Dynamic Learning-Based Synchronization and Multifractal Characteristics of Memristive Hindmarsh–Rose Neuronal Networks in Complex Topologies

To address the poor adaptability of traditional fixed control-gain methods to complex topological dynamics, we propose a Dynamic Learning-based Synchronization (DLS) strategy for memristive Hindmarsh–Rose (HR) neuronal networks. By integrating Multifractal Detrended Fluctuation Analysis, we investigate the synchronization mechanisms and multifractal characteristics of the neuronal networks under four topologies: regular, random, small-world, and scale-free. The results demonstrate that the proposed DLS strategy can significantly enhance the steady-state synchronization factor [Formula: see text]. The regulation of topological structure on the synchronization performance shows obvious heterogeneity, among which the small-world network that balances local aggregation and global efficiency has the best steady-state synchronization performance, while the synchronization stability of the regular network is relatively poor. It is observed that the synchronization factor [Formula: see text] tends to decrease as the singularity spectrum width increases across the four topologies examined. Moreover, the noise experiment shows that a topology with strong synchronization robustness does not necessarily have better fractal robustness. This study clarifies the regulatory mechanism of topological structure on the synchronization behavior and multifractal characteristics of HR neuronal networks, which provides theoretical support for the design of brain-like network control strategies and the evaluation of the network synchronization states.