Spatial complexity in multi-layer cellular neural networks

Journal of Differential Equations, 2012

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 (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.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2002

It is shown that first-order autonomous space-invariant cellular neural networks (CNNs) may exhibit a complex dynamic behavior (i.e., equilibrium point and limit cycle bifurcation, strange and chaotic attractors). The most significant limit cycle bifurcation processes, leading to chaos, are investigated through the computation of the corresponding Floquet's multipliers and Lyapunov exponents. It is worth noting that most practical CNN implementations exploit first-order cells and space-invariant templates: so far no example of complex dynamics has been shown in first-order autonomous space-invariant CNNs.

Proceedings of the 2002 7th IEEE International Workshop on Cellular Neural Networks and Their Applications

The stability and dynamics of a class of Cellular Neural Networks (CNN's) in the central linear pan is investigated using the decoupling lechnique based on discrete spatial transforms, Nyquist and root locus techniques.

ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196), 2001

The occurrence of complex dynamic behavior (i.e bifurcation processes, strange and chaotic attractors) in autonomous space-invariant cellular neural networks (CNNs) is investigated. Firstly some sufficient conditions for the instability of CNNs are provided; then some classes of unstable template are identified. Finally it is shown that unstable CNNs often exhibit complex dynamics and for a case study the most significant bifurcation processes are described. It is worth noting that most CNN implementations exploit spaceinvariant templates and so far no example of complex dynamics has been shown in autonomous space-invariant CNNs.

Proceedings of the 2002 International Joint Conference on Neural Networks. IJCNN'02 (Cat. No.02CH37290), 2002

Cellular neural networks (CNNs) are analog dynamic processors that have found several applications for the solution of complex computational problems. The mathematical model of a CNN consists in a large set of coupled nonlinear differential equations that have been mainly studied through numerical simulations; the knowledge of the dynamic behavior is essential for developing rigorous design methods and for establishing new applications. CNNs can be divided in two classes: stable CNNs, with the property that each trajectory (with the exception of a set of measure zero) converges towards an equilibrium point; unstable CNNs with either a periodic or a non/periodic (possibly complex) behavior. The manuscript is devoted to the comparison of the dynamic behavior of two CNN models: the original Chua-Yang model and the Full Range model, that was exploited for VLSI implementations.

2001

It is shown that first-order autonomous space-invariant cellular neural networks (CNNs) may exhibit a complex dynamic behavior (i.e. equilibrium point and limit cycle bifurcation, strange and chaotic attractors). The most significant limit cycle bifurcation processes, leading to chaos, are investigated through the computation of the corresponding Floquet's multipliers and Lyapunov exponents. It is worth noting that most practical CNN implementations exploit first order cells and spaceinvariant templates: so far no example of complex dynamics has been shown in first-order autonomous space-invariant CNNs.

International Journal of Intelligent Computing and Cybernetics, 2011

PurposeThe purpose of this paper is to develop a methodology for the design of cellular neural networks with interconnection topologies optimized and suitable for spatially distributed implementation.Design/methodology/approachThe authors perform combinatorial optimization on the neural network's topology to obtain a sparser network, in which the links between the components of the network that reside in different physical locations are minimized. The approach builds on existing computationally efficient tools for the design of cellular neural networks and uses the concept of the network's stability parameters to assess the performance of the network prior to testing.FindingsIt turns out that the sparser cellular neural networks thus produced exhibit performance that can be on par with that of networks with full connectivity, and that for implementations of modest size, communication delays are not that significant to affect the stability of the dynamical system.Originality/...

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

In this paper, a simple system showing chaotic behavior is introduced. It is based on the well-known concept of cellular neural networks (CNNs), which have already given good results in generating complex dynamics. The peculiarity of the CNN model consists in the fact that it replaces the traditional first-order cell with a noninteger-order one. The introduction of the fractional cell, with a suitable choice of the coupling parameters, leads to the onset of chaos in a simple two-cell system. A theoretical approach, based on the harmonic balance theory, has been used to investigate the existence of chaos. A circuit realization of the proposed fractional two-cell chaotic CNN is reported and the corresponding strange attractor is also shown.

Neural Networks, 2013

This manuscript considers the learning problem of multi-layer neural networks (MNNs) with an activation function which comes from cellular neural networks. A systematic investigation of the partition of the parameter space is provided. Furthermore, the recursive formula of the transition matrix of an MNN is obtained. By implementing the well-developed tools in the symbolic dynamical systems, the topological entropy of an MNN can be computed explicitly. A novel phenomenon, the asymmetry of a topological diagram that was seen in Ban, [J. Differential Equations 246, pp. 552-580, 2009], is revealed.

2021

Graph theory is a discrete branch of mathematics for designing and predicting a network. Some topological invariants are mathematical tools for the analysis of connection properties of a particular network. The Cellular Neural Network (CNN) is a computer paradigm in the field of machine learning and computer science. In this article we have given a close expression to dominating invariants computed by the dominating degree for a cellular neural network. Moreover, we have also presented a 3D comparison between dominating invariants and classical degree-based indices to show that, in some cases, dominating invariants give a better correlation on the cellular neural network as compared to classical indices.