On the structure of multi-layer cellular neural networks

Let Y⊆{-1,1} ^Z∞×n be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say Y ⁽¹⁾, Y ⁽²⁾,..., Y ⁽ⁿ⁾, 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 1≤i≤n. This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y ⁽ⁱ⁾ and Y ^(j) are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layer's structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2≤k≤n, and demonstrates each subspace's structure.