Existence and stability of a unique equilibrium in continuous-valued discrete-time asynchronous Hopfield neural networks

On the Working Principle of the Hopfield Neural Networks and Its Equivalence to the GADIA in Optimization

Hopfield neural networks (HNNs) are one of the most well-known and widely used kinds of neural networks in optimization. In this article, the author focuses on building a deeper understanding of the working principle of the HNN during an optimization process. Our investigations yield several novel results giving some important insights into the working principle of both continuous and discrete HNNs. This article shows that what the traditional HNN actually does as energy function decreases is to divide the neurons into two classes in such a way that the sum of biased class volumes is minimized (or maximized) regardless of the types of the optimization problems. Introducing neuron-specific class labels, the author concludes that the traditional discrete HNN is actually a special case of the greedy asynchronous distributed interference avoidance algorithm (GADIA) [17] of Babadi and Tarokh for the 2-class optimization problems. The computer results confirm the findings.

Stability analysis of Hopfield-type neural networks

The paper applies several concepts in robust control research such as linear matrix inequalities, edge theorem, parameter-dependent Lyapunov function, and Popov criteria to investigate the stability property of Hopfield-type neural networks. The existence and uniqueness of an equilibrium is formulated as a matrix determinant problem. An induction scheme is used to find the equilibrium. To verify whether the determinant is nozero for a class of matrix, a numerical range test is proposed. Several robust control techniques in particular linear matrix inequalities are used to characterize the local stability of the neural networks around the equilibrium. The global stability of the Hopfield neural networks is then addressed using a parameter-dependent Lyapunov function technique. All these results are shown to generalize existing results in verifying the existence/uniqueness of the equilibrium and local/global stability of Hopfield-type neural networks.

Cellular neural field and its convergence analysis

A new continuum model complementary to traditional cellular neural networks is introduced in this note. We consider a cellular neural field formed by infinitely many cellular neurons and modelled by an integrodifferential equation. Then, using LaSalle's invariance principle on Banach space, we show that the field quantity will asymptotically converge to an equilibrium state under the condition that all equilibria of the system are isolated. From the practical sense, the convergence indicates the essential capability of retrieving message from original raw data.

Multiperiodicity of Discrete-Time Delayed Neural Networks Evoked by Periodic External Inputs

In this paper, the multiperiodicity of a general class of discrete-time delayed neural networks (DTDNNs) is formulated and studied. Several sufficient conditions are obtained to ensure n-neuron DTDNNs can have 2n periodic orbits and these periodic orbits are locally attractive. In addition, we give the conditions for a periodic orbit to be locally or globally attractive when the periodic orbit locates in a designated region. As two typical representatives, the Hopfield neural network and the cellular neural network are examined in detail. These conditions improve and extend the existing stability results in the literature. Simulations results are also discussed in three illustrative examples.

Basins of attraction in fully asynchronous discrete-time discrete-state dynamic networks

This paper gives a formulation of the basins of fixed point states of fully asynchronous discrete-time discrete-state dynamic networks. That formulation provides two advantages. The first one is to point out the different behaviors between synchronous and asynchronous modes and the second one is to allow us to easily deduce an algorithm which determines the behavior of a network for a given initialization. In the context of this study, we consider networks of a large number of neurons (or units, processors, etc.), whose dynamic is fully asynchronous with overlapping updates. We suppose that the neurons take a finite number of discrete states and that the updating scheme is discrete in time. We make no hypothesis on the activation functions of the nodes, so that the dynamic of the network may have multiple cycles and/or basins. Our results are illustrated on a simple example of a fully asynchronous Hopfield neural network.

Exponential Stability of Impulsive High-Order Hopfield-Type Neural Networks With Time-Varying Delays

This paper considers the problems of global exponential stability and exponential convergence rate for impulsive high-order Hopfield-type neural networks with time-varying delays. By using the method of Lyapunov functions, some sufficient conditions for ensuring global exponential stability of these networks are derived, and the estimated exponential convergence rate is also obtained. As an illustration, an numerical example is worked out using the results obtained.