# Z

Authors: Baodong Zheng; Haidong Yin
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

We study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric groupΓ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3. We discuss the spatiotemporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling; that is, different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase, antiphase, T/3,2T/3,4T/3,5T/6, or 7T/6 periods out of phase depending on the delay. Some numerical simulations support our analysis results.


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## Body

## 1. Introduction

The theory of spatiotemporal pattern formation in systems of coupled nonlinear oscillators with symmetry has grown extensively in recent years. Its impact has been felt in a wide variety of fields of applied science. Coupled networks of nonlinear dynamical systems have become important models for studying the behavior of large complex systems. These models allow us to investigate fundamental features of physical systems, biological systems, and so on. The central question is to understand how specific properties of the individual behavior and the coupling architecture can give rise to the emergence of new collective phenomena [1–5]. Couple can lead to oscillators’ synchronization, chaos, symmetric bifurcation, and so on [6].Networks with a ring topology, where locally coupled oscillators or oscillatory populations form a closed loop of signal transmission, appear to be relevant for many practical situations. These systems sometimes show symmetric properties. In general, symmetric systems typically exhibit more complicated bifurcations than nonsymmetric systems, and as well they may increase the dimension of the space and the number of variables involved. Some bifurcations can have a smaller codimension in a class of systems with specified symmetries. Other bifurcations, on the contrary, may not occur in the presence of certain symmetries [7, 8].Time delays have been incorporated into coupled models by many authors, since in real systems the signal inevitably propagates from one oscillator to the next over a finite distance and with a finite speed; a time delay can not be negligible. From the mathematical point of view, the presence of delays makes the problem harder to handle. In fact, the state vector characterizing a nonlinear delayed system evolves in an infinite dimensional functional space. Networks with interacting loops and time delays are common in physiological systems. For example, there are many interacting loops and feedback systems in the model of brain’s motor circuitry [9, 10].In this paper, we focus on the simplest Hopfield network with delays. This model consists of two coupling unidirectional rings, each with three oscillators. See Figure1.Figure 1
The architecture of the model (1).The case leads to the following system of delay differential equations:(1)x˙1(t)=-x1(t)+btanh(x2(t))+ctanh(x4(t-τ)),x˙2(t)=-x2(t)+btanh(x3(t))+ctanh(x5(t-τ)),x˙3(t)=-x3(t)+btanh(x1(t))+ctanh(x6(t-τ)),x˙4(t)=-x4(t)+btanh(x5(t))+ctanh(x1(t-τ)),x˙5(t)=-x5(t)+btanh(x6(t))+ctanh(x2(t-τ)),x˙6(t)=-x6(t)+btanh(x4(t))+ctanh(x3(t-τ)),
where τ≥0 is the time delay. Let X=(x1,x2,x3,x4,x5,x6) represent the state variables. For Γ=Z3×Z2, where Z3 and Z2 are the cycle group, the action on X follows ρ∈Z3, κ∈Z2:
(2)ρ(x1,x2,x3,x4,x5,x6)=(x2,x3,x1,x5,x6,x4),κ(x1,x2,x3,x4,x5,x6)=(x4,x5,x6,x1,x2,x3).We will determine the effects of symmetric coupling between parallel copies of a network structure in the presence of delays. In the following, we focus on the symmetric properties of (1). Let C([-τ,0],R6) denote the Banach space of continuous mapping from [-τ,0] to R6 equipped with the supremum norm φ=sup⁡-τ≤θ≤0|φ(θ)| for φ∈C([-τ,0],R6). Let σ∈R,A≥0, X:[σ-τ,σ+A]→R6, t∈[σ,σ+A] be defined by Xt(θ)=X(t+θ) for -τ≤θ≤0. Define the mapping f:C([-τ,0],R6)→R6 by
(3)fφ=-φ10+tanh⁡φ20+tanh⁡φ4-τ-φ20+tanh⁡φ30+tanh⁡φ5-τ-φ30+tanh⁡φ10+tanh⁡φ6-τ-φ40+tanh⁡φ50+tanh⁡φ1-τ-φ50+tanh⁡φ60+tanh⁡φ2-τ-φ60+tanh⁡φ40+tanh⁡φ3-τ,
where φ∈C([-τ,0],R6).It is clear that (1) has symmetric group Γ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3.In the next section we focus on the linear stability analysis of the trivial equilibrium. This then leads us to a discussion of the bifurcations of the trivial equilibrium. In Section3, we present a characterization of all possible periodic solutions, their twisted isotropy subgroups, and corresponding fixed-point subspaces. We obtain some important results about spontaneous bifurcations of multiple branches of periodic solutions and their spatiotemporal patterns, which describe the oscillatory mode of each neuron. Finally, some numerical simulations are carried out to support the analysis results.


