Jung-Chao Ban

On the structure of multi-layer cellular neural networks

Abstract 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 layerʼs structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2 ⩽ k ⩽ n , and demonstrates each subspaceʼs structure.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

Spatial disorder of cellular neural networks with biased term

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

DEVIL'S STAIRCASE OF GAP MAPS

This study demonstrates the devil’s staircase structure of topological entropy functions for onedimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

Spatial complexity in multi-layer cellular neural networks

This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.

On the Structure of Two-Layer Cellular Neural Networks

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.

Complexity of two dimensional multi-layer cellular neural networks

This study investigates the complexity of the global set of output patterns for two-dimensional multi-layer cellular neural networks. Applying labeling to the output space produces a two-dimensional sofic shift space. The ordering matrices and symbolic transition matrices are introduced to study the spatial entropy of the output space.

Cellular neural networks and zeta functions

This talk is concerned with zeta functions of two-dimensional shifts of finite type. The zeta function is an important invariant, which combines information of all periodic patterns. The zeta function can be explicitly expressed as a reciprocal of an infinite product of polynomials by patterns generation approaches. The methods can apply to two-dimensional cellular neural networks.