Autonomous Cellular Neural Networks: A Unified Paradigm for Pattern Formation and Active Wave Propagation

This tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ecology, engineering, physics, etc. Each CNN is defined mathematically by its cell dynamics (e.g., state equations) and synaptic law, which specifies each cell’s interaction with its neighbors. We focus in this paper on reaction-diffusion CNNs having a linear synaptic law that approximates a spatial Laplacian operator. Such a synaptic law can be realized by one or more layers of linear resistor couplings. An autonomous CNN made of third-order universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves, target (concentric) waves, spiral waves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell to its neighboring cells, the resulting CNN can be used to generate various turing patterns. Although the equations describing these autonomous CNNs represent an excellent approximation to the nonlinear partial differential equations describing reaction-diffusion systems if the number of cells is sufficiently large, they can exhibit new phenomena (e.g., propagation failure) that can not be obtained from their limiting partial differential equations. This demonstrates that the autonomous CNN is in some sense more general than its associated nonlinear partial differential equations. To demonstrate how an autonomous CNN can serve as a unifying paradigm for pattern formation and active wave propagation, several well-known examples chosen from different disciplines are mapped into a generic reaction-diffusion CNN made of third-order cells. Finally, several examples that can not be modeled by reaction-diffusion equations are mapped into other classes of autonomous CNNs in order to illustrate the universality of the CNN paradigm.