AUTONOMOUS CELLULAR NEURAL NETWORKS - A UNIFIED PARADIGM FOR PATTERN-FORMATION AND ACTIVE WAVE-PROPAGATION

This tutorial paper proposes a subclass of cellular neural networks (C NN) having no inputs (i.e., autonomous) as a universal active substrat e or medium for modeling and generating many pattern formation and non linear wave phenomena from numerous disciplines, including biology, ch emistry, ecology, engineering, physics, etc. Each CNN is defined mathe matically by its cell dynamics (e.g., state equations) and synaptic la w, which specifies each cell's interaction with its neighbors. We focu s in this paper on reaction-diffusion CNNs having a linear synaptic la w 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 eac h other by only one layer of linear resistors provides a unified activ e medium for generating trigger (autowave) waves, target (concentric) waves, spiral leaves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell t o its neighboring cells, the resulting CNN can be used to generate var ious turing patterns. Although the equations describing these autonomo us CNNs represent an excellent approximation to the nonlinear partial differential equations describing reaction-diffusion systems if the nu mber 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 parti al differential equations. To demonstrate how an autonomous CNN can se rve as a unifying paradigm for pattern formation and active wave propa gation, several well-known examples chosen from different disciplines are mapped into a generic reaction-diffusion CNN made of third-order c ells. Finally, several examples that can not be modeled by reaction-di ffusion equations are mapped into other classes of autonomous CNNs in order to illustrate the universality of the CNN paradigm.