Characterization and Dynamics of Pattern Formation in Cellular Neural Networks

We study some properties of pattern formation arising in large arrays of locally coupled first-order nonlinear dynamical systems, namely Cellular Neural Networks (CNNs). We will present exact results to analyze spatial patterns for symmetric coupling and to analyze spatio-temporal patterns for anti-symmetric coupling in one-dimensional lattices, which will then be completed by approximative results based on a spatial and/or temporal frequency approach. We will discuss the validity of these approximations, which bring a lot of insight. This spectral approach becomes very convenient for the two-dimensional lattice, as exact results get more complicated to establish. In this second part, we will only consider a symmetric coupling between cells. We will show what kinds of motifs can be found in the patterns generated by 3 × 3 templates. Then, we will discuss the dynamics of pattern formation starting from initial conditions which are a small random noise added to the unstable equilibrium: ...