Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system
Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system
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DOI：
10.1016/j.physd.2005.12.008
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发表时间：
2006-02-15
影响因子：
4
通讯作者：
Wu, J
中科院分区：
文献类型：
--
作者：
Campbell, SA;Ncube, I;Wu, J
我们考虑一个具有多个离散信号传输延迟的三个相同神经元的网络。这种网络的模型是一个非线性时滞微分方程组。在考虑了系统的绝对同步性和零解的全局吸引性之后，我们详细讨论了平凡解的稳定域的边界。这使我们能够确定系统中可能发生的余维1分叉。特别地，我们证明了产生同步周期解的标准Hopf分支和产生三种周期解的D-3等变Hopf分支的存在性：锁相、镜面反射和驻波。Hopf-Hopf和Hopf-稳态分叉的相互作用被证明是存在的，并且导致稳定的同步解和去同步解共存。利用微扰技术和Floquet理论相结合的方法来确定锁相振荡的稳定性。(C)2006爱思唯尔B.V.保留所有权利。
We consider a network of three identical neurons with multiple discrete signal transmission delays. The model for such a network is a system of nonlinear delay differential equations. After some consideration of the absolute synchronization of the system and the global attractivity of the zero solution, we present a detailed discussion about the boundaries of the stability region of the trivial solution. This allows us to determine the possible codimension one bifurcations which occur in the system. In particular, we show the existence of standard Hopf bifurcations giving rise to synchronized periodic solutions and of D-3 equivariant Hopf bifurcations giving rise to three types of periodic solutions: phase-locked, mirror-reflecting, and standing waves. Hopf-Hopf and Hopf-steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions. Perturbation techniques coupled with the Floquet theory are used to determine the stability of the phase-locked oscillations. (c) 2006 Elsevier B.V. All rights reserved.