Pattern formation properties of autonomous Cellular Neural Networks

We use the Cellular Neural Network (CNN) to study the pattern formation properties of large scale spatially distributed systems. We have found that the Cellular Neural Network can produce patterns similar to those found in Ising spin glass systems, discrete bistable systems, and the reaction-diffusion system. A thorough analysis of a 1-D CNN whose cells are coupled to immediate neighbors allows us to completely characterize the patterns that can exist as stable equilibria, and to measure their complexity thanks to an entropy function. In the 2-D case, we do not restrict the symmetric coupling between cells to be with immediate neighbors only or to have a special diffusive form. When larger neighborhoods and generalized diffusion coupling are allowed, it is found that some new and unique patterns can be formed that do not fit the standard ferro-antiferromagnetic paradigms. We have begun to develop a theoretical generalization of these paradigms which can be used to predict the pattern formation properties of given templates. We give many examples. It is our opinion that the Cellular Neural Network model provides a method to control the critical instabilities needed for pattern formation without obfuscating parameterizations, complex nonlinearities, or high-order cell states, and which will allow a general and convenient investigation of the essence of the pattern formation properties of these systems