Stability Of Recurrent Neural Networks Research Articles

Design and analysis of associative memories based on external inputs of delayed recurrent neural networks

This paper presents two new design procedures for synthesizing autoassociative memory and heteroassociative memory based on recurrent neural networks with different external inputs and mixed delays. Sufficient criteria are established to guarantee the global exponential stability for recurrent neural networks with mixed delays. The design procedures are dependent on external inputs, which ensure to restore accurate memories. The stored patterns are associated with their retrieval probes internally in a robust and fault tolerant way. The obtained results improve and extend some related works. Two illustrative examples are given to verify the effectiveness of the theoretical results.

Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays

In this paper, input-to-state stability problems for a class of recurrent neural networks model with multiple time-varying delays are concerned with. By utilizing the Lyapunov-Krasovskii functional method and linear matrix inequalities techniques, some sufficient conditions ensuring the exponential input-to-state stability of delayed network systems are firstly obtained. Two numerical examples and its simulations are given to illustrate the efficiency of the derived results.

Novel Global Exponential Stability Criterion for Recurrent Neural Networks with Time-Varying Delay

The problem of global exponential stability for recurrent neural networks with time-varying delay is investigated. By dividing the time delay interval [0,τ(t)] intoK+1dynamical subintervals, a new Lyapunov-Krasovskii functional is introduced; then, a novel linear-matrix-inequality (LMI-) based delay-dependent exponential stability criterion is derived, which is less conservative than some previous literatures (Zhang et al., 2005; He et al., 2006; and Wu et al., 2008). An illustrate example is finally provided to show the effectiveness and the advantage of the proposed result.

Two algebraic criteria for input-to-state stability of recurrent neural networks with time-varying delays

This paper presents two algebraic criteria for the input-to-state stability of recurrent neural networks with time-varying delays. The criteria which also ensure global exponential stability when the input u(t) is equal to 0 and is easy to be verified only with the connection weights of the recurrent neural networks. Two numerical examples are given to demonstrate the effectiveness of the proposed criteria.

Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances.

In recent years, the global stability of recurrent neural networks (RNNs) has been investigated extensively. It is well known that time delays and external disturbances can derail the stability of RNNs. In this paper, we analyze the robustness of global stability of RNNs subject to time delays and random disturbances. Given a globally exponentially stable neural network, the problem to be addressed here is how much time delay and noise the RNN can withstand to be globally exponentially stable in the presence of delay and noise. The upper bounds of the time delay and noise intensity are characterized by using transcendental equations for the RNNs to sustain global exponential stability. Moreover, we prove theoretically that, for any globally exponentially stable RNNs, if additive noises and time delays are smaller than the derived lower bounds arrived at here, then the perturbed RNNs are guaranteed to also be globally exponentially stable. Three numerical examples are provided to substantiate the theoretical results.

Fine-Tuning and the Stability of Recurrent Neural Networks

A central criticism of standard theoretical approaches to constructing stable, recurrent model networks is that the synaptic connection weights need to be finely-tuned. This criticism is severe because proposed rules for learning these weights have been shown to have various limitations to their biological plausibility. Hence it is unlikely that such rules are used to continuously fine-tune the network in vivo. We describe a learning rule that is able to tune synaptic weights in a biologically plausible manner. We demonstrate and test this rule in the context of the oculomotor integrator, showing that only known neural signals are needed to tune the weights. We demonstrate that the rule appropriately accounts for a wide variety of experimental results, and is robust under several kinds of perturbation. Furthermore, we show that the rule is able to achieve stability as good as or better than that provided by the linearly optimal weights often used in recurrent models of the integrator. Finally, we discuss how this rule can be generalized to tune a wide variety of recurrent attractor networks, such as those found in head direction and path integration systems, suggesting that it may be used to tune a wide variety of stable neural systems.

Further Stability Criterion on Delayed Recurrent Neural Networks Based on Reciprocal Convex Technique

Together with Lyapunov‐Krasovskii functional theory and reciprocal convex technique, a new sufficient condition is derived to guarantee the global stability for recurrent neural networks with both time‐varying and continuously distributed delays, in which one improved delay‐partitioning technique is employed. The LMI‐based criterion heavily depends on both the upper and lower bounds on state delay and its derivative, which is different from the existent ones and has more application areas as the lower bound of delay derivative is available. Finally, some numerical examples can illustrate the reduced conservatism of the derived results by thinning the delay interval.

Robust stability of recurrent neural networks with ISS learning algorithm

In this paper, an input-to-state stability (ISS) approach is used to derive a new robust weight learning algorithm for dynamic neural networks with external disturbance. Based on linear matrix inequality (LMI) formulation, the ISS learning algorithm is presented to not only guarantee exponential stability but also reduce the effect of an external disturbance. It is shown that the design of the ISS learning algorithm can be achieved by solving LMI, which can be easily facilitated by using some standard numerical packages. A numerical example is presented to demonstrate the validity of the proposed learning algorithm.

On the P-critical dynamics analysis of projection recurrent neural networks

Dynamics research of recurrent neural networks is very meaningful in both theoretical importance and practical significance. Recently, the study on the critical dynamics behaviors of such networks has drawn especial attention because of its application requirements. In this paper, new criteria are found to ascertain the global convergence and asymptotic stability of recurrent neural networks under the generally P-critical conditions, i.e., a discriminant matrix M ( L , Γ ) + P is nonnegative definite, where M ( L , Γ ) is a matrix related with the network and P is an arbitrary nonnegative definite matrix. The analysis results given in this paper improve substantially upon the existing relevant convergence and stability results in literature, including both the non-critical conclusions, i.e., the dynamics analysis under the conditions that M ( L , Γ ) is positive definite, and the special critical discuss when M ( L , Γ ) is nonnegative definite.

