Fixed points as regulatory hubs in discrete memristive neural networks: An analysis of the FitzHugh–Nagumo model

This study investigates the dynamics of discrete memristive FitzHugh–Nagumo (FHN) neural networks. We introduce a discrete memristor with hyperbolic tangent nonlinearity and incorporate it into neuron models ranging from single neurons and coupled pairs to complex networks with ring and small-world topologies. Stability and bifurcation analyses reveal transitions from periodic to chaotic dynamics. A key contribution is the identification of a constant fixed point that remains invariant across periodic, weakly chaotic, and chaotic regimes. Linear stability analysis of this fixed point provides a fundamental basis for understanding the system’s dynamical evolution. The fixed point theory explains how memristive coupling induces diverse synchronization patterns, including stable phase-locking and synchronization–desynchronization transitions, and further accounts for the emergence of chimera states in ring networks as well as their alteration in small-world networks owing to long-range connections. Field-programmable gate array (FPGA) implementation successfully validates the mathematical models, confirming the feasibility of hardware realization. Overall, this work establishes a theoretical framework linking fixed point properties with firing mechanisms and synchronization dynamics in discrete memristive FHN neural networks, providing insights into potential applications in neuromorphic computing.