## 2. Elementary Analysis

It is clear that (0,0,0,0,0,0) is an equilibrium point of (1). The linearization of (1) at the origin leads to
(4)x˙1(t)=-x1(t)+bx2(t)+cx4(t-τ),x˙2(t)=-x2(t)+bx3(t)+cx5(t-τ),x˙3(t)=-x3(t)+bx1(t)+cx6(t-τ),x˙4(t)=-x4(t)+bx5(t)+cx1(t-τ),x˙5(t)=-x5(t)+bx6(t)+cx2(t-τ),x˙6(t)=-x6(t)+bx4(t)+cx3(t-τ).
The associated characteristic equation of (4) takes the form
(5)det⁡(Δ(λ,τ))=0,
where
(6)Δλ,τ=λI6-CircA0,A1,A0=0b000bb00,A1=c000c000ce-λτ.Rewrite (4) as
(7)U˙(t)=LU(t)
with
(8)Lφ=-φ1(0)+bφ2(0)+cφ4(-τ)-φ2(0)+bφ3(0)+cφ5(-τ)-φ3(0)+bφ1(0)+cφ6(-τ)-φ4(0)+bφ5(0)+cφ1(-τ)-φ5(0)+bφ6(0)+cφ2(-τ)-φ6(0)+bφ4(0)+cφ3(-τ)T.
The infinitesimal generator of the C0-semigroup generated by linear system (4) is A(τ) with
(9)Aτϕ=ϕ˙,ϕ∈Dom⁡Aτ,Dom⁡Aτ=ϕ∈C,ϕ˙∈C,ϕ˙0=Lτϕ.Regardingτ as the parameter, we determine when the infinitesimal generator A(τ) of the C0-semigroup generated by linear system (7) has a pair of pure imaginary eigenvalues.Using Lemma 2.1 in [11], the characteristic equation then factors as
(10)Δλ=λ+1-b-ce-λτλ+1-be2π/3i-ce-λτ×λ+1-be4π/3i-ce-λτ×λ+1-b+ce-λτ[λ+1-be2π/3i+ce-λτ]×λ+1-be4π/3i+ce-λτ=Δ1Δ2Δ3Δ4Δ5Δ6=0.It is not difficult to verify thata+bi is a root of Δ2=0 or Δ5=0 if and only if a-bi is a root of Δ4=0 or Δ6=0.In order to study the distribution of zeros of (10), it is sufficient to investigate Δ1=0, Δ2=0, Δ4=0, and Δ5=0. We make the following assumption:(H1):b-1<c;(H2):1+b/2<c.If the assumptions(H1), (H2) hold, then the roots of Δ1=0, Δ2=0, Δ4=0, and Δ5=0 have negative real parts when τ=0. In the sequel, we consider the distribution of zeros of Δ=0.Case 1 (Δ1=0).
Letiω(ω>0) be a zero of Δ1; then the critical frequency is identified as
(11)ω1=c2-1-b2,
and the critical delay is
(12)τk1=1ω12kπ+2π-arccos⁡1-bc,1-b>0;1ω12kπ+π+arccos1-bc,1-b<0;k=0,1,2,….
Moreover, we differentiate the equality Δ1=0 with respect to τ to get
(13)Redλdττ=τk1,ω=ω1=1-b1-b2+ω12>0,1-b>0;<0,1-b<0;hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,….
Next, we consider the generalized eigenspace corresponding to pure imaginary eigenvalues ofA(τ).
Let assumptions(H1) and (H2) hold such that (10) has roots ±iω1 when τ=τ1k. Using Theorem 2.1 in [11], we have the generalized eigenspace U±iω1 consisting of eigenvectors of A(τ1k) corresponding to ±iω1 is
(14)U±iω1=∑r=12‍xrςr,xr∈R,
where
