Stability analysis of Hopfield-type neural networks

Dynamics of Hopfield-Type Neural Networks with Modulo Periodic Unpredictable Synaptic Connections, Rates and Inputs

In this paper, we rigorously prove that unpredictable oscillations take place in the dynamics of Hopfield-type neural networks (HNNs) when synaptic connections, rates and external inputs are modulo periodic unpredictable. The synaptic connections, rates and inputs are synchronized to obtain the convergence of outputs on the compact subsets of the real axis. The existence, uniqueness, and exponential stability of such motions are discussed. The method of included intervals and the contraction mapping principle are applied to attain the theoretical results. In addition to the analysis, we have provided strong simulation arguments, considering that all the assumed conditions are satisfied. It is shown how a new parameter, degree of periodicity, affects the dynamics of the neural network.

A reference model approach to stability analysis of neural networks

In this paper, a novel methodology called a reference model approach to stability analysis of neural networks is proposed. The core of the new approach is to study a neural network model with reference to other related models, so that different modeling approaches can be combinatively used and powerfully cross-fertilized. Focused on two representative neural network modeling approaches (the neuron state modeling approach and the local field modeling approach), we establish a rigorous theoretical basis on the feasibility and efficiency of the reference model approach. The new approach has been used to develop a series of new, generic stability theories for various neural network models. These results have been applied to several typical neural network systems including the Hopfield-type neural networks, the recurrent back-propagation neural networks, the BSB-type neural networks, the bound-constraints optimization neural networks, and the cellular neural networks. The results obtained unify, sharpen or generalize most of the existing stability assertions, and illustrate the feasibility and power of the new method.

Dynamical Analysis of Continuous Higher-Order Hopfield Networks for Combinatorial Optimization

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.

Some generalized sufficient convergence criteria for nonlinear continuous neural networks

A reason for applying the direct method of Lyapunov to artificial neural networks (ANNs) is to design dynamical neural networks so that they exhibit global asymptotic stability. Lyapunov functions that frequently appear in the ANN literature include the quadratic function, the Persidskii function, and the Luré-Postnikov function. This contribution revisits the quadratic function and shows that via Krasovskii-like stability criteria, it is possible to have a very simple and systematic procedure to obtain not only new and generalized results but also well-known sufficient conditions for convergence established recently by non-Lyapunov methods, such as the matrix measure and nonlinear measure.

A comparative study of two modeling approaches in neural networks

The neuron state modeling and the local field modeling provides two fundamental modeling approaches to neural network research, based on which a neural network system can be called either as a static neural network model or as a local field neural network model. These two models are theoretically compared in terms of their trajectory transformation property, equilibrium correspondence property, nontrivial attractive manifold property, global convergence as well as stability in many different senses. The comparison reveals an important stability invariance property of the two models in the sense that the stability (in any sense) of the static model is equivalent to that of a subsystem deduced from the local field model when restricted to a specific manifold. Such stability invariance property lays a sound theoretical foundation of validity of a useful, cross-fertilization type stability analysis methodology for various neural network models.

Global exponential stability of continuous-time interval neural networks

This paper addresses global robust stability of a class of continuous-time interval neural networks that contain time-invariant uncertain parameters with their values being unknown but bounded in given compact sets. We first introduce the concept of diagonally constrained interval neural networks and present a necessary and sufficient condition for global exponential stability of these interval neural networks irregardless of any bounds of non-diagonal uncertain parameters in connection weight matrices. Then we extend the robust stability result to general interval neural networks by giving a sufficient condition. Simulation results illustrate the characteristics of the main results.