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

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay di ﬀ erential 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.


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][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][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: where x i (i = 1, 2) denotes the input voltage of the ith neuron, x j ( j = 3, 4) is the output of the jth neuron, ξ > 0 is the damping factor and A i (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 is τ) in the response of the output voltages to changes in the input, then system (1) can be revised as follows:

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: holds, 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: whose characteristic equation has the form namely, In order to investigate the distribution of roots of the transcendental equation (5), the following Lemma is necessary.
Advances in Artificial Neural Systems 3 Lemma 2. If (H1) and (H2) hold, then all roots of (5) have a negative real part when τ ∈ [0, τ 0 ) and (5) admits a pair of purely imaginary roots ±ω k when τ = τ Substituting λ(τ) into the left hand side of (5) and taking derivative with respect to τ, we have Then where We assume that the following condition holds: According to above analysis and the results of Kuang [29] and Hale [30], we have the following theorem

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 = τ k 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 where respectively, where φ(θ) = (φ 1 (θ), φ 2 (θ), φ 3 (θ), φ 4 (θ)) T ∈ C. From the discussion in Section 2, 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 ±ω k τ ( j) k . By the representation theorem, there is a matrix function with bounded variation components η(θ, μ), θ ∈ [−1, 0] such that    Figure 1: (a)-(p) The dynamical behavior of system (37) with τ = 0.7 < τ 0 ≈ 0.9 and the initial value (0.01, 0.02, 0.02, 0.01). The equilibrium E 0 (0, 0, 0, 0) is asymptotically stable.
In fact, we can choose where δ is the Dirac delta function.
Then (18) is equivalent to the abstract differential equatioṅ For φ ∈ C([−1, 0], R 4 ) and ψ ∈ C([0, 1], (R 4 ) * ), define the bilinear form where η(θ) = η(θ, 0), the A = A(0) and A * are adjoint operators. By the discussions in Section 2, we know that ±iω k τ ( j) k are eigenvalues of A(0), and they are also eigenvalues of A * corresponding to iω k τ ( j) k and −iω k τ ( j) k respectively. By direct computation, we can obtain where 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 C 0 at μ = 0. Let u t be the solution of (18) when μ = 0. Define on the center manifold C 0 , and we have Advances in Artificial Neural Systems