一类具有对称性的神经网络模型的分支
Bifurcations of a neural network model with symmetry
摘要：
我们分析了一个家庭的集群兴奋抑制神经网络和潜在的分岔结构，因为在网络中的置换对称性作为全球耦合强度$g$是不同的。我们主要考虑两个网络拓扑结构：一个是排除自连接的全对全连接网络，另一个是兴奋性细胞被分成大小相等的簇的网络。虽然在这两种情况下的分岔结构是由系统中的对称性，两个系统的行为是定性不同的。在全对全连通网络中，系统会发生Hopf分岔，从而得到周期轨道解;值得注意的是，对于大的$g$，存在一个稳定的周期轨道解，并且没有稳定的不动点。相比之下，在集群网络，有没有Hopf分支，有一个家庭的稳定不动点大$g$。
英文摘要：
We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength $g$ is varied. We primarily consider two network topologies: an all-to-all connected network which excludes self-connections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large $g$, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large $g$.
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