Z 2 × Z 3 Equivariant Bifurcation in Coupled Two Neural Network Rings

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 group Γ = Z 3 ×Z 2 , whichmeans the global symmetryZ 2 and internal symmetryZ 3 .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.


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][2][3][4][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 Figure 1.The case leads to the following system of delay differential equations: where  ≥ 0 is the time delay.Let  = ( 1 ,  ( 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 ([−, 0],  ) ) ) , where  ∈ ([−, 0],  6 ).
It is clear that (1) has symmetric group Γ =  3 × 2 , which means the global symmetry  2 and internal symmetry  3 .
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 Section 3, we present a characterization of all possible periodic solutions, their twisted isotropy subgroups, and corresponding fixedpoint 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.

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 The associated characteristic equation of (4) takes the form det (Δ (, )) = 0, where Rewrite (4) as with The infinitesimal generator of the  0 -semigroup generated by linear system (4) is A() with Regarding  as the parameter, we determine when the infinitesimal generator () of the  0 -semigroup generated by linear system (7) has a pair of pure imaginary eigenvalues.
Case 1 (Δ 1 = 0).Let  ( > 0) be a zero of Δ 1 ; then the critical frequency is identified as and the critical delay is Moreover, we differentiate the equality Δ 1 = 0 with respect to  to get Next, we consider the generalized eigenspace corresponding to pure imaginary eigenvalues of A().
Let assumptions ( 1 ) and ( 2 ) hold such that (10) has roots ± 1 when  =   1 .Using Theorem 2.1 in [11], we have the generalized eigenspace  ± 1 consisting of eigenvectors of A(  1 ) corresponding to ± 1 is where Case 2 (Δ 2 = 0).Letting  ( ̸ = 0) be a zero of Δ 2 , then For further analysis, we found that the transversality conditions are met: The generalized eigenspace  ± 2 consisting of eigenvectors of A( ±  2 ) corresponding to ± ± 2 is where In a similar manner it can be shown that, for the fourth factor, Δ 4 = 0, and fifth factor, Δ 5 = 0, we have the following.
Case 3 (Δ 4 = 0).In this case, and the transversality conditions are also met: The generalized eigenspace  ± 4 consisting of eigenvectors of A( 4  ) corresponding to ± 4 is where Case 4 (Δ 5 = 0).Using the same method of case two, we have where

Multiple Hopf Bifurcations
In order to study the Hopf bifurcation of the origin, we consider the action of Γ ×  1 , where Γ =  2 ×  3 and  1 is the temporal.The action of the group  1 is defined as follows: where  ∈  1 .It is clear that For fixed , , let  = 2/ ± .Denote by   the Banach space of all continuous -periodic solutions.Then Γ ×  1  In the following, by discussing the isotropy subgroup and fixed-point subspaces, we will give the possible bifurcating solutions.From Section 2, we have obtained the generalized eigenspace corresponding to pure imaginary eigenvalues of A ( ±  ,  = 1, 2, 4, 5;  = 0, 1, . . ..).Hence, we know their corresponding isotropy subgroup; see Table 1.
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.
For Case 3, Δ 4 = 0 means the purely imaginary eigenvalues associated with Hopf bifurcation are simple, and the That means neurons in different rings are /2 out of phase with each other, and all neurons are 2/3 out of phase with the adjacent behaving identically in the same ring.
6) denote the Banach space of continuous mapping from [−, 0] to  6 equipped with