On the Structure of Two-Layer Cellular Neural Networks

Abstract

Let $\mathbf{Y} \subseteq \{-1,{1\}}^{\mathbb{Z}_{\infty \times 2}}$ be the mosaic solution space of a two-layer cellular neural network (TCNN). We decouple Y into two subspaces, say Y ⁽¹⁾ and Y ⁽²⁾, and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y ⁽ⁱ⁾ is a sofic shift for i = 1,2. This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y ⁽¹⁾ and Y ⁽²⁾ are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a TCNN, each layer’s structure.