Department of Mathematical Sciences, National Chengchi University, Wenshan District, Taipei City 11605, Taiwan.
Neural Netw. 2017 Feb;86:18-31. doi: 10.1016/j.neunet.2016.07.011. Epub 2016 Aug 3.
This investigation establishes the global cluster synchronization of complex networks with a community structure based on an iterative approach. The units comprising the network are described by differential equations, and can be non-autonomous and involve time delays. In addition, units in the different communities can be governed by different equations. The coupling configuration of the network is rather general. The coupling terms can be non-diffusive, nonlinear, asymmetric, and with heterogeneous coupling delays. Based on this approach, both delay-dependent and delay-independent criteria for global cluster synchronization are derived. We implement the present approach for a nonlinearly coupled neural network with heterogeneous coupling delays. Two numerical examples are given to show that neural networks can behave in a variety of new collective ways under the synchronization criteria. These examples also demonstrate that neural networks remain synchronized in spite of coupling delays between neurons across different communities; however, they may lose synchrony if the coupling delays between the neurons within the same community are too large, such that the synchronization criteria are violated.
本研究基于迭代方法，建立了具有社区结构的复杂网络的全局集群同步。网络中的单元由微分方程描述，可以是非自治的并且涉及时滞。此外，不同社区中的单元可以由不同的方程控制。网络的耦合结构相当通用。耦合项可以是非扩散的、非线性的、非对称的，并且具有异质的耦合延迟。基于这种方法，推导出了全局集群同步的时滞相关和时滞无关判据。我们将该方法应用于具有异质耦合延迟的非线性耦合神经网络。给出了两个数值例子，以说明在同步判据下神经网络可以以多种新的集体方式表现。这些例子还表明，尽管不同社区之间的神经元之间存在耦合延迟，但神经网络仍然可以保持同步；然而，如果同一社区内的神经元之间的耦合延迟太大，从而违反了同步判据，则神经网络可能会失去同步。
