(0,W(z,z¯,θ)+2Re{zq(θ)})=defiωkτk(j)z+q¯*(0)f0.
That is,(32)ż(t)=iωkτk(j)z+g(z,z¯),
where(33)g(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
Hence we have(34)g(z,z¯)=q¯*(0)f0(z,z¯)=q¯*(0)f(0,ut)=τk(j)D¯(1,α*¯,β*¯,γ*¯)(00A2x2t3(-1)+h.o.t.A1x1t3(-1)+h.o.t.)=D¯τk(j)[3β*e-iωkτk(j)+3γ*α2α¯e-2iωkτk(j)]z2z¯+h.o.t.
and we obtain(35)g20=g11=g02=0,g21=2D¯τk(j)[3β*e-iωkτk(j)+3γ*α2α¯e-2iωkτk(j)].
Therefore, we can compute the following values:(36)c1(0)=i2ωkτk(j)(g20g11-2|g11|2-|g02|23)+g212,μ2=-Re{c1(0)}Re{λ′(τk(j))},β2=2Re(c1(0)),T2=-Im⁡{c1(0)}+μ2Im⁡{λ′(τk(j))}ωkτk(j),
which determine the quantities of bifurcation periodic solutions on the center manifold at the critical value τk(j)(k=1,2,3,…,8;j=0,1,2,3,…), μ2 determines the direction of the Hopf bifurcation: If μ2>0  (μ2<0), then the Hopf bifurcation is supercritical (subcritical); β2 determines the stability of the bifurcating periodic solutions: the periodic solutions are stable (unstable) if β2<0  (β2>0); and T2 determines the period of the bifurcating periodic solutions: the period is increases (decreases) if T2>0  (T2<0).


## 4. Numerical Examples

To illustrate the analytical results found, letξ=0.1,A1=0.4,A2=0.2, then system (2) becomes(37)dx1dt=x3,dx2dt=x4,dx3dt=-0.2x3-x1+0.2tanh(x2(t-τ)),dx4dt=-0.2x4-x2+0.4tanh(x1(t-τ)),
which has a unique equilibrium E0(0,0,0,0) and satisfies the conditions indicated in Theorem 3. The equilibrium E0(0,0,0,0) is asymptotically stable for τ=0. For j=0, using the software Matlab, we derive ω0≈3.4233,τ0≈0.9,λ′(τ0)≈0.2556-2.3588i,g21≈-0.2409-3.7451i. Thus by the algorithm (36) derived in Section 3, we get c1(0)≈-0.1205-1.8726i,μ2≈-0.4714,β2≈-0.2410,T2≈0.9687. Thus the equilibrium E0(0,0,0,0) is stable when τ<τ0≈0.9. Figures 1(a)–1(j) show that the equilibrium E0(0,0,0,0) is asymptotically stable when τ=0.7<τ0≈0.9. When τ passes through the critical value τ0≈0.9, the equilibrium E0(0,0,0,0) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from the equilibrium E0(0,0,0,0). Since μ2>0 and β2<0, the direction of the Hopf bifurcation is τ>τ0≈0.9, and these bifurcating periodic solutions from E0(0,0,0,0) at τ0≈0.9 are stable. Figures 1(j)–2(d) suggest that Hopf bifurcation occurs from the equilibrium E0(0,0,0,0) when τ=1.5>τ0≈0.9.(a)–(p) The dynamical behavior of system (37) with τ=0.7<τ0≈0.9 and the initial value (0.01,0.02,0.02,0.01). The equilibrium E0(0,0,0,0) is asymptotically stable.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(a)–(p) The dynamical behavior of system (37) with τ=1.5>τ0≈0.9 and the initial value (0.01,0.02,0.02,0.01). Hopf bifurcation occurs from the equilibrium E0(0,0,0,0).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)



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