Kenneth R. Crounse

Pattern formation properties of autonomous Cellular Neural Networks

We use the Cellular Neural Network (CNN) to study the pattern formation properties of large scale spatially distributed systems. We have found that the Cellular Neural Network can produce patterns similar to those found in Ising spin glass systems, discrete bistable systems, and the reaction-diffusion system. A thorough analysis of a 1-D CNN whose cells are coupled to immediate neighbors allows us to completely characterize the patterns that can exist as stable equilibria, and to measure their complexity thanks to an entropy function. In the 2-D case, we do not restrict the symmetric coupling between cells to be with immediate neighbors only or to have a special diffusive form. When larger neighborhoods and generalized diffusion coupling are allowed, it is found that some new and unique patterns can be formed that do not fit the standard ferro-antiferromagnetic paradigms. We have begun to develop a theoretical generalization of these paradigms which can be used to predict the pattern formation properties of given templates. We give many examples. It is our opinion that the Cellular Neural Network model provides a method to control the critical instabilities needed for pattern formation without obfuscating parameterizations, complex nonlinearities, or high-order cell states, and which will allow a general and convenient investigation of the essence of the pattern formation properties of these systems. >

CHARACTERIZATION AND DYNAMICS OF PATTERN FORMATION IN CELLULAR NEURAL NETWORKS

We study some properties of pattern formation arising in large arrays of locally coupled first-order nonlinear dynamical systems, namely Cellular Neural Networks (CNNs). We will present exact results to analyze spatial patterns for symmetric coupling and to analyze spatio-temporal patterns for anti-symmetric coupling in one-dimensional lattices, which will then be completed by approximative results based on a spatial and/or temporal frequency approach. We will discuss the validity of these approximations, which bring a lot of insight. This spectral approach becomes very convenient for the two-dimensional lattice, as exact results get more complicated to establish. In this second part, we will only consider a symmetric coupling between cells. We will show what kinds of motifs can be found in the patterns generated by 3×3 templates. Then, we will discuss the dynamics of pattern formation starting from initial conditions which are a small random noise added to the unstable equilibrium: this can generally be well predicted by the spatial frequency approach. We will also study whether a defect in a pure pattern can propagate or not through the whole lattice, starting from initial conditions being a localized perturbation of a stable pattern: this phenomenon is no longer correctly predicted by the spatial frequency approach. We also show that patterns such as spirals and targets can be formed by “seed” initial conditions — localized, non-random perturbations of an unstable equilibrium. Finally, the effects on the patterns formed of a bias term in the dynamics are demonstrated.

Pyramidal cells: a novel class of adaptive coupling cells and their applications for cellular neural networks

A significant increase in the information processing abilities of CNNs demands powerful information processing at the cell level. In this paper, the defining formula, the main properties, and several applications of a novel coupling cell are presented. Since it is able to implement any Boolean function, its functionality expands on those of digital RAMs by adding new capabilities such as learning and interpolation. While it is able to embed all previously accumulated knowledge regarding useful binary information processing tasks performed by standard CNNs, the pyramidal universal cell provides a broader context for defining other useful processing tasks, including extended gray scale or color image processing as well. Examples of applications in image processing are provided in this paper. Implementation issues are also considered. Assuming some compromise between area and speed, a VLSI implementation of CNNs based on pyramidal cells offers a speedup of up to one million times when compared to corresponding software implementations.

ANALYTICAL CRITERIA FOR LOCAL ACTIVITY OF REACTION–DIFFUSION CNN WITH FOUR STATE VARIABLES AND APPLICATIONS TO THE HODGKIN–HUXLEY EQUATION

This paper presents analytical criteria for local activity in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin–Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the long-lasting action and pace-maker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin–Huxley CNNs supply a possible explanation for why a heart with a normal heart-rate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin–Huxley CNN by a computer. As a second application, we present a smoothed Chuas circuit (SCC) CNN. The bifurcation diagrams of the SCC CNNs show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixed-cell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

