%0 Journal Article
%T Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay
%A Changjin Xu
%A Peiluan Li
%J Advances in Artificial Neural Systems
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/397146
%X A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential 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. 1. 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 -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¨C23]. 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 denotes the input voltage of the th neuron, is the output of the th neuron, is the damping factor and 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 ) in the response of the output voltages to changes in the input, then system (1) can be revised as follows: As is known to us that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation is very critical. When delays are incorporated into the network models, stability, and Hopf bifurcation analysis become much complex. To obtain a deep and clear understanding of dynamics of neural network model with delays, we will make a investigation on system (2), that is, we study the stability, the local Hopf bifurcation for system (2). The remainder of the paper is organized as follows. In
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