Dissipativity and Synchronization of Generalized BAM Neural Networks With Multivariate Discontinuous Activations

This paper is concerned with the dissipativity and synchronization problems of a class of delayed bidirectional associative memory (BAM) neural networks in which neuron activations are modeled by discontinuous bivariate functions. First, the concept of the Filippov solution is extended to functional differential equations with discontinuous right-hand sides and mixed delays via functional differential inclusions. The global dissipativity of the Filippov solution to the considered BAM neural networks is proven using generalized Halanay inequalities and matrix measure approaches. Second, to realize global exponential complete synchronization of BAM neural networks with multivariate discontinuous activations, discontinuous state feedback controllers are designed using functional differential inclusions theory and nonsmooth analysis theory with generalized Lyapunov functional method. Finally, several numerical examples are provided to demonstrate the applicability and effectiveness of our proposed results.

Winner-take-all in a phase oscillator system with adaptation

We consider a system of generalized phase oscillators with a central element and radial connections. In contrast to conventional phase oscillators of the Kuramoto type, the dynamic variables in our system include not only the phase of each oscillator but also the natural frequency of the central oscillator, and the connection strengths from the peripheral oscillators to the central oscillator. With appropriate parameter values the system demonstrates winner-take-all behavior in terms of the competition between peripheral oscillators for the synchronization with the central oscillator. Conditions for the winner-take-all regime are derived for stationary and non-stationary types of system dynamics. Bifurcation analysis of the transition from stationary to non-stationary winner-take-all dynamics is presented. A new bifurcation type called a Saddle Node on Invariant Torus (SNIT) bifurcation was observed and is described in detail. Computer simulations of the system allow an optimal choice of parameters for winner-take-all implementation.

A Consistent Model for Lazzaro Winner-Take-All Circuit With Invariant Subthreshold Behavior

This paper considers the basic Lazzaro winner-take-all analog computing circuit with N current inputs, N voltage outputs, and a bias current. Motivated by low-power applications, we find a mathematical accurate model of the network when all MOS devices remain inside the subthreshold region all along the transient. This involves analytical inequalities relating the range of admissible input currents to transistor parameters, to supply voltage, and to bias current. The restrictions are sufficiently weak to allow extra demands of functionality or performance. The technical novelty here is that by a slight cut of the maximal subthreshold domain and by choosing proper coordinates, we get a ordinary differential equation invariance problem on a rectangle of ℜ^N+1 space which is analytically tractable. A more precise localization of the asymptotically stable steady state inside the region of interest is also inferred. Although the work is mainly theoretical, an effort to infer simple, interpretable formulas useful for synthesis has been made. The results are numerically verified and their feasibility is discussed in detail. The subject matter is new and it is worth extending it to larger classes of circuits with regional behavior.

Solving nonlinear equality constrained multiobjective optimization problems using neural networks

This paper develops a neural network architecture and a new processing method for solving in real time, the nonlinear equality constrained multiobjective optimization problem (NECMOP), where several nonlinear objective functions must be optimized in a conflicting situation. In this processing method, the NECMOP is converted to an equivalent scalar optimization problem (SOP). The SOP is then decomposed into several-separable subproblems processable in parallel and in a reasonable time by multiplexing switched capacitor circuits. The approach which we propose makes use of a decomposition-coordination principle that allows nonlinearity to be treated at a local level and where coordination is achieved through the use of Lagrange multipliers. The modularity and the regularity of the neural networks architecture herein proposed make it suitable for very large scale integration implementation. An application to the resolution of a physical problem is given to show that the approach used here possesses some advantages of the point of algorithmic view, and provides processes of resolution often simpler than the usual techniques.

Lateral inhibition pyramidal neural network for image classification

The human visual system is one of the most fascinating and complex mechanisms of the central nervous system that enables our capacity to see. It is through the visual system that we are able to accomplish from the most simple task such as object recognition to the most complex visual interpretation, understanding and perception. Inspired by this sophisticated system, two models based on the properties of the human visual system are proposed. These models are designed based on the concepts of receptive and inhibitory fields. The first model is a pyramidal neural network with lateral inhibition, called lateral inhibition pyramidal neural network. The second proposed model is a supervised image segmentation system, called segmentation and classification based on receptive fields. This work shows that the combination of these two models is beneficial, and the results obtained are better than that of other state-of-the-art methods.

Selective positive-negative feedback produces the winner-take-all competition in recurrent neural networks

The winner-take-all (WTA) competition is widely observed in both inanimate and biological media and society. Many mathematical models are proposed to describe the phenomena discovered in different fields. These models are capable of demonstrating the WTA competition. However, they are often very complicated due to the compromise with experimental realities in the particular fields; it is often difficult to explain the underlying mechanism of such a competition from the perspective of feedback based on those sophisticate models. In this paper, we make steps in that direction and present a simple model, which produces the WTA competition by taking advantage of selective positive-negative feedback through the interaction of neurons via p-norm. Compared to existing models, this model has an explicit explanation of the competition mechanism. The ultimate convergence behavior of this model is proven analytically. The convergence rate is discussed and simulations are conducted in both static and dynamic competition scenarios. Both theoretical and numerical results validate the effectiveness of the dynamic equation in describing the nonlinear phenomena of WTA competition.

New accurate and flexible design procedure for a stable KWTA continuous time network

The classical continuous time recurrent (Hopfield) network is considered and adapted to K -winner-take-all operation. The neurons are of sigmoidal type with a controllable gain G, an amplitude m and interconnected by the conductance p. The network is intended to process one by one a sequence of lists, each of them with N distinct elements, each of them squeezed to [0,I] admission interval, each of them having an imposed minimum separation between elements z(min). The network carries out: 1) a matching dynamic process between the order of list elements and the order of outputs, and 2) a binary type steady-state separation between K and K+1 outputs, the former surpassing a +ξ threshold and the later falling under the -ξ threshold. As a result, the machine will signal the ranks of the K largest elements of the list. To achieve 1), the initial condition of processing phase has to be placed in a computable θ -vicinity of zero-state. This requires a resetting procedure after each list. To achieve 2) the bias current M has to be within a certain interval computable from circuit parameters. In addition, the steady-state should be asymptotically stable. To these goals, we work with high gain and exploit the sigmoid properties and network symmetry. The various inequality type constraints between parameters are shown to be compatible and a neat synthesis procedure, simple and flexible, is given for the tanh sigmoid. It starts with the given parameters N, K, I, z(min), m and computes simple bounds of p, G, ξ, θ, and M. Numerical tests and comments reveal qualities and shortcomings of the method.