On the structure of multi-layer cellular neural networks

Let Y⊆Z∞×n{−1,1} 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 layerʼs structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2⩽k⩽n, and demonstrates each subspaceʼs structure.