# Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling

Authors: Changjin Xu; Peiluan Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results.


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## Body

## 1. Introduction

In recent years, a number of different classes of neural networks with or without delays, including Hopfield networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associate memory neural networks have been active research topic as [1], and substantial efforts have been made in neural network models, for example, Huang et al. [2] studied the global exponential stability and the existence of periodic solution of a class of cellular neural networks with delays, Guo and Huang [3] investigated the Hopf bifurcation natures of a ring of neurons with delays, Yan [4] analyzed the stability and bifurcation of a delayed tri-neuron network model, Hajihosseini et al. [5] made a discussion on the Hopf bifurcation of a delayed recurrent neural network in the frequency domain, and Liao et al. [6] did a theoretical and empirical investigation of a two-neuron system with distributed delays in the frequency domain. Agranovich et al. [7] considered the impulsive control of a hysteresis cellular neural network model. For more information, one can see [8–24]. In 1986 and 1987, Babcock and Westervelt [25, 26] had investigated the stability and dynamics of the following simple neural network model of two neurons with inertial coupling:
(1.1)dx1dt=x3,dx2dt=x4,dx3dt=-2ξx3-x1+A2tanh(x2),dx4dt=-2ξx4-x2+A1tanh(x1),
where xi(i=1,2) is the input voltage of the ith neuron, xj(j=3,4) denotes 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 [25, 26]. In 1997, Lin and Li [27] made a detail discussion on the bifurcation direction of periodic solution for system (1.1).From applications point of view, considering that there is a time delay (we assume that it isτ) in the response of the output voltages to changes in the input, that is, there exists a feedback delay of the input voltage of the ith neuron to the growth of the output of the jth neuron, then we modify system (1.1) as follows:
(1.2)dx1dt=x3,dx2dt=x4,dx3dt=-2ξx3-x1(t-τ)+A2tanh(x2(t-τ)),dx4dt=-2ξx4-x2(t-τ)+A1tanh(x1(t-τ)).
It is well known 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 more difficult. To obtain a deep and clear understanding of dynamics of neural network model of two neurons with inertial coupling, we will make a discussion on system (1.2), that is, we study the stability, the local Hopf bifurcation for system (1.2).The remainder of this paper is organized as follows. In Section2, we investigate the stability of the equilibrium and the occurrence of local Hopf bifurcations. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results.


