Dynamics of Structured Complex-Valued Hopfield Neural Networks

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. This work was supported in part by the National Council for Scientific and Technological Development (CNPq) under grant no 315820/2021-7, the S ao Paulo Research Foundation (FAPESP) under grant no 2023/03368-0, and the Postdoctoral Researcher Program (PPPD) at the Universidade Estadual de Campinas (UNICAMP). Keywords-- Hopfield neural network, complex-valued neural network, associative memory, braided Hermitian matrix. 1 Introduction Artificial neural networks have been conceived as emulators of the biological neural network synapse process. Their processing units, the artificial neurons, usually act based on input signals received from other neurons or cells. Like a biological neuron firing an electric impulse in the presence of specific chemical components in appropriate concentrations, an artificial neuron fires when certain mathematical conditions are satisfied.