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

An oscillatory criterion for a time delayed neural ring network model

The effects of delays on dynamical networks and the stability analysis of time delayed systems have received a notable attention over the past decades. In this paper, the effects of delays on the oscillatory properties of a neural ring networks model are considered. The existence of oscillations for a specific type of recurrent neural network with time delays between neural interconnections is investigated. By using Chafee's closed orbit theory, some sufficient conditions for permanent oscillations are obtained. Simple and practical criteria for selecting the range of parameters in this network model are also derived. Among other things, the solutions that we provide can be applied to various activation functions. A few computer simulations are presented to support our analysis. The present study can be applied to analyze under which conditions a ring network could be exploited as an oscillatory pattern generator.

Multistability of neural networks with time-varying delays and concave-convex characteristics

In this paper, stability of multiple equilibria of neural networks with time-varying delays and concave-convex characteristics is formulated and studied. Some sufficient conditions are obtained to ensure that an n-neuron neural network with concave-convex characteristics can have a fixed point located in the appointed region. By means of an appropriate partition of the n-dimensional state space, when nonlinear activation functions of an n-neuron neural network are concave or convex in 2k+2m-1 intervals, this neural network can have (2k+2m-1)n equilibrium points. This result can be applied to the multiobjective optimal control and associative memory. In particular, several succinct criteria are given to ascertain multistability of cellular neural networks. These stability conditions are the improvement and extension of the existing stability results in the literature. A numerical example is given to illustrate the theoretical findings via computer simulations.

Stability and dissipativity analysis of static neural networks with time delay

This paper is concerned with the problems of stability and dissipativity analysis for static neural networks (NNs) with time delay. Some improved delay-dependent stability criteria are established for static NNs with time-varying or time-invariant delay using the delay partitioning technique. Based on these criteria, several delay-dependent sufficient conditions are given to guarantee the dissipativity of static NNs with time delay. All the given results in this paper are not only dependent upon the time delay but also upon the number of delay partitions. Some examples are given to illustrate the effectiveness and reduced conservatism of the proposed results.

Synchronization of Markovian coupled neural networks with nonidentical node-delays and random coupling strengths

In this paper, a general model of coupled neural networks with Markovian jumping and random coupling strengths is introduced. In the process of evolution, the proposed model switches from one mode to another according to a Markovian chain, and all the modes have different constant time-delays. The coupling strengths are characterized by mutually independent random variables. When compared with most of existing dynamical network models which share common time-delay for all modes and have constant coupling strengths, our model is more practical because different chaotic neural network models can have different time-delays and coupling strength of complex networks may randomly vary around a constant due to environmental and artificial factors. By designing a novel Lyapunov functional and using some inequalities and the properties of random variables, we derive several new sufficient synchronization criteria formulated by linear matrix inequalities. The obtained criteria depend on mode-delays and mathematical expectations and variances of the random coupling strengths as well. Numerical examples are given to demonstrate the effectiveness of the theoretical results, meanwhile right-continuous Markovian chain is also presented.

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.

Estimating the ultimate bound and positively invariant set for a class of Hopfield networks

In this paper, we investigate the ultimate bound and positively invariant set for a class of Hopfield neural networks (HNNs) based on the Lyapunov stability criterion and Lagrange multiplier method. It is shown that a hyperelliptic estimate of the ultimate bound and positively invariant set for the HNNs can be calculated by solving a linear matrix inequality (LMI). Furthermore, the global stability of the unique equilibrium and the instability region of the HNNs are analyzed, respectively. Finally, the most accurate estimate of the ultimate bound and positively invariant set can be derived by solving the corresponding optimization problems involving the LMI constraints. Some numerical examples are given to illustrate the effectiveness of the proposed results.

Theoretic design of differential minimax controllers for stochastic cellular neural networks

This paper presents a theoretical design of how a minimax equilibrium of differential game is achieved in stochastic cellular neural networks. In order to realize the equilibrium, two opposing players are selected for the model of stochastic cellular neural networks. One is the vector of external inputs and the other is the vector of internal noises. The design procedure follows the nonlinear H infinity optimal control methodology to accomplish the best rational stabilization in probability for stochastic cellular neural networks, and to attenuate noises to a predefined level with stability margins. Three numerical examples are given to demonstrate the effectiveness of the proposed approach.

Delay-independent stability of genetic regulatory networks

Genetic regulatory networks can be described by nonlinear differential equations with time delays. In this paper, we study both locally and globally delay-independent stability of genetic regulatory networks, taking messenger ribonucleic acid alternative splicing into consideration. Based on nonnegative matrix theory, we first develop necessary and sufficient conditions for locally delay-independent stability of genetic regulatory networks with multiple time delays. Compared to the previous results, these conditions are easy to verify. Then we develop sufficient conditions for global delay-independent stability for genetic regulatory networks. Compared to the previous results, this sufficient condition is less conservative. To illustrate theorems developed in this paper, we analyze delay-independent stability of two genetic regulatory networks: a real-life repressilatory network with three genes and three proteins, and a synthetic gene regulatory network with five genes and seven proteins. The simulation results show that the theorems developed in this paper can effectively determine the delay-independent stability of genetic regulatory networks.

Stability and L2 performance analysis of stochastic delayed neural networks

This brief focuses on the robust mean-square exponential stability and L(2) performance analysis for a class of uncertain time-delay neural networks perturbed by both additive and multiplicative stochastic noises. New mean-square exponential stability and L(2) performance criteria are developed based on the delay partition Lyapunov-Krasovskii functional method and generalized Finsler lemma which is applicable to stochastic systems. The analytical results are established without involving any model transformation, estimation for cross terms, additional free-weighting matrices, or tuning parameters. Numerical examples are presented to verify that the proposed approach is both less conservative and less computationally complex than the existing ones.

Novel exponential stability criteria of high-order neural networks with time-varying delays

The global exponential stability is analyzed for a class of high-order Hopfield-type neural networks with time-varying delays. Based on the Lyapunov stability theory, together with the linear matrix inequality approach and free-weighting matrix method, some less conservative delay-independent and delay-dependent sufficient conditions are presented for the global exponential stability of the equilibrium point of the considered neural networks. Two numerical examples are provided to demonstrate the effectiveness of the proposed stability criteria.

Segmented-memory recurrent neural networks

Conventional recurrent neural networks (RNNs) have difficulties in learning long-term dependencies. To tackle this problem, we propose an architecture called segmented-memory recurrent neural network (SMRNN). A symbolic sequence is broken into segments and then presented as inputs to