Dynamical behavior analysis for small-world scale-free neural networks

Abstract The neural networks of the human brain exhibit dual structural characteristics of small-world and scale-free networks, and their electrical activity is readily modulated by electromagnetic fields. Constructing neural network models that mimic biological structures is crucial for elucidating the brain’s information processing mechanisms and the pathological basis of neurological disorders. This paper constructs a small-world scale-free neural network (SWSFNN) model under electromagnetic effects using discrete memristors. By optimizing network topology via graph theory, we systematically investigate how memristor initial values and electromagnetic induction intensity influence the network dynamics. Numerical simulations reveal that memristor initial values affect neuronal firing patterns and regulate network synchronization. We further find a spontaneous “synchronization–cluster synchronization–synchronization” transition under constant parameters. This finding demonstrates that, even in the absence of parameter variations, the inherent nonlinear interactions within the neural network system can drive spontaneous state transitions, thereby generating rich dynamical behaviors. Furthermore, increasing electromagnetic induction intensity also enhances network synchronization. This study provides a theoretical foundation for understanding the nonlinear dynamical mechanisms and synchronization control of neural networks in electromagnetic environments, offering insights for neural computation and information processing.