A large number of experiments have proved that the ring structure is a common phenomenon in neural networks. Nevertheless, a few works have been devoted to studying the neurodynamics of networks with only one ring. Little is known about the dynamics of neural networks with multiple rings. Consequently, the study of neural networks with multiring structure is of more practical significance. In this article, a class of high-dimensional neural networks with three rings and multiple delays is proposed. Such network has an asymmetric structure, which entails that each ring has a different number of neurons. Simultaneously, three rings share a common node. Selecting the time delay as the bifurcation parameter, the stability switches are ascertained and the sufficient condition of Hopf bifurcation is derived. It is further revealed that both the number of neurons in the ring and the total number of neurons have obvious influences on the stability and bifurcation of the neural network. Ultimately, some numerical simulations are given to illustrate our qualitative results and to underpin the discussion.

中文翻译：

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具有不对称三环结构的大规模神经网络：稳定性、非线性振荡和 Hopf 分岔

大量实验证明，环状结构是神经网络中的普遍现象。尽管如此，还是有一些作品致力于研究只有一个环的网络的神经动力学。对具有多个环的神经网络的动力学知之甚少。因此，研究具有多环结构的神经网络更具有现实意义。在本文中，提出了一类具有三环和多延迟的高维神经网络。这种网络具有不对称结构，这意味着每个环具有不同数量的神经元。同时，三个环共享一个公共节点。选择时延作为分岔参数，确定稳定性开关，推导出Hopf分岔的充分条件。进一步揭示了环中神经元的个数和神经元的总数对神经网络的稳定性和分岔都有明显的影响。最后，给出了一些数值模拟来说明我们的定性结果并支持讨论。