A class of discrete-time system modelling a network with two neurons is considered. First, we investigate the global stability of the given system. Next, we study the local stability by techniques developed by Kuznetsov to discrete-time systems. It is found that Neimark-Sacker bifurcation (or Hopf bifurcation for map) will occur when the bifurcation parameter exceeds a critical value. A formula determining the direction and stability of Neimark-Sacker bifurcation by applying normal form theory and center manifold theorem is given. Finally, some numerical simulations for justifying the theoretical results are also provided.

Since one of the models with electric circuit implementation was proposed by Hopfield [

For computer simulation, experimental or computational purposes, it is common to discrete the continuous-time neural networks. Certainly, the discrete-time analog inherits the dynamical characteristics of the continuous-time neural networks under mild or no restriction on the discretional step size and also remains functionally similar to the continuous-time system and any physical or biological reality that the continuous-time system has. We refer to [

It will be pointed that two neurons have the same transfer function

The discrete-time system (

In this paper, we investigate the nonlinear dynamical behavior of a discrete-time system of two neurons, namely, (

The organization of this paper is as follows. In Section

In this section, we discuss the global and local stability of the equilibrium

(H1)

Let

It follows from system (

Next, we will analyze the local stability of the equilibrium

(H2)

For the sake of simplicity and the need of discussion, we define the following parameters:

The zero solution of (

Under (H2), using Taylor expansion, we can expand the right-hand side of system

Obviously, we obtained that the modulus of eigenvalues

Combining case 1 with case 2 yields that the the eigenvalues

In what follows, we will choose

If (H2) and

where

Under the assumption

By Lemma 2.2 in [

Suppose that (H2) and

If

If

The Neimark-Sacker bifurcation occurs at

Obviously, we have

In the above section, we have shown that Neimark-Sacker bifurcation occurs at some value

(H3)

Now system (

where

where

Thus

Suppose that condition (H3) holds and

This method is introduced by Kuznetsov in [

In this section, we give numerical simulations to support our theoretical analysis. Let

The equilibrium

The equilibrium

The equilibrium

An invariant closed circle bifurcates from equilibrium

An invariant closed circle bifurcates from equilibrium

An invariant closed circle bifurcates from equilibrium

The discrete-time delay system of neural networks provides some dynamical behaviors which enrich the theory of continuous system and have potential applications in neural networks. Although the system discussed in this paper is quite simple, it is potentially useful applications as the complexity which has been carried over to the other models with delay. By choosing a proper bifurcation parameter, we have shown that a Neimark-Sacker bifurcation occurs when this parameter passes through a critical value. We have also determined the direction of the Neimark-Sacker bifurcation and the stability of periodic solutions by applying the normal form theory and the center manifold reduction. Our simulation results have verified and demonstrated the correctness of the theoretical results. Our work is a excellent complementary to the known results [

This work is supported by National Natural Science Foundation of China (no. 10961008), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province (2012), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).