Standard Cellular Neural Network Research Articles

Complete stability of feedback CNNs with dynamic memristors and second‐order cells

SummaryThe paper considers a feedback cellular neural network (CNN) obtained by interconnecting elementary cells with an ideal capacitor and an ideal flux‐controlled memristor. It is supposed that during the analogue computation of the CNN the memristors behave as dynamic elements, so that each dynamic memristor (DM)‐CNN cell is described by a second‐order differential system in the state variables given by the capacitor voltage and the memristor flux. The proposed networks are called DM‐CNNs, that is CNNs using a dynamic (D) memristor (M). After giving a foundation to the DM‐CNN model, the paper establishes a fundamental result on complete stability, that is convergence of solutions toward equilibrium points, when the DM‐CNN has symmetric interconnections. Because of the presence of dynamic memristors, a DM‐CNN displays peculiar and basically different dynamic properties with respect to standard CNNs. First of all a DM‐CNN computes during the time evolution of the memristor fluxes, instead of the capacitor voltages as for a standard CNN. Furthermore, when a steady state is reached, the memristors keep in memory the result of the computation, that is the limiting values of the fluxes, while all memristor currents and voltages, as well as all currents, voltages, and power in the DM‐CNN vanish. Instead, for standard CNNs, currents, voltages, and power do not drop off when a steady state is reached. Copyright © 2016 John Wiley & Sons, Ltd.

A new design for reconfigurable XOR function based on cellular neural networks

We have described a new method to construct the reconfigurable XOR logic circuit by using the modification of the standard uncoupled cellular neural network (CNN) cells. The modification of the cell is easier to implement in engineering applications. The scheme proposed in this paper, using the modification of standard uncoupled CNN cells, allows less hardware consumption in comparison to the utilisation of chaos computing system or harnessing piecewise-linear systems. The template parameters of the modified cell have been discussed, and the physical circuit implementing the reconfigurable two-input and three-input XOR function has also been presented.

Convergence of a Class of Cooperative Standard Cellular Neural Network Arrays

This paper considers a nonsymmetric standard (S) cellular neural network (CNN) array with cooperative (nonnegative) interconnections between neurons and a typical three-segment piecewise-linear (PL) neuron activation. The CNN is defined by a one-dimensional cell-linking (irreducible) cloning template with nearest-neighbor interconnections and has periodic boundary conditions. The flow generated by the SCNN is monotone but, due to the squashing effect of the horizontal segments in the PL activations, is not eventually strongly monotone (ESM). A new method for addressing convergence of the cooperative SCNN array is developed, which is based on the two main tools: (1) the concept of a frozen saddle, i.e., an unstable saddle-type equilibrium point (EP) enjoying certain dynamical properties that hold also for an asymptotically stable EP (a sink); (2) the analysis of the order relations satisfied by the sinks and saddle-type EPs. The analysis permits to show a fundamental result according to which any pair of ordered EPs of the SCNN contains at least a sink or a frozen saddle. On this basis it is shown that the flow generated by the SCNN enjoys a Limit Set Dichotomy and convergence properties analogous to those valid for ESM flows. Such results hold in the case where the SCNN displays either a local diffusion or a global propagation behavior.

THE ŁOJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

REALIZATION AND BIFURCATION OF BOOLEAN FUNCTIONS VIA CELLULAR NEURAL NETWORKS

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

Elementary linear filtering tasks using CNNs with minimum-size templates

In this paper we investigate the linear filtering capabilities of the standard cellular neural network in the general case of non-symmetric templates. We approached here systematically the CNNs with minimum-size templates (1x3), analyzing in detail their filtering capabilities in the one-dimensional case. Starting from a general form of the spatial transfer function, we emphasize some useful filtering functions that can be obtained. For each filter type, we derive the relations which give the template parameter values, in order to design a given CNN filter with specified characteristics-like central frequency, bandwidth, selectivity etc. Filters with symmetric templates are treated as a particular case. For each type of filtering the characteristics are shown and simulation results are presented as well. Some of these results are then extended to 2-D CNNs and several simulations of useful filtering tasks are presented on real images.

RTD-BASED CELLULAR NEURAL NETWORKS WITH MULTIPLE STEADY STATES

In this paper, we study the relationship between the standard cellular neural network (CNN) and the resonant tunneling diode (RTD)-based CNN. We investigate the functional and advanced capabilities of a new generation of CNNs that exploit the multiplicity of steady states. We also include in the analysis higher order CNNs. Furthermore, some methods for designing RTD-based CNNs with multiple steady states are presented.

Ω-GLOBALLY ATTRACTIVE EQUILIBRIUM POINTS OF THE CNN

The dynamic behavior of the standard Cellular Neural Network (CNN) was studied via explicit solutions of the CNN vector state equations for certain piecewise linear domains Ω. The concepts of Ω-globally attractive domain, Ω-globally attractive equilibrium point, and Ω-globally expelling domain for the CNN state equations are presented. The terminology "Ω-global" indicates that the property is global relative to a domain Ω. It is shown that every "saturated domain" D (i.e. for every vector X∈ D, each corresponding component of X has the same sign as all others and its absolute value is greater than or equal to 1), is either Ω-globally attractive or Ω-globally expelling. Attractive equilibrium points in another special domain were also studied. Sufficient conditions for convergence to such equilibria are given, along with concrete new computer simulation examples which demonstrate this theory. Finally, several definitions and simulation results are given to explicate the complex dynamical relationships which were observed between initial states in the linear domain Ω (0) (i.e. every vector X∈Ω(0) has the property that the absolute value of each component is less that 1) and Ω-globally attractive equilibrium points in the saturated domain.

Methods for image processing and pattern formation in Cellular Neural Networks: a tutorial

In this paper, we demonstrate that many image processing and pattern formation effects of the simple Cellular Neural Network (CNN) can be understood by means of a common approach. By examining the dynamics in the frequency domain, when all CNN cells are in the linear region, the mechanisms for IIR spatial filtering, pattern formation, morphogenesis, and synergetics can be shown to be present, even though each cell has only first-order dynamics. In addition, the method allows many of the standard CNN templates, such as the nonlinear averaging, halftoning, and diffusion templates to be explained in a new light. Through many examples, it is shown how generalizations of these templates can be used to design linear and nonlinear filters for image processing tasks such as low-pass filtering, time-varying spatial filtering, and fingerprint enhancement. >

On the applications of the cellular neural network paradigm in mechanical vibrating systems

Cellular neural network (CNNs) are cellular, analog, programmable multidimensional processing arrays with distributed logic and memory. The CNN paradigm is applied to compute the transient response of mechanical vibrating systems. Owing to the fact that here the displacement field has to be computed, the spatially discretized equations of motion are of second order. In this case the standard CNN cell has to be augmented by another capacitance to have second-order cells. The transient response of a mechanical vibrating system with a few thousand degrees of freedom can be computed in real time. >