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The paper considers a class of discrete-time cellular neural networks (DT-CNNs) obtained by applying Euler's discretization scheme to standard CNNs. Let T be the DT-CNN interconnection matrix which is defined by the feedback cloning template. The paper shows that a DT-CNN is convergent, i.e. each solution tends to an equilibrium point, when T is symmetric and, in the case where T + E n is not positive-semidefinite, the step size of Euler's discretization scheme does not exceed a given bound ( E n is the n × n unit matrix). It is shown that two relevant properties hold as a consequence of the local and space-invariant interconnecting structure of a DT-CNN, namely: (1) the bound on the step size can be easily estimated via the elements of the DT-CNN feedback cloning template only; (2) the bound is independent of the DT-CNN dimension. These two properties make DT-CNNs very effective in view of computer simulations and for the practical applications to high-dimensional processing tasks. The obtained results are proved via Lyapunov approach and LaSalle's Invariance Principle in combination with some fundamental inequalities enjoyed by the projection operator on a convex set. The results are compared with previous ones in the literature on the convergence of DT-CNNs and also with those obtained for different neural network models as the Brain-State-in-a-Box model. Finally, the results on convergence are illustrated via the application to some relevant 2D and 1D DT-CNNs for image processing tasks. [ABSTRACT FROM AUTHOR]

The paper considers a neural network with a class of real extended memristors obtained via the parallel connection of an ideal memristor and a nonlinear resistor. The resistor has the same rectifying characteristic for the current as that used in relevant models in the literature to account for diode-like effects at the interface between the memristor metal and insulating material. The paper proves some fundamental results on the trajectory convergence of this class of real memristor neural networks under the assumption that the interconnection matrix satisfies some symmetry conditions. First of all, the paper shows that, while in the case of neural networks with ideal memristors, it is possible to explicitly find functions of the state variables that are invariants of motions, the same functions can be used as Lyapunov functions that decrease along the trajectories in the case of real memristors with rectifying characteristics. This fundamental property is then used to study convergence by means of a reduction-of-order technique in combination with a Lyapunov approach. The theoretical predictions are verified via numerical simulations, and the convergence results are illustrated via the applications of real memristor neural networks to the solution of some image processing tasks in real time. [ABSTRACT FROM AUTHOR]

Neural networks with memristors are promising candidates to overcome the limitations of traditional von Neumann machines via the implementation of novel analog and parallel computation schemes based on the in-memory computing principle. Of special importance are neural networks with generic or extended memristor models that are suited to accurately describe real memristor devices. The manuscript considers a general class of delayed neural networks where the memristors obey the relevant and widely used generic memristor model, the voltage threshold adaptive memristor (VTEAM) model. Due to physical limitations, the memristor state variables evolve in a closed compact subset of the space; therefore, the network can be mathematically described by a special class of differential inclusions named differential variational inequalities (DVIs). By using the theory of DVI, and the Lyapunov approach, the paper proves some fundamental results on convergence of solutions toward equilibrium points, a dynamic property that is extremely useful in neural network applications to content addressable memories and signal-processing in real time. The conditions for convergence, which hold in the general nonsymmetric case and for any constant delay, are given in the form of a linear matrix inequality (LMI) and can be readily checked numerically. To the authors knowledge, the obtained results are the only ones available in the literature on the convergence of neural networks with real generic memristors. [ABSTRACT FROM AUTHOR]

Convergence in the presence of multiple equilibrium points is one of the most fundamental dynamical properties of a neural network (NN). Goal of the paper is to investigate convergence for the classic Brain-State-in-a-Box (BSB) NN model and some of its relevant generalizations named Brain-State-in-a-Convex-Body (BSCB). In particular, BSCB is a class of discrete-time NNs obtained by projecting a linear system onto a convex body of R n. The main result in the paper is that the BSCB is convergent when the matrix of the linear system is symmetric and positive semidefinite or, otherwise, it is symmetric and the step size does not exceed a given bound depending only on the minimum eigenvalue of the matrix. This result generalizes previous results in the literature for BSB and BSCB and it gives a solid foundation for the use of BSCB as a content addressable memory (CAM). The result is proved via Lyapunov method and LaSalle's Invariance Principle for discrete-time systems and by using some fundamental inequalities enjoyed by the projection operator onto convex sets as Bourbaki–Cheney–Goldstein inequality. [ABSTRACT FROM AUTHOR]

