Turing patterns in CNNs. I. Once over lightly

International Journal of Circuit Theory and Applications, 1998

In order to be able to take full advantage of the great application potential that lies in cellular neural networks (CNNs) we need to have successful design and learning techniques as well. In almost any analogic CNN algorithm that performs an image processing task, binary CNNs play an important role. We observed that all binary CNNs reported in the literature, except for a connected component detector, exhibit monotonic dynamics. In the paper we show that the local stability of a monotonic binary CNN represents su cient condition for its functionality, i.e. convergence of all initial states to the prescribed global stable equilibria. Based on this ÿnding, we propose a rigorous design method, which results in a set of design constraints in the form of linear inequalities. These are obtained from simple local rules similar to that in elementary cellular automata without having to worry about continuous dynamics of a CNN. In the end we utilize our method to design a new CNN template for detecting holes in a 2D object.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1995

This tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ecology, engineering, physics, etc. Each CNN is defined mathematically by its cell dynamics (e.g., state equations) and synaptic law, which specifies each cell's interaction with its neighbors. We focus in this paper on reaction4iffusion CNNs having a linear synaptic law that approximates a spatial Laplacian operator. Such a synaptic law can be realized by one or more layers of linear resistor couplings. An autonomous CNN made of third-order universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves, target (concentric) waves, spiral waves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell to its neighboring cells, the resulting CNN can be used to generate various turingpatterns. Although the equations describing these autonomous CNNs represent an excellent approximation to the nonlinear partial differential equations describing reaction-diffusion systems if the number of cells is sufficiently large, they can exhibit new phenomena (e.g., propagation failure) that can not be obtained from their limiting partial differential equations. This demonstrates that the autonomous CNN is in some sense more general than its associated nonlinear partial differential equations. To demonstrate how an autonomous CNN can serve as a unifying paradigm for pattern formation and active wave propagation, several well-known examples chosen from different disciplines are mapped into a generic reaction-diffusion CNN made of thirdorder cells. Finally, several examples that can not be modeled by reaction-diffusion equations are mapped into other classes of autonomous CNNs in order to illustrate the universality of the CNN paradigm. 'We use the term cell to mean an artificial neuron in this paper.

IEEE Transactions on Circuits and Systems I: Regular Papers, 2008

Stable patterns that can be realized by a class of 1-D two-layer cellular neural networks (CNNs) are studied in this paper. We first introduce the notions of potentially stable pattern, potentially stable local pattern, and local pattern set. We then show that all of 256 possible sets can be realized as the local pattern set of the two-layer CNN, while only 59 sets can be realized as the local pattern set of the single-layer CNN. We also propose a simple way to optimize the template values of the CNN, which is formulated as a set of linear programming problems, and present the obtained values for all of 256 sets.

In this paper we study and characterize the controllability of a constant 2-cells CNN (Cellular Neural Network) with feedback resembling a symmetric or antisymmetric matrix and input with all entries set to zero except its first element . We characterize and give a precise description of the control in each case. This problem has been attacked already in order to study complete stability and in the seek of chaotic attractor; but this time the controllability is addressed.

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.

2019

Universality in cellular automata theory is a central problem studied and developed from their origins by John von Neumann. In this paper, we present an algorithm where any Turing machine can be converted to one-dimensional cellular automaton with a 2-linear time and display its spatial dynamics. Three particular Turing machines are converted in three universal one-dimensional cellular automata, they are: binary sum, rule 110 and a universal reversible Turing machine.

2016

It is well known that simple reaction–diffusion systems can display very rich pattern formation behaviour. Here we have studied two examples of such systems in three dimensions. First we investigate the morphology and stability of a generic Turing system in three dimensions and then the well-known Gray–Scott model. In the latter case, we added a small number of morphogen sources in the system in order to study its robustness and the formation of connections between the sources. Our results raise the question of whether Turing patterning can produce an inductive signalling mechanism for neuronal growth.

International Journal of Circuit Theory and Applications, 2005

This paper presents a cellular neural network (CNN) scheme employing a new non-linear activation function, called trapezoidal activation function (TAF). The new CNN structure can classify linearly non-separable data points and realize Boolean operations (including eXclusive OR) by using only a single-layer CNN. In order to simplify the stability analysis, a feedback matrix W is deÿned as a function of the feedback template A and 2D equations are converted to 1D equations. The stability conditions of CNN with TAF are investigated and a su cient condition for the existence of a unique equilibrium and global asymptotic stability is derived. By processing several examples of synthetic images, the analytically derived stability condition is also conÿrmed. 394 E. BILGILI,İ. C. G OKNAR AND O. N. UCAN CNN stability is analysed for the standard activation function as in References , (v) global exponential stability conditions of CNN via a new Lyapunov function are stated in Reference [9]. It is well known that the standard uncoupled CNN single-layer structures, extremely useful for realizing Boolean functions, are not capable of classifying linearly nonseparable data. The parity is a binary function of the inputs, which returns a high output if the number of inputs set to 1 is odd and a low output if that number is even. Therefore, for n inputs, the parity problem consists of being able to divide the n-dimensional input space into disjoint decision regions such that all input patterns in the same region yield the same output and, thus is linearly non-separable. Uncoupled CNN can only classify linearly separable data, that is can only separate the input space with hyper-planes [10]. Recently, a single perceptron-like cell with: (i) double threshold, (ii) implemented using only ÿve MOS transistors, (iii) capable of classifying data which are not linearly separable has been reported in References .

Information Sciences, 1993

This paper is a review of recent published work in the application of automata networks as part of a pattern or image recognition system. The principal requirements were to integrate model-based and data-driven approaches within a connectionist framework and to allow full parallelism. In particular, we construct a network of probabilistic cellular automata (PCAs) for iteratively resolving ambiguities and conflicts in pattern recognition. A natural implementation of inductive inference rules in such a network results in a d~amics that is sensitive to nons~metric couplings (s~aptic weights), unlike that of the more common models inspired by statistical physics (e.g., the Boltzmann machine). This, along with full parallelism, means that object recognition must be achieved through the intermediate-time rather than "infinite''-time behavior of the system. Another, more technology-driven, feature includes using local inferences insofar as possible. The framework is translation-invariant, which is natural for image re~gnition. This leads to a different architecture for describing the model from that used in Bayesian inference networks. ' We discuss only bilinear couplings here. Inference rules are often in the form, for example, that feature Q occurs only in conjunction with both features y and 6 in appropriate positions. The most direct way lo incorporate that these would be with trilinear and higher order couplings. An alternative would bc to have hidden units but keep the couplings bilinear (see [281X

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.