Synchronization of the memristor-coupled memristive complex-valued FHN neurons

Complex-valued neurons can encode both amplitude and phase simultaneously, providing a biologically plausible way to model rich signal dynamics in real neural systems. Synaptic coupling is central to coordinating neuronal activity and transmitting information, and memristors—serving as synapse-like devices with internal memory—offer an effective mechanism for mediating its energy-based modulation. In this study, a sinusoidal flux-controlled memristor is used as a biological synapse to connect two memristive complex-valued FitzHugh-Nagumo (mCV-FHN) neurons, forming memristor-coupled mCV-FHN neurons (MC-mCV-FHNs), and their synchronization dynamics are investigated. The system has no equilibrium point and exhibits rich hidden dynamics, including hyperchaotic extreme multistability, multi-scroll attractors with scroll-growth, and pattern migration in firing activities. Based on Lyapunov stability theory in complex space, a rigorous and practical synchronization criterion is derived and numerically validated. The results show that complex synchronization transitions depend critically on both the memristor-coupling strength and the initial state of the memristive coupling channel. Additionally, the Hamilton energy of the mCV-FHN neuron is formulated using Helmholtz’s theorem, explaining the synchronization transition mechanism from an energy-balance perspective. This work establishes a foundation for scaling compact memristive neuron models to larger brain-inspired systems and offers a theoretical basis for future developments in neuromorphic computing and energy-efficient neural hardware.