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

Neural synchronization is associated with various brain disorders, so it is necessary to study the intrinsic factors that affect the synchronization of coupled neural networks. In this paper, a minimal architecture is proposed as a prototype, comprising two bi-neuron Hopfield neural networks (HNNs) coupled via a memristor. The coupling elevates the original two bi-neuron HNNs into a five-dimensional system, which has an unstable line equilibrium set and rich dynamics absent in the uncoupled case. The results claim that changing the coupling strength and the memristor initial state can trigger periodic, chaotic, hyperchaotic, and quasi-periodic oscillations, along with initial offset-regulating multistability. Sufficient conditions for achieving exponential synchronization are derived, and multiple synchronous regimes with transitions strongly dependent on the initial states are identified. FPGA implementation verifies 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 on neuromorphic circuits and synchronization.