A Synchronized Hopf Bifurcation Analysis Of Neural Nets Model With Time Delays

The interactions between neurons through synapses lead to synchronization and phase-locked state in each levels of neural system and thus make the brain becomes a harmonious unity in brain neural nets. In response to incoming signals, the neuron produces an output in the form of axonal pulses. Because time delay occurs between pulses, when we study the stability of brain neural network, it needs to add delays into previous mathematic model. Thus, we describe accurately the real neural network. There are some essential differences between delayed network and non-delayed network. The stability of the system with delays becomes very complex. Therefore, it is very necessary that we study the stability of neural network with delays.In this paper, we analyze the stability and qualitative properties of the synchronized model of brain neural network. At first, we give the formulation of the model. Secondly, we analyze the stability of the synchronized model by using eigenvalue method. We study the stability of the model by linearizing locally the nonlinear system at the equilibrium. The specific statements are as follows: in the first part, we takeÏ„₁ as a parameter whenÏ„₂ is equal to zero, by analyzing the roots of the characteristic equation, and obtain the stability of the zero solution and the existence of the Hopf bifurcation. In the second part, by using the same method, whenÏ„₁ is in its stable interval, we takeÏ„₂ as a parameter, obtaining the stability of the zero solution and the existence of the Hopf bifurcation. Then, we analyze the Hopf bifurcation properties for the synchronized model of neural network. Finally, numerical simulations are given for two examples to illustrate the previous results.