Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.


Introduction
In the past several decades, the dynamics of neural networks have been extensively investigated.
The artificial neural network has been used widely in various fields such as signal processing, pattern recognition, optimization, associative memories, automatic control engineering, artificial intelligence, and fault diagnosis, because it has the characteristics of self-adaption, self-organization, and self-learning.
Most of the phenomena occurring in real-world complex systems do not have an immediate effect but appear with some delay; for example, there exist time delays in the information processing of neurons.Therefore, time delays have been inserted into mathematical models and in particular in models of the applied sciences based on ordinary differential equations.The delayed axonal signal transmissions in the neural network models make the dynamical behaviors become more complicated, because a time delay into an ordinary differential equation could change the stability of the equilibrium (stable equilibrium becomes unstable) and could cause fluctuations, and Hopf bifurcation can occur (see [1]).And in [1] we can know the time delays' effects from the work by Carlo Bianca, Massimiliano Ferrara, and Luca Guerrini.So, the delay is an important control parameter.
In addition, we must consider that the activations vary in space as well as in time, because the electrons move in asymmetric electromagnetic fields, and there exists diffusion in neural network (see [2]).
In the past, the main work was to research local field neural networks, and static neural networks were rarely studied.Considering the fact that the problem of generalized neural network is more general in many aspects; in this paper, we will investigate a class of generalized neural networks which combine local field neural networks and static neural networks.
In order to study the effect of time delays and diffusion on the dynamics of a neural network model, in [3], Gan and Xu considered the following neural network model: Motivated by the works of Gan and Xu, in this paper, we are concerned with the following neural network system with time delay and reaction-diffusion: with initial and boundary conditions (Neumann boundary conditions): where  1 ,  The organization of this paper is as follows.In Section 2, by analyzing the corresponding characteristic equations, we discuss the local stability of trivial uniform steady state and the existence of Hopf bifurcations of ( 2) and (3).In Section 3, by applying the normal form and the center manifold theorem, closed-form expressions are derived which allow us to determine the direction of the Hopf bifurcations and the stability of the periodic solutions in (2) and (3) (see [2]).In Section 4, numerical simulations are carried out to illustrate the main theoretical results.

Local Stability and Hopf Bifurcation
Obviously, we can easily show that system (2) always has a trivial uniform steady state  * = (0, 0).
For  = 1, if  1 < 0, then (12) has a unique positive root  0 , where It means that the characteristic equation ( 6) admits a pair of purely imaginary roots of the form ± 0 for  = 1.
. Obviously, (12) holds if and only if  = 1.Now, we define that Then, for  = 1, when  =  0 , (6) has a pair of purely imaginary roots ± 0 and all roots of it have negative real parts for  ≥ 2. It is easy to see that if (1) holds, the trivial uniform steady state  * is locally stable for  = 0. Hence, on the basis of the general theory on characteristic equations of delay-differential equations from [3, Theorem 4.1], we can know that  * remains stable when  <  0 , where  0 =  00 .Now, we claim that This will mean that there exists at least one eigenvalue with positive real part when  >  0 .In addition, the conditions for the existence of a Hopf bifurcation [2] are then satisfied generating a periodic solution.To this end, we differentiate (6) about ; then, So, we know that Therefore, By (11), we can obtain that Hence, the transversal condition holds and a Hopf bifurcation occurs when  =  0 and  =  0 .
Consequently, we gain the following results.

Direction and Stability of Hopf Bifurcation
In Section 2, we have demonstrated that systems (2) and ( 3) undergo a train of periodic solutions bifurcating from the trivial uniform steady state  * at the critical value of .In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at critical value  0 by using the normal form theory and center manifold reduction for PFDEs.In this section, we also let the condition (1) hold and  1 < 0. And the work of Bianca and Guerrini in papers [4][5][6][7] is the founder of the method in this section.
Set  =  +  0 .We first should normalize the delay  by the time-scaling  → /.Then, (2) can be rewritten in the fixed phase space ℓ * = ([−1, 0], ) as where  * :ℓ * ×  + →  2 is defined by where  = ( 1 ,  2 )  ∈ ℓ * .By the discussion in Section 2, we can know that the origin (0, 0) is a steady state of (24) and Λ 0 = {− 0  0 ,  0  0 } are a pair of simple purely imaginary eigenvalues of the linear equation and the functional differential equation On the basis of the Riesz representation theorem, there exists a function (, ) of bounded variation for  ∈ [−1, 0] such that Here, we choose that where  is the Dirac delta function.

Numerical Simulations
In this section, in order to illustrate the results above, we will give two examples.
What should be remarked is that we choose the parameter values stochastically under the condition  2 < 0 in order to ensure the existence of Hopf bifurcation at  * when  =  0 .So,  0 = 1.9371 and  0 = 0.2939.Then, we can know on the basis of Theorem 1 that the trivial uniform steady state  * = (0, 0) is asymptotically stable when 0 ≤  <  0 .When  >  0 , the steady state is unstable and a Hopf bifurcation is arising from the steady state.The numerical simulations in Figures 1 and 2 illustrate the facts.
The similar Hopf bifurcation phenomenon is illustrated by the numerical simulations in Figures 3 and 4.

Discussion and Research Perspective
This section is devoted to a summary of discussion and research perspective for the generalized reaction-diffusion neural network model.The model is based on the assumption that the signal transmission is of a digital (McCulloch-Pitts) nature; the model then described a combination of analog and digital signal processing in the network [12].Depending on the modeling approaches, neural networks can be modeled either as a static neural network model or as a local field neural network model.In order to let the problem be more general in many aspects, we build a generalized reactiondiffusion neural network model which includes reactiondiffusion local field neural networks and reaction-diffusion static neural networks.For a delayed neural network, an important issue is the dynamical behaviors of the network [13].Thus, there has been a large body of work discussing the stability and bifurcation in delayed neural network models.By analyzing the characteristic equation, we discussed the local stability of the trivial uniform of system (2) [14].It was shown that when the delay  varies, the trivial uniform steady state exchanges its stability and Hopf bifurcations occur.Numerical simulations illustrated the occurrence of the bifurcate periodic solutions when the delay  passes the critical value  0 .
A research perspective includes the problem of determining the bifurcating periodic solutions and the stability and directions of the Hopf bifurcation using the normal form theory and the center manifold reaction.A challenging perspective is the comparison of the generalized model introduced in the present paper with the experimentally measurable quantities.Indeed, the mathematical models should reproduce both qualitatively and quantitatively empirical data (see [4]).