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

In this section, we shall study the stability of the equilibrium and the existence of local Hopf bifurcations. For simplification, we only consider the zero equilibrium. One can check that if the following condition:(2.1)(H1)A1A2<1
holds, then (1.2) has a unique equilibrium E(0,0,0,0). The linearization of (1.2) at E(0,0,0,0) is given by
(2.2)dx1dt=x3,dx2dt=x4,dx3dt=-2ξx3-x1(t-τ)+A2x2(t-τ),dx4dt=-2ξx4-x2(t-τ)+A1x1(t-τ),
whose characteristic equation takes the form of
(2.3)det⁡(λ0-100λ0-1e-λτ-A2e-λτλ+2ξ0-A1e-λτe-λτ0λ+2ξ)=0,
that is,
(2.4)λ4+4ξλ3+4ξ2λ2+(2λ2+4ξλ)e-λτ+(1-A1A2)e-2λτ=0.
Multiplying eλτ on both sides of (2.4), it is easy to obtain
(2.5)(λ4+4ξλ3+4ξ2λ2)eλτ+2λ2+4ξλ+(1-A1A2)e-λτ=0.
In order to investigate the distribution of roots of the transcendental equation (2.5), the following lemma is helpful.Lemma 2.1 (see [28]).
For the following transcendental equation:(2.6)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, (2.5) becomes
(2.7)λ4+4ξλ3+(4ξ2+2)λ2+4ξλ+1-A1A2=0.
In view of Routh-Hurwitz criteria, we know that all roots of (2.7) have a negative real part if the following condition:
(2.8)(H2)4ξ2+A1A2>0
is fulfilled.Forω>0,iω is a root of (2.5) if and only if
(2.9)(ω4-4ξω3i-4ξ2ω2)(cos⁡ωτ+isin⁡ωτ)-2ω2+4ξωi+(1-A1A2)(cos⁡ωτ-isin⁡ωτ)=0.
Separating the real and imaginary parts gives
(2.10)(ω4-4ξ2ω2+1-A1A2)cos⁡ωτ+4ξω3sin⁡ωτ=2ω2,(ω4-4ξ2ω2-1+A1A2)sin⁡ωτ-4ξω3cos⁡ωτ=-4ξω.
Then, we obtain
(2.11)sin⁡ωτ=4ξω5+16ξ3ω3-(1-A1A2)(ω4+4ξ2ω2)2-(1-A1A2)2,(2.12)cos⁡ωτ=2ω6+8ξ2ω4-(1-A1A2)(ω4+4ξ2ω2)2-(1-A1A2)2.
In view of sin2ωτ+cos2ωτ=1, then, we have
(2.13)[4ξω5+16ξ3ω3-(1-A1A2)]2+[2ω6+8ξ2ω4-(1-A1A2)]2=[(ω4+4ξ2ω2)2-(1-A1A2)2]2,
which is equivalent to
(2.14)ω16+l1ω14+l2ω12+l3ω10+l4ω8+l5ω6+l6ω5+l7ω4+l8ω3+l9=0,
wherel1=16ξ2,l2=96ξ4-4,l3=256ξ6-48ξ2,l4=256ξ8-64ξ4-128ξ3-2(1-A1A2)2,l5=2(1-A1A2)-16(1-A1A2)2ξ2-256ξ6,l6=2ξ(1-A1A2),l7=16ξ2(1-A1A2)-32ξ4(1-A1A2)2,l8=32ξ3(1-A1A2),l9=-[(1-A1A2)4+(1-A1A2)2].Denote(2.15)h(ω)=ω16+l1ω14+l2ω12+l3ω10+l4ω8+l5ω6+l6ω5+l7ω4+l8ω3+l9.
Since l9<0 and limω→+∞h(ω)=+∞, then we can conclude that (2.14) has at least one positive root. Without loss of generality, we assume that (2.14) has sixteen positive roots, denoted by ωk(k=1,2,3,…,16). Then, by (2.12), we have
(2.16)τk(j)=1ωk{arccos[2ω6+8ξ2ω4-(1-A1A2)(ω4+4ξ2ω2)2-(1-A1A2)2]+2jπ},
where k=1,2,3,…,16;j=0,1,…, then ±iωk are a pair of purely imaginary roots of (2.4) with τk(j). Define
(2.17)τ0=τk0(0)=min⁡k∈{1,2,3,…,16}{τk(0)}.
The above analysis leads to the following result.Lemma 2.2.
If(H1) and (H2) hold, then all roots of (2.4) have a negative real part when τ∈[0,τ0) and (2.4) admits a pair of purely imaginary roots ±ωk when τ=τk(j)(k=1,2,3,…,16;j=0,1,2,…).Letλ(τ)=α(τ)+iω(τ) be a root of (2.5) near τ=τk(j), α(τk(j))=0, and ω(τk(j))=ωk. Due to functional differential equation theory, for every τk(j),k=1,2,3,…,16;j=0,1,2,…, there exists ε>0 such that λ(τ) is continuously differentiable in τ for |τ-τk(j)|<ε. Substituting λ(τ) into the left hand side of (2.5) and taking derivative with respect to τ, we have
(2.18)[dλdτ]-1=(4λ3+12ξλ2+8ξ2λ)eλτ+4λ+4ξλ(1-A1A2)e-λτ-λ(λ4+4ξλ3+4ξ2λ2)eλτ-τλ.
Noting that
(2.19)[4λ3+12ξλ2+8ξ2λ)eλτ+4λ+4ξ]τ=τk(j)=K1+K2i,[λ(1-A1A2)e-λτ-λ(λ4+4ξλ3+4ξ2λ2)eλτ]τ=τk(j)=P1+P2i,
whereK1=4ξ-12ξωk2cosωkτk(j)-(18ξ2ωk-4ωk3)sinωkτk(j),K2=4ωk+(8ξ2ωk-4ωk3)cosωkτk(j)-12ξωk2sinωkτk(j).P1=4ξωk4cosωkτk(j)+[(1-A1A2)ωk-ωk5-4ξ2ωk3]sinωkτk(j)P2=[(1-A1A2)ωk-ωk5-4ξ2ωk3]cosωkτk(j)-4ξωk4sinωkτk(j),we derive(2.20)[d(Reλ(τ))dτ]τ=τk(j)-1=Re{(4λ3+12ξλ2+8ξ2λ)eλτ+4λ+4ξλ(1-A1A2)e-λτ-λ(λ4+4ξλ3+4ξ2λ2)eλτ}τ=τk(j)=Re{K1+K2iP1+P2i}=K1P1-K2P2P12+P22.
We assume that the following condition holds:
(2.21)(H3)K1P1≠K2P2.
According to above analysis and the results of Kuang [29] and Hale [30], we have the following.Theorem 2.3.
If(H1) and (H2) hold, then the equilibrium E(0,0,0,0) of system (1.2) is asymptotically stable for τ∈[0,τ0). Under the conditions (H1) and (H2), if the condition (H3) holds, then system (1.2) undergoes a Hopf bifurcation at the equilibrium E(0,0,0,0) when τ=τk(j), k=1,2,3,…,16; j=0,1,2,….


