1,720,961 research outputs found
Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube
No full textThe paper considers nonsmooth neural networks described by a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs include the relevant class of neural networks, introduced by Li, Michel and Porod, described by linear systems evolving in a closed hypercube of Rn. The main result in the paper is a necessary and sufficient condition for multistability of DVIs with nonsymmetric and cooperative (nonnegative) interconnections between neurons. The condition is easily checkable and provides a sharp bound between DVIs that can store multiple patterns, as asymptotically stable equilibria, and those for which this is not possible. Numerical examples and simulations are presented to confirm and illustrate the theoretic findings
Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks
Full text linkThe paper addresses convergence of solutions for a class of differential inclusions termed differential variational inequalities (DVIs). Each DVI describes the dynamics of a neural network (NN) evolving in a closed hypercube of and defined by a continuously differentiable, {\em cooperative\/} and (possibly) nonreciprocal vector field . The main result in the paper is that under a new condition on , which is called strong Kamke-Muller condition, the solution semiflow generated by the DVI is strongly order preserving (SOP) and hence it satisfies a {\sc Limit Set Dichotomy} and enjoys generic convergence properties. A characterization of the SKM condition is given in terms of the interconnection properties of the Jacobian matrix of . In the case where is an affine, or a linear, vector field the considered DVIs include two relevant classes of NNs, namely, the linear systems operating on a closed hypercube, also known as linear systems in saturated mode (LSSMs), and the full-range (FR) model of cellular neural networks (CNNs). By applying the results to LSSMs it is obtained that any cooperative LSSM with a (possibly) nonsymmetric and fully interconnected matrix is generically convergent. Analogous results hold for FRCNNs. All the obtained convergence results hold in the general case where the DVIs, and the LSSMs and FRCNNs, possess multiple equilibrium points
Limit set dichotomy and multistability for a class of cooperative neural networks with delays
No full textRecent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points (EPs) (multistability) for neural networks (NNs) with nonsymmetric cooperative (nonnegative) interconnections and neuron activations modeled by piecewise linear (PL) functions. One basic difficulty is that the semiflows generated by such NNs are monotone but, due to the horizontal segments in the PL functions, are not eventually strongly monotone (ESM). This notwithstanding, it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows, although they are not ESM, enjoy convergence properties similar to those of ESM semiflows. The results obtained so far concern the case of cooperative NNs without delays. The goal of this paper is to extend some of the existing results to the relevant case of NNs with delays. More specifically, this paper considers a class of NNs with PL neuron activations, concentrated delays, and a nonsymmetric cooperative interconnection matrix A and delay interconnection matrix Aτ. The main result is that when A+Aτ satisfies a full interconnection condition, then the generated semiflows, which are monotone but not ESM, satisfy a limit set dichotomy analogous to that valid for ESM semiflows. It follows that there is an open and dense set of initial conditions, in the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the general case where the NNs possess multiple EPs, i.e., is a result on multistability, and is valid for any constant value of the delays
The dichotomy of omega-limit sets fails for cooperative standard CNNs
No full textThe paper investigates some basic aspects of the solution semiflow associated to a class of cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment pwl neuron activation. It is assumed that the SCNN neuron interconnection matrix is irreducible. By means of two counterexamples the following basic facts are shown: 1) in general the semiflow associated to the SCNN is not eventually strongly monotone; 2) in the general case also the fundamental property of the omega-limit set dichotomy fails. The consequences of these results are discussed in the context of the existing methods for addressing convergence of cooperative dynamical systems
Lojasiewicz inequality and exponential convergence of the full-range model of CNNs
No full textThis paper considers the Full-range (FR) model of Cellular Neural Networks (CNNs) in the case where the signal range is delimited by an ideal hard-limiter nonlinearity with two vertical segments in the i−v characteristic. A Łojasiewicz inequality around any equilibrium point, for a FRCNN with a symmetric interconnection matrix, is proved. It is also shown that the Łojasiewicz exponent is equal to 1/2. The main consequence is that any forward solution of a symmetric FRCNN has finite length and is exponentially convergent toward an equilibrium point, even in degenerate situations where the FRCNN possesses non-isolated equilibrium points. The obtained results are shown to improve the previous results in literature on convergence or almost convergence of symmetric FRCNNs
Multistability of delayed neural networks with hard-limiter saturation nonlinearities
No full textThe paper considers a class of nonsmooth neural networks where hard-limiter saturation nonlinearities are used to constrain solutions of a linear system with concentrated and distributed delays to evolve within a closed hypercube of Rn. Such networks are termed delayed linear systems in saturated mode (D-LSSMs) and they are a generalization to the delayed case of a relevant class of neural networks previously introduced in the literature. The paper gives a rigorous foundation to the D-LSSM model and then it provides a fundamental result on convergence of solutions toward equilibrium points in the case where there are nonsymmetric cooperative (nonnegative) interconnections between neurons. The result ensures convergence for any finite value of the maximum delay and is physically robust with respect to perturbations of the interconnections. More importantly, it encompasses situations where there exist multiple stable equilibria, thus guaranteeing multistability of cooperative D-LSSMs. From an application viewpoint the delays in combination with the property of multistability make D-LSSMs potentially useful in the fields of associative memories, motion detection and processing of temporal patterns
On global exponential stability of standard and Full-Range CNNs
No full textThis paper compares the dynamical behaviour of the standard (S) cellular neural networks (CNNs) and the full-range (FR) CNNs, when the two CNN models are characterized by the same set of parameters (interconnections and inputs). The FR-CNNs are assumed to be characterized by ideal hard-limiter nonlinearities with two vertical segments in the i-υ characteristic. The main result is that some basic conditions ensuring global exponential stability (GES) of the unique equilibrium point of S-CNNs, with or without delay, continue to ensure the same property for FR-CNNs for the same set of parameters. The significance of this result is discussed with respect to the results in a paper by Corinto and Gilli addressing the similarity of the qualitative behaviour of S-CNNs and FR-CNNs. FR-CNNs are analysed in this paper from a rigorous mathematical viewpoint by means of theoretical tools from set-valued analysis and differential inclusions. In particular, GES is investigated via an extended Lyapunov approach that is applicable to the differential inclusion describing the dynamics of FR-CNNs
A study on semiflows generated by cooperative full-range CNNs
No full textThe paper considers the full-range (FR) model of cellular neural networks (CNNs) characterized by ideal hard-limiter nonlinearities with two vertical segments in the current–voltage characteristic. It is shown that when the FRCNNs are cooperative, i.e., there are excitatory interconnections between distinct neurons, the generated solution semiflow is monotone and that monotonicity implies some fundamental restrictions on the geometry of omega-limit sets. The result on monotonicity is a generalization to the class of differential inclusions describing the dynamics of FRCNNs of a classic result due to Kamke for cooperative ordinary differential equations. The paper also points out difficulties to use the standard theory of eventually strongly monotone (ESM) semiflows for addressing convergence of FRCNNs. By means of counterexamples, it is shown that, even assuming the irreducibility of the interconnections, the semiflow generated by a cooperative FRCNN is not ESM; furthermore, also the limit set dichotomy can be violated
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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