Synchronization of multiplex and directed reaction-diffusion neural networks with hybrid coupling

The present paper addresses synchronization matter of multiplex and directed reaction-diffusion neural networks with hybrid coupling (MDHCRDNN). Hybrid coupling indicates that state information and spatial information are mixed to promote synchronization together by different channels corresponding to multiple outer matrices. In comparison with reported literature of multiweighted networks in the past, they often suppose outer matrices are symmetric, cooperative with strong connectedness, while outer matrices in this study are asymmetric, competitive and even not connected. Also, negative elements can be involved in inner matrices. A novel approach is developed for the multiplex network by transforming state coupling into spatial coupling and integrating all matrices of spatial diffusion with state into new added matrices. Then, we obtain that synchronization can be guaranteed exponentially if the combined groups of these added matrices are strongly connected. In addition, more issues of hybrid control and adaptive strength are settled. Simulations are provided to verify the effectiveness of these derived results.