## 3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained conditions for Hopf bifurcation to occur whenτ=τk(j),k=1,2,3,…,16;j=0,1,2,…. In this section, we shall obtain the explicit formulae for determining the direction, stability, and periods of these periodic solutions bifurcating from the equilibrium E(0,0,0,0) at these critical value of τ, by using techniques from normal form and center manifold theory [31]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcation at the equilibrium E(0,0,0,0) for τ=τk(j),k=1,2,3,…,16;j=0,1,2,…, and then ±iωk are corresponding purely imaginary roots of the characteristic equation at the equilibrium E(0,0,0,0).For convenience, letx-i(t)=xi(τt)(i=1,2,3,4) and τ=τk(j)+μ, where τk(j) is defined by (2.16) and μ∈R, drop the bar for the simplification of notations, then system (2.2) can be written as an FDE in C=C([-1,0]),R4) as
(3.1)u˙(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
(3.2)Lμϕ=(τk(j)+μ)(0010000100-2ξ0000-2ξ)(ϕ1(0)ϕ2(0)ϕ3(0)ϕ4(0))+(τk(j)+μ)(00000000-1A200A1-100)(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 (3.1) undergoes a Hopf bifurcation at the equilibrium E(0, 0, 0, 0) and the associated characteristic equation of system (3.1) 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
(3.3)Lμϕ=∫-10dη(θ,μ)ϕ(θ)forϕ∈C.
In fact, we can choose
(3.4)η(θ,μ)=(τk(j)+μ)(0010000100-2ξ0000-2ξ)δ(θ)-(τk(j)+μ)(00000000-1A200A1-100)δ(θ+1),
where δ is the Dirac delta function.Forϕ∈C([-1,0],R4), define
(3.5)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, (3.1) is equivalent to the following abstract differential equation:
(3.6)u˙t=A(μ)ut+R(μ)ut,
where ut(θ)=u(t+θ),θ∈[-1,0]. For ψ∈C([0,1],(R4)*), define
(3.7)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 following bilinear form:
(3.8)〈ψ,ϕ〉=ψ-(0)ϕ(0)-∫-10∫ξ=0θψT(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0), and 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
(3.9)q(θ)=(1,α,β,γ)Teiωkτk(j)θ,q*(s)=D(1,α*,β*,γ*)eiωkτk(j)s,
where
(3.10)α=iωk(iωk+2ξ)+e-iωkτk(j)A2e-iωkτk(j),β=-iωk,γ=iωke-iωkτk(j)-ωk2(iωk+2ξ)A2e-iωkτk(j),α*=iωk(2ξ-iωk)-e-iωkτk(j)A1e-iωkτk(j),β*=12ξ-iωk,γ*=ωk2(iωk+2ξ)-iωke-iωkτk(j)A2e-iωkτk(j),D=11+α-α*+β-β*+γ-γ*+τk(j)[γ-(A1-α*)-β-(A2α*+1)]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. [31] and we first compute the coordinates to describe the center manifold C0 at μ=0. Let ut be the solution of (3.1), when μ=0.Define(3.11)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)},
on the center manifold C0, and we have
(3.12)W(t,θ)=W(z(t),z-(t),θ),
where
(3.13)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 (3.1), we have
(3.14)z˙(t)=iωkτk(j)z+q-*(θ)f(0,W(z,z-,θ)+2Re{zq(θ)})≝iωkτk(j)z+q-*(0)f0.
That is,
(3.15)z˙(t)=iωkτk(j)z+g(z,z-),
where
(3.16)g(z,z-)=g20z22+g11zz-+g02z-22+g21z2z-2+⋯.
Hence, we have
(3.17)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
(3.18)g20=g11=g02=0,(3.19)g21=2D-τk(j)[3β*e-iωkτk(j)+3γ*α2α-e-2iωkτk(j)].
Thus, we derive the following values:
(3.20)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 of (3.1) on the center manifold at the critical value τ=τk(j),(k=1,2,3,…,16;j=0,1,2,3,…). Summarizing the results obtained above leads to the following theorem.Theorem 3.1.
The periodic solution is forward (backward) ifμ2>0(μ2<0). The bifurcating periodic solutions on the center manifold are orbitally asymptotically stable with asymptotical phase (unstable) if β2<0(β2>0). The periods of the bifurcating periodic solutions increase (decrease) ifT2>0(T2<0).


