In this paper, we explore the dynamics of structured complex-valued Hopfield neural networks (CvHNNs), which arise when the synaptic weight matrix possesses specific structural properties. We begin by analyzing CvHNNs with a Hermitian synaptic weight matrix and establish the existence of four-cycle dynamics in CvHNNs with skew-Hermitian weight matrices operating synchronously. Furthermore, we introduce two new classes of complex-valued matrices: braided Hermitian and braided skew-Hermitian matrices. We demonstrate that CvHNNs utilizing these matrix types exhibit cycles of length eight when operating in full parallel update mode. Finally, we conduct extensive computational experiments on synchronous CvHNNs, exploring other synaptic weight matrix structures. The findings provide a comprehensive overview of the dynamics of structured CvHNNs, offering insights that may contribute to developing improved associative memory models when integrated with suitable learning rules.

翻译：本文研究了具有特定结构特性的突触权重矩阵所引发的结构化复值Hopfield神经网络（CvHNNs）的动力学行为。我们首先分析了具有厄米特突触权重矩阵的CvHNNs，并证明了在具有斜厄米特权重矩阵的同步运行CvHNNs中存在四周期动力学。此外，我们引入了两类新的复值矩阵：编织厄米特矩阵和编织斜厄米特矩阵。我们证明了采用这些矩阵类型的CvHNNs在全并行更新模式下会呈现长度为八的周期。最后，我们对同步CvHNNs进行了大量计算实验，探索了其他突触权重矩阵结构。这些发现为结构化CvHNNs的动力学提供了全面概述，所获得的见解在与合适的学习规则结合时，可能有助于开发改进的联想记忆模型。