Deblurring of images by cellular neural networks with applications to microscopy

In this paper it is shown how the Cellular Neural Network (CNN) can be used to perform image and volume deblurring, with particular emphases on applications to microscopy. We discuss the basic linear theory of the CNN including issues of stability and template size. It is observed that a CNN with a small template can be used to implement an Infinite Impulse Response filter. It is then shown how general deblurring problems can be addressed with a CNN when the blurring operator is known. The proposed application is to solve the basic 3-D confocal image reconstruction task about the form of the blurring operator, confocal behavior in microscope images can be obtained with only 3-5 acquired image planes. In addition, the stored program capability of the CNN Universal Machine would provide integration of several image processing and detection tasks in the same architecture.<<ETX>>

Pseudo-random sequence generation using the CNN universal machine with applications to cryptography

A good source of reproducible random-looking data is important in many applications ranging from simulation of physical systems, communications, and cryptography. It is demonstrated that the cellular neural network (CNN) universal machine (or the discrete-time CNN) is capable of producing a two-dimensional pseudo-random bit stream at high speeds by means of cellular automata (CA). First, the random properties of some irreversible two-dimensional CA rules, selected by applying mean-field theory, are analyzed by a battery of statistical tests. Second, a special class of reversible CA are considered for random number generation and are shown to have some of the desirable properties of physics-like models. Finally, as an example application for random number generation on the CNNUM, some cryptographic schemes are proposed.

Analog combinatorics and cellular automata-key algorithms and layout design

This paper demonstrates how certain logic and combinatorial tasks can be solved using CNNs. The most important application generalizes a shortest path algorithm to design the layout of printed circuit boards. Besides, it is shown how cellular automata can be simulated on CNN, and tasks, such as sorting, parity analysis, histogram calculation of black-and-white images, and computing minimum Hamming distance are also solved.<<ETX>>

An extended class of synaptic operators with application for efficient VLSI implementation of cellular neural networks

A synaptic operator based on multiplication requires a large amount of hardware, particularly in digital implementations. In this brief, we introduce an extended class of synaptic operators which includes the standard multiplication as a particular case. The properties of the extended class of operators are established. Among these, it was found that the global stability theorem of cellular neural networks (CNNs) is applicable to the extended class of synaptic operator as well as for the multiplier-based synapse. This is an important property which allows for the replacement of the multiplication-based synaptic operator with another specific member of the extended class, here referred to as a comparative synapse, without changing the functionality of the overall CNN system. Instead of multiplication, which has an implementation complexity of O(n/sup 2/), the comparative synapse has a complexity of only O(n) in a digital implementation (where n is the resolution of the fixed-point implementation). The effectiveness of this new operator is demonstrated by a few examples of discrete-time CNN operating in all possible dynamic modes (equilibrium, periodic and chaotic).

ANALOGUE COMBINATORICS AND CELLULAR AUTOMATA—KEY ALGORITHMS AND LAY-OUT DESIGN†

This paper demonstrates how certain logic and combinatorial tasks can be solved using CNNs. A design method is proposed for solving combinatorial tasks on a CNN. It can be used to simulate cellular automata on a CNN, to prove the self-reproducing capability of the CNNUM and for sorting, histogram calculation, parity analysis and minimum Hamming distance computation. These solutions are especially useful since they can serve as subroutines of more complex CNNUM algorithms. As an important real-life application the lay-out of printed circuit boards is designed with the CNNUM at an extremely high speed.

SPHERICAL CELLULAR NONLINEAR NETWORKS

We present the structures and behaviors of cellular nonlinear networks (CNN) on spheres (spherical CNN for short). Although the cells in a spherical CNN can have any number of neighbors, in this paper we study the cases where each cell has only three neighbors. We present the simulation results of symmetric and asymmetric spherical CNNs. In our simulations, we use a Bucky-ball spherical CNN structure with 60 cells and an irregular spherical CNN structure with 100 cells. Since a spherical CNN has no boundary, we use an asymmetric spherical CNN to show how information propagates on spheres via local couplings. We also show that if a spherical CNN is locally active then different patterns can be found in its stable output.