神经网络时滞系统非共振双Hopf分岔及其广义同步

本文建立了具有自连接和抑制-兴奋型他连接的两个同性神经元模型。其中自连接是由于兴奋型的突触产生，而他连接则分别对应于两神经元兴奋、抑制型的突触。发现如果有兴奋型自连接就会有双Hopf分岔，而没有时滞自连接时双Hopf分岔就会消失，因此自连接引起了双Hopf分岔。作为一个例子，通过变动连接中的时滞和他连接中的比重，1／√2双Hopf分岔得到了详细研究。通过中心流形约化，分岔点邻域内各种不同的动力学行为得到了分类，并以解析形式表出。神经元活动的分岔路径得以表明。从得到的解析近似解可以发现，本文所研究的具有兴奋一抑制型他连接的两相同神经元的节律不能完全同步而只能广义同步。时滞也可以使其节律消失，两神经元变为非活动的。这些结果在控制神经网络关联记忆和设计人工神经网络方面有着潜在的应用。

The Nonresonant Double Hopf Bifurcation and the Generalized Synchronization in the Delayed Neural Network

A two-neuron network with self/neighbor-delayed-connections is investigated in this paper. The self-connections are due to excitatory synapses. The neighbor-connections for two neurons are distinct, corresponding to excitatory and inhibitory synapses. The two first-order delayed differential equations are proposed to model such network. It is found that the excitatory self-connections can lead to non-resonant double Hopf bifurcations since the double Hopf bifurcation disappears without self delayed connections. As an example, an 1/2 double Hopf bifurcation is studied in detail with varying the delay in connections and the weight of neighbor connections. Various dynamic behaviors are classified in the neighborhood of the bifurcation point and are expressed approximately in a closed form by the center manifold reduction. The bifurcating routes for spiking of the neurons are displayed. It follows from the obtained approximate solutions that spiking of two identical neurons with neighbor connections can't be synchronized completely. Namely, there is only so-called generalized synchronization for the spiking in the considered network. As well as, the time delay can also make the spiking disappear such that the neurons in the network are non active. These results have some potential applications, such as controlling the associative memory and designing the neural network.