Recent work has considered a class of cellular neural networks (CNNs) where each cell contains an ideal capacitor and an ideal flux-controlled memristor. One main feature is that during the analog computation the memristor is assumed to be a dynamic element, hence each cell is second-order with state variables given by the capacitor voltage and the memristor flux. Such CNNs, named dynamic memristor (DM)-CNNs, were proved to be convergent when a symmetry condition for the cell interconnections is satisfied. The goal of this paper is to investigate convergence and multistability of DM-CNNs in the general case of nonsymmetric interconnections. The main result is that convergence holds when there are (possibly) nonsymmetric, non-negative interconnections between cells and an irreducibility assumption is satisfied. This result appears to be similar to the classic convergence result for standard (S)-CNNs with positive cell-linking templates. Yet, due to the presence of DMs, a DM-CNN displays some basically different and peculiar dynamical properties with respect to S-CNNs. One key difference is that the DM-CNN processing is based on the time evolution of memristor fluxes instead of capacitor voltages as it happens for S-CNNs. Moreover, when a steady state is reached, all voltages and currents, and hence power consumption of a DM-CNN vanish. This notwithstanding the memristors are able to store in a nonvolatile way the result of the processing. Voltages, currents and power instead do not vanish when an S-CNN reaches a steady state. [ABSTRACT FROM PUBLISHER]

The paper introduces a class of memristor neural networks (NNs) that are characterized by the following salient features. (a) The processing of signals takes place in the flux–charge domain and is based on the time evolution of memristor charges. The processing result is given by the constant asymptotic values of charges that are stored in the memristors acting as non-volatile memories in steady state. (b) The dynamic equations describing the memristor NNs in the flux–charge domain are analogous to those describing, in the traditional voltage–current domain, the dynamics of a standard (S) cellular (C) NN, and are implemented by using a realistic model of memristors as that proposed by HP. This analogy makes it possible to use the bulk of results in the SCNN literature for designing memristor NNs to solve processing tasks in real time. Convergence of memristor NNs in the presence of multiple asymptotically stable equilibrium points is addressed and some applications to image processing tasks are presented to illustrate the real-time processing capabilities. Computing in the flux–charge domain is shown to have significant advantages with respect to computing in the voltage–current domain. One advantage is that, when a steady state is reached, currents, voltages and hence power in a memristor NN vanish, whereas memristors keep in memory the processing result. This is basically different from SCNNs for which currents, voltages and power do not vanish at a steady state, and batteries are needed to keep in memory the processing result. [ABSTRACT FROM AUTHOR]

This brief considers a class of delayed full-range (FR) cellular neural networks (CNNs) with uncertain interconnections between neurons modeled by means of intervalized matrices. Using mathematical tools from the theory of differential inclusions, a fundamental result on global robust stability of standard (S) CNNs is extended to prove global robust exponential stability for the corresponding class (same interconnection weights and inputs) of FR-CNNs. The result is of theoretical interest since, in general, the equivalence between the dynamical behavior of FR-CNNs and S-CNNs is not guaranteed. [ABSTRACT FROM AUTHOR]

The paper analyzes some fundamental properties of the solution semiflow of nonsymmetric cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment piecewise-linear (pwl) neuron activation. Two relevant subclasses of SCNNs, corresponding to one-dimensional circular SCNNs with two-sided or single-sided positive interconnections between nearest neighboring neurons only, are considered. For these subclasses it is shown that the associated solution semiflow satisfies the fundamental properties of the CONVERGENCE CRITERION, the NONORDERING OF LIMIT SETS and the LIMIT SET DICHOTOMY, and that this is true although the semiflow is not eventually strongly monotone. As a consequence such CNNs are almost convergent, i.e., almost all solutions converge toward an equilibrium point as time tends to infinity. To the authors' knowledge the paper is the first rigorous investigation on the geometry of limit sets and convergence properties of cooperative SCNNs with a pwl neuron activation. All available convergence results in the literature indeed concern a modified cooperative CNN model where the original pwl activation of the SCNN model is replaced by a continuously differentiable strictly increasing sigmoid function. The main results in the paper are established by conducting a deep analysis of the properties of the omega-limit sets of the solution semiflow defined by the considered subclasses of SCNNs. In doing so the paper exploits and extends some mathematical tools for monotone systems in order that they can be applied to pwl vector fields that govern the dynamics of SCNNs. By using some transformations and referring to specific examples it is also shown that the treatment in the paper can be extended to other subclasses of SCNNs. [ABSTRACT FROM AUTHOR]