(15)ς1θ=cos⁡ω1θReV1-sinω1θIm⁡V1,ς2θ=sin⁡ω1θReV1+cos⁡⁡ω1θIm⁡V1,V1=1,1,1,1,1,1T,θ∈[-1,0].Case 2 (Δ2=0).
Lettingiω(ω≠0) be a zero of Δ2, then
(16)ω2+=32b+c2-1+b22;ω2-=32b-c2-1+b22,(17)τk2+=1ω2+2kπ+2π-arccos1+b/2c,1+b2>0;1ω2+2kπ+π+arccos1+b/2c,1+b2<0;hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,…,τk2-=1ω2-2kπ+arccos⁡1+b/2-c,1+b2>0;1ω2-2kπ+π-arccos⁡1+b/2-c,1+b2<0;hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,….
For further analysis, we found that the transversality conditions are met:(18)Redλdττ=τk2+,ω=ω2+=ω2+-3/2bω2+1+b/22+ω2+-3/2b2>0,hhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,…,Redλdττ=τk2-,ω=ω2-=ω2--3/2bω2-1+b/22+ω2--3/2b2<0,hhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,….
The generalized eigenspaceU±iω2 consisting of eigenvectors of A(τ2k±) corresponding to ±iω2± is
(19)U±iω2±=∑r=14‍xrεr,xr∈R,
where
(20)ε1θ=cos⁡ω2±θReV2-sinω2±θIm⁡V2,ε2θ=sinω2±θReV2+cos⁡ω2±θIm⁡V2,ε3θ=cos⁡ω2±θReV3-sinω3±θIm⁡V2,ε4θ=sinω2±θReV3+cos⁡ω3±θIm⁡V3,V2=1,e-2π/3e-4π/3,1,e-2π/3,e-4π/3T,V3=1,e-4π/3,e-8π/3,1,e-4π/3,e-8π/3T,θ∈-1,0.
In a similar manner it can be shown that, for the fourth factor,Δ4=0, and fifth factor, Δ5=0, we have the following.Case 3 (Δ4=0).
In this case,(21)ω4=c2-1-b2;τk4=1ω42kπ+π-arccosc1-b,1-b>0;1ω42kπ+arccos-c1-b,1-b<0;k=0,1,2,…,
and the transversality conditions are also met:
(22)Redλdττ=τk4,ω=ω4=1-b1-b2+ω42>0,1-b>0;<0,1-b<0;k=0,1,2,….
The generalized eigenspace U±iω4 consisting of eigenvectors of A(τk4) corresponding to ±iω4 is
(23)U±iω4=∑r=12‍xrςr,xr∈R,
where
(24)ς1θ=cos⁡ω4θReV4-sinω4θIm⁡V4,ς2θ=sinω4θReV4+cos⁡ω4θIm⁡V4,V4=1,1,1,1,1,1T,θ∈-1,0.Case 4 (Δ5=0).
Using the same method of case two, we have(25)ω5+=32b+c2-1+b22,ω5-=32b-c2-1+b22,τk5+=1ω5+2kπ+π-arccos1+b/2-c,1+b2>0;1ω5+2kπ+arccos1+b/2-c,1+b2<0;hhhhhhhhhhhhhhhhhhhhk=0,1,2,…,τk5-=1ω5-2kπ+2π-arccos⁡1+b/2-c,1+b2>0;1ω5-2kπ+π+arccos1+b/2-c,1+b2<0;hhhhhhhhhhhihhhhhhhhhk=0,1,2,…,Redλdττ=τk5+,ω=ω5+=ω5+-3/2bω5+1+b/22+ω5+-3/2b2>0,hhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,…,Redλdττ=τk5-,ω=ω5-=ω5--3/2bω5-1+b/22+ω5--3/2b2<0,hhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,2,….U±iω5±=∑r=14‍xrϵr,xr∈R,
where
(26)ϵ1θ=cos⁡ω5±θReV5-sinω5±θIm⁡V5,ϵ2θ=sinω5±θReV5+cos⁡ω5±θIm⁡V5,ϵ3θ=cos⁡ω5±θReV6-sinω5±θIm⁡V6,ϵ4θ=sinω5±θReV6+cos⁡ω5±θIm⁡V6,V5=1,e-2π/3,e-4π/3,-1,-e-2π/3,-e-4π/3T,V6=1,e-4π/3,e-8π/3,-1,-e-4π/3,-e-8π/3T,θ∈-1,0.


