A second-order memristive Rulkov neuron: Megastability, extreme events, and collective dynamics

In this paper, a novel discrete-time memristive neuron model is presented by coupling the Rulkov neuron with a newly developed second-order memristor. The proposed model introduces richer nonlinear dynamics while preserving a simple mathematical structure suitable for numerical investigations. The dynamical properties of the model are systematically analyzed using one- and two-dimensional bifurcation diagrams together with the largest Lyapunov exponent. The results show that the proposed system exhibits a wide variety of dynamics, including multiple periodic oscillations, chaotic attractors, and transitions between different dynamical regimes under parameter variations. Furthermore, the model is shown to possess megastability, where numerous coexisting attractors are obtained by varying the initial conditions of the memristor state. The occurrence of extreme events is then investigated. Several parameter regions producing rare high-amplitude events are identified, and their statistical properties are analyzed through peak-amplitude distributions and precursor averaging. The obtained precursor evolution reveals a characteristic increase in the average trajectory prior to the occurrence of extreme events, indicating the existence of a statistically distinguishable precursor signature within the model. Finally, the collective dynamics of a nonlocally coupled ring network composed of the proposed neurons are studied. The synchronization behavior, coupling-induced extreme events, and spatiotemporal patterns are analyzed for different coupling strengths and system parameters. The results show that increasing the coupling strength can destabilize synchronization and induce extreme events even when isolated neurons do not exhibit such behavior. Moreover, phase diagrams reveal distinct regions corresponding to synchronization, local extreme events, global extreme events, and divergence, while the spatiotemporal evolution demonstrates the emergence of chimera-like patterns accompanied by localized extreme events. The obtained results indicate that the proposed memristive neuron provides a rich platform for investigating nonlinear dynamics, megastability, synchronization transitions, and extreme-event phenomena in complex neuronal networks.