## 4. Numerical Examples

In this section, we present some numerical results of system (1.2) to verify the analytical predictions obtained in the previous section. Let us consider the following system:
(4.1)dx1dt=x3,dx2dt=x4,dx3dt=-0.8x3-x1(t-τ)+0.2tanh(x2(t-τ)),dx4dt=-0.8x4-x2(t-τ)+0.4tanh(x1(t-τ)),
which has an equilibrium E0(0,0,0,0) and satisfies the conditions indicated in Theorem 2.3. The equilibrium E0(0,0,0,0) is asymptotically stable for τ=0. Using the software MATLAB (here take j=0 for example), we derive ω0≈4.1208,τ0≈0.6801,λ′(τ0)≈0.3307-3.1524i,g21≈-1.4203-4.5518i. Thus by algorithm (3.20) derived in Section 3, we have c1(0)≈-0.7102-2.1609i,  μ2≈-2.1476, β2≈-1.4202,  T2≈3.1867. Furthermore, it follows that μ2>0 and β2<0. Thus, the equilibrium E0(0,0,0,0) is stable when τ<τ0≈0.6801. Figures 1(a)–1(j) show that the equilibrium E0(0,0,0,0) is asymptotically stable when τ=0.65<τ0≈0.6801. It is observed from Figures 1(a)–1(j) that the input voltage of the i(i=1,2)th neuron and the output of the j(j=3,4)th neuron converge to their steady states in finite time. If we gradually increase the value of τ and keep other parameters fixed, when τ passes through the critical value τ0≈0.6801, the equilibrium E0(0,0,0,0) loses its stability and a Hopf bifurcation occurs, that is, the input voltage of the i(i=1,2)th neuron the output of the j(j=3,4)th neuron will keep an oscillary mode near the equilibrium E0(0,0,0,0). Due to μ2>0 and β2<0, the direction of the Hopf bifurcation is τ>τ0≈0.6801, and these bifurcating periodic solutions from E0(0,0,0,0) at τ0≈0.6801 are stable. Figures 1(j)–2(d) suggest that Hopf bifurcation occurs from the equilibrium E0(0,0,0,0) when τ=0.8>τ0≈0.6801.The time histories and phase trajectories of system (4.1) with τ=0.65<τ0≈0.6801 and the initial value (0.05, 0.05, 0.05, 0.025). The equilibrium E0(0,0,0,0) is asymptotically stable.
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(p)The time histories and phase trajectories of system (4.1) with τ=0.8>τ0≈0.6801 and the initial value (0.05, 0.05, 0.05, 0.025). Hopf bifurcation occurs from the equilibrium E0(0,0,0,0).
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## 5. Conclusions

In this paper, we have studied the bifurcation natures of a delayed neural network model of two neurons with inertial coupling. Regarding delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the equilibriumE0(0,0,0,0), we proposed the conditions to define the parameters for the occurrence of Hopf bifurcation and the oscillatory solutions of the models equations. It is shown that if conditions (H1) and (H2) hold, the equilibrium E0(0,0,0,0) of system (1.2) is asymptotically stable for all τ∈[0,τ0). Under conditions (H1) and (H2), if condition (H3) is satisfied, as the delay τ increases and crosses a threshold value τk(j), the equilibrium loses its stability and the delayed network model of two neurons with inertial coupling enters into a Hopf bifurcation. In addition, using the normal form method and center manifold theorem, explicit formulaes for determining the properties of periodic solutions are worked out. Simulations are included to verify the theoretical findings. The obtained findings are useful in applications of network control.



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