Memristor-coupled dynamics and synchronization in two bi-neuron Hopfield neural networks

Neural synchronization is associated with various brain disorders, making it essential to investigate the intrinsic factors that influence the synchronization of coupled neural networks. In this paper, we propose a minimal architecture as a prototype, consisting of two bi-neuron Hopfield neural networks (HNNs) coupled via a memristor. This coupling elevates the original two bi-neuron HNNs into a five-dimensional system, featuring an unstable line equilibrium set and rich dynamics absent in the uncoupled case. Our results show that varying the coupling strength and the initial state of the memristor can induce periodic, chaotic, hyperchaotic, and quasi-periodic oscillations, as well as initial-offset-regulated multistability. We derive sufficient conditions for achieving exponential synchronization and identify multiple synchronous regimes with transitions that strongly depend on the initial states. Field-programmable gate array (FPGA) implementation confirms the predicted dynamics and synchronization in real time, demonstrating that the memristive coupler enables complex dynamics and controllable synchronization in the most compact Hopfield architecture, with implications for the study of neuromorphic circuits and synchronization.