## 3. Multiple Hopf Bifurcations

In order to study the Hopf bifurcation of the origin, we consider the action ofΓ×S1, where Γ=Z2×Z3 and S1 is the temporal. The action of the group S1 is defined as follows:
(27)θx1,x2,x3,x4,x5,x6=eiθx1,eiθx2,eiθx3,eiθx4,eiθx5,eiθx6,
where θ∈S1. It is clear that
(28)Γ=Z3×Z2=1,ρ,ρ2,κ,κρ,κρ2.For fixedk,j, let T=2π/ω±. Denote by PT the Banach space of all continuous T-periodic solutions. Then Γ×S1 acts on PT by
(29)(γ,θ)x(t)=γx(t+θ),(γ,θ)∈Γ×S1,x∈PT.
Denote by SPT the subspace of PT consisting of all T-periodic solutions of (4) with τ=τkj±(j=1,2,4,5). Then, for each subgroup Σ≤Γ×S1,
(30)Fix⁡(Σ,SPT)={x∈SPT;(r,θ)x=x∀(γ,θ)∈Σ}
is a subspace.In the following, by discussing the isotropy subgroup and fixed-point subspaces, we will give the possible bifurcating solutions. From Section2, we have obtained the generalized eigenspace corresponding to pure imaginary eigenvalues of A(τkj±,j=1,2,4,5;k=0,1,….). Hence, we know their corresponding isotropy subgroup; see Table 1.Table 1
The twisted isotropy subgroups forΓ-equivariant system (1).
τ
Group action
Twisted isotropy subgroups
Fixed-point subspaces
τ
k
1
ρ
z
=
z
;
κ
z
=
z
∑
(
ρ
,
κ
)
ς
1
(
θ
)
,
ς
2
(
θ
)
τ
k
2
±
ρ
z
=
e
i
2
π
/
3
z
;
κ
z
=
z
;
ρ
z
=
e
i
4
π
/
3
z
;
κ
z
=
z
;
∑
(
ρ
e
i
2
π
/
3
,
κ
)
;
∑
(
ρ
e
i
4
π
/
3
,
κ
)
;
ε
1
(
θ
)
,
ε
2
(
θ
)
,
ε
3
(
θ
)
,
ε
4
(
θ
).
τ
k
4
±
ρ
z
=
z
;
κ
z
=
-
z
∑
(
ρ
,
-
κ
)
ς
3
(
θ
)
,
ς
4
(
θ
)
τ
k
5
±
ρ
z
=
e
i
2
π
/
3
z
;
κ
z
=
-
z
;
ρ
z
=
e
i
4
π
/
3
z
;
κ
z
=
-
z
;
∑
(
ρ
e
i
2
π
/
3
,
-
κ
)
;
∑
(
ρ
e
i
4
π
/
3
,
-
κ
)
;
ϵ
1
(
θ
)
,
ϵ
2
(
θ
)
,
ϵ
3
(
θ
)
,
ϵ
4
(
θ
)The equivariant bifurcation theorem asserts the existence of branches of small amplitude periodic solutions to system (1), whose spatiotemporal symmetries can be completely characterized by isotropy subgroup.In case one,Δ1=0 implies that the purely imaginary eigenvalues associated with Hopf bifurcation are simple. It follows that the action of Z2×Z3×S1 is given by ρz=z;κz=z. Obviously, the maximal isotropy subgroup is Z2×Z3, which corresponds to standard Hopf bifurcation and is preserved. Thus, all neurons in two rings are synchronous:(1)
(x(t),x(t),x(t),x(t),x(t),x(t)).Similar to the analysis in Case2, Δ2=0 implies that the purely imaginary eigenvalues associated with Hopf bifurcation are double. ∑‍(ρei2π/3,κ) and ∑‍(ρei4π/3,κ) are maximal isotropy subgroups of Z2×Z3 which are generated by ρz=ei2π/3z;κz=z and ρz=ei4π/3z;κz=z. Two types of symmetric periodic solutions are generated:(2)
(x(t),x(t+T/3),x(t+2T/3),x(t),x(t+T/3),x(t2T/3));(3)
(x(t),x(t+2T/3),x(t+4T/3),x(t),x(t+2T/3),x(t4T/3)).For Case3, Δ4=0 means the purely imaginary eigenvalues associated with Hopf bifurcation are simple, and the maximal isotropy subgroup is ∑‍(ρ,-κ) and the symmetric periodic solutions have the form(4)
(x(t),x(t),x(t),-x(t),-x(t),-x(t)).That means neurons in different rings areT/2 out of phase with each other, and all neurons are 2T/3 out of phase with the adjacent behaving identically in the same ring.The fourth case,Δ5=0, gives purely imaginary with double. The maximal isotropy subgroup has two types: ∑‍(ρei2π/3,-κ) and ∑‍(ρei4π/3,-κ), so the symmetric periodic solutions have the form(5)
(x(t),x(t+T/3),x(t+2T/3),x(t+T/2),x(t+5T/6),x(t7T/6));(6)
(x(t),x(t+2T/3),x(t+T/3),x(t+T/2),x(t+7T/6),x(t+5T/6)).In summary, we write the results in Table2.Table 2
Bifurcating periodic solutions.
Twisted isotropy subgroups∑‍
Periodic solutions
∑
(
ρ
,
κ
)
(
x
(
t
)
,
x
(
t
)
,
x
(
t
)
,
x
(
t
)
,
x
(
t
)
,
x
(
t
)
)
∑
(
ρ
e
i
2
π
/
3
,
κ
)
(
x
(
t
)
,
x
(
t
+
T
3
)
,
x
(
t
+
2
T
3
)
,
x
(
t
)
,
x
(
t
+
T
3
)
,
x
(
t
2
T
3
)
)
∑
(
ρ
e
i
4
π
/
3
,
κ
)
(
x
(
t
)
,
x
(
t
+
2
T
3
)
,
x
(
t
+
4
T
3
)
,
x
(
t
)
,
x
(
t
+
2
T
3
)
,
x
(
t
4
T
3
)
)
∑
(
ρ
,
-
κ
)
(
x
(
t
)
,
x
(
t
)
,
x
(
t
)
,
-
x
(
t
)
,
-
x
(
t
)
,
-
x
(
t
)
)
∑
(
ρ
e
i
2
π
/
3
,
-
κ
)
(
x
(
t
)
,
x
(
t
+
T
3
)
,
x
(
t
+
2
T
3
)
,
x
(
t
+
T
2
)
,
x
(
t
+
5
T
6
)
,
x
(
t
7
T
6
)
)
∑
(
ρ
e
i
4
π
/
3
,
-
κ
)
(
x
(
t
)
,
x
(
t
+
2
T
3
)
,
x
(
t
+
T
3
)
,
x
(
t
+
T
2
)
,
x
(
t
+
7
T
6
)
,
x
(
t
+
5
T
6
)
)


