Facilitating and determining Turing patterns in 3-D memristor cellular neural networks

Turing patterns in diffusion neural networks are strongly associated with the performance of artificial intelligence model. However, two practical considerations are rarely mentioned in the mechanism research on the problem of diffusion-coupled cellular neural networks (CNNs): the effects of memristor and three-dimensional (3-D) network structure. Furthermore, facilitating the formation of the determined Turing pattern is expected to make the neural network exhibit the desired intelligence. This paper first realizes the 3-D diffusive networking for a primary cell circuit with memristor characteristics and utilizes a proportional-derivative (PD) control strategy to drive the pattern formation. Next, the characteristic equation is derived using the spatial eigenfunction-based decoupling method. Then, by tracing the root distribution of the characteristic equation, some analytical conditions for the stability or forming Turing patterns are deduced. In addition, the central manifold reduction and linear analysis approaches are utilized to derive the amplitude equations of 3-D Turing patterns and investigate their stability. At last, some numerical simulations are provided to illustrate the results. It is also demonstrated that PD control and memristor occupy dominant positions in various CNNs' Turing patterns. After determining the pattern stability, the implementation of PD control can be considered an effective means to facilitate stable networks to experience Turing instability and generate switchable 3-D pattern.