Solubility and stability of recurrent neural networks with nonlinearity or time-varying delays

Because to apply a deterministic RNN to a noisy time series and the existence of a linear approximation are doubtful, we reconsider the solubility and stability of a recurrent neural network (RNN). Simpler methods are proposed to replace the complicated nonsingular M-matrix method when nonlinearities exist and to replace the complicated linear matrix inequality method when time-varying delays exist.

New delay-dependent criterion for the stability of recurrent neural networks with time-varying delay

This paper is concerned with the global asymptotic stability of a class of recurrent neural networks with interval time-varying delay. By constructing a suitable Lyapunov functional, a new criterion is established to ensure the global asymptotic stability of the concerned neural networks, which can be expressed in the form of linear matrix inequality and independent of the size of derivative of time varying delay. Two numerical examples show the effectiveness of the obtained results.

Almost Sure Exponential Stability of Recurrent Neural Networks With Markovian Switching

This paper presents new stability results for recurrent neural networks with Markovian switching. First, algebraic criteria for the almost sure exponential stability of recurrent neural networks with Markovian switching and without time delays are derived. The results show that the almost sure exponential stability of such a neural network does not require the stability of the neural network at every individual parametric configuration. Next, both delay-dependent and delay-independent criteria for the almost sure exponential stability of recurrent neural networks with time-varying delays and Markovian-switching parameters are derived by means of a generalized stochastic Halanay inequality. The results herein include existing ones for recurrent neural networks without Markovian switching as special cases. Finally, simulation results in three numerical examples are discussed to illustrate the theoretical results.

Stability analysis of static recurrent neural networks using delay-partitioning and projection

This paper introduces an effective approach to studying the stability of recurrent neural networks with a time-invariant delay. By employing a new Lyapunov-Krasovskii functional form based on delay partitioning, novel delay-dependent stability criteria are established to guarantee the global asymptotic stability of static neural networks. These conditions are expressed in the framework of linear matrix inequalities, which can be verified easily by means of standard software. It is shown, by comparing with existing approaches, that the delay-partitioning projection approach can largely reduce the conservatism of the stability results. Finally, two examples are given to show the effectiveness of the theoretical results.

Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations

This paper investigates the dynamics of a class of recurrent neural networks where the neural activations are modeled by discontinuous functions. Without presuming the boundedness of activation functions, the sufficient conditions to ensure the existence, uniqueness, global exponential stability and global convergence of state equilibrium point and output equilibrium point are derived, respectively. Furthermore, under certain conditions we prove that the system is convergent globally in finite time. The analysis in the paper is based on the properties of M -matrix, Lyapunov-like approach, and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend previous works on global stability of recurrent neural networks with not only Lipschitz continuous but also discontinuous neural activation functions.

Delay-Dependent Stability for Recurrent Neural Networks With Time-Varying Delays

This brief is concerned with the stability for static neural networks with time-varying delays. Delay-independent conditions are proposed to ensure the asymptotic stability of the neural network. The delay-independent conditions are less conservative than existing ones. To further reduce the conservatism, delay-dependent conditions are also derived, which can be applied to fast time-varying delays. Expressed in linear matrix inequalities, both delay-independent and delay-dependent stability conditions can be checked using the recently developed algorithms. Examples are provided to illustrate the effectiveness and the reduced conservatism of the proposed result.

IMPROVED SUFFICIENT CONDITIONS FOR GLOBAL EXPONENTIAL STABILITY OF RECURRENT NEURAL NETWORKS WITH DISTRIBUTED DELAYS

In this paper, the globally exponential stability of recurrent neural networks with continuously distributed delays is investigated. New theoretical results are presented in the presence of external stimuli. It is shown that the recurrent neural network is globally exponentially stable, and the estimated location of the equilibrium point can be obtained. As typical representatives, the Hopfield neural network (HNN) and the cellular neural network (CNN) are examined in detail. Comparison between our results and the previous results admits the improvement of our results.

Inverse optimal noise‐to‐state stabilization of stochastic recurrent neural networks driven by noise of unknown covariance

AbstractIn this paper, we extend our previous research results regarding the stabilization of recurrent neural networks from the concept of input‐to‐state stability to noise‐to‐state stability, and present a new approach to achieve noise‐to‐state stabilization in probability for stochastic recurrent neural networks driven by the noise of unknown covariance. This approach is developed by using the Lyapunov technique, inverse optimality, differential game theory, and the Hamilton–Jacobi–Isaacs equation. Numerical examples demonstrate the effectiveness of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.

Absolute Exponential Stability of Recurrent Neural Networks With Generalized Activation Function

In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results.

Delay-Dependent Approaches to Globally Exponential Stability for Recurrent Neural Networks

This brief deals with the stability analysis problem for recurrent neural networks with delay. An improved stability condition is derived to guarantee the existence of the unique equilibrium point and its globally exponential stability, which is shown with novel methods. Both delay-dependent and delay-independent stability conditions are obtained. Expressed in terms of LMIs, they can be checked using the numerically efficient Matlab LMI toolbox. Examples are provided to demonstrate the effectiveness and the reduced conservatism of the analysis results.

Global Asymptotic Stability of Recurrent Neural Networks With Multiple Time-Varying Delays

In this paper, several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed in our investigation. The results are shown to be generalizations of some previously published results and are less conservative than existing results. The present results are also applied to recurrent neural networks with constant time delays.