## 4. Computer Simulation

To illustrate the analytical results found, in the following we consider the following particular case of (1).Letb=-0.5, c=2. Then ω2+=ω5+=1.421, τ02+=3.585,τ05+=1.375.From Table2, the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay. See Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13.Figure 2
Three adjacent neuronsx1(t), x2(t), x3(t) are 2T/3 out of phase with τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 3
Three adjacent neuronsx4(t), x5(t), x6(t) are 2T/3 out of phase with τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 4
Two neuronsx1(t), x4(t) in different rings are T/2 out of phase with τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 5
Two neuronsx2(t), x5(t) in different rings are T/2 out of phase with τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 6
Two neuronsx3(t), x6(t) in different rings are T/2 out of phase with τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 7
Neurons in different rings areT/2 out of phase with each other, and each neuron is 2T/3 out of phase with the adjacent neuron when τ=1.45 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 8
Three adjacent neuronsx1(t), x2(t), x3(t) are 2T/3 out of phase with τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 9
Three adjacent neuronsx4(t), x5(t), x6(t) are 2T/3 out of phase with τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 10
Two neuronsx1(t), x4(t) in different rings behave identically with τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 11
Two neuronsx2(t), x5(t) in different rings behave identically with τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 12
Two neuronsx3(t), x6(t) in different rings behave identically with τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).Figure 13
Neurons in different rings behave identically, and each neuron is2T/3 out of phase with the adjacent neuron when τ=3.3 and initial condition (2, 1.5, −1, −0.4, −1.5, 1.8).



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*Source: 971520-2014-12-30.xml*