Adaptive Anti-Synchronization of Multi-layer Reaction-Diffusion Neural Networks

In this paper, an antisynchronization problem is considered for an array of linearly coupled reaction-diffusion neural networks with cooperative-competitive interactions and time-varying coupling delays. The interaction topology among the neural nodes is modeled by a multilayer signed graph. The state evolution of a neuron in each layer of the coupled neural network is described by a reaction-diffusion equation (RDE) with Dirichlet boundary conditions. Then, the collective dynamics of the multilayer neural network are modeled by coupled RDEs with both spatial diffusion coupling and state coupling. An edge-based adaptive antisynchronization strategy is proposed for each neural node to achieve antisynchronization by using only local information of neighboring nodes. Furthermore, when the activation functions of the neural nodes are unknown, a linearly parameterized adaptive antisynchronization strategy is also proposed. The convergence of the antisynchronization errors of the nodes is analyzed by using a Lyapunov-Krasovskii functional method and a structural balance condition. Finally, some numerical simulations are presented to demonstrate the effectiveness of the proposed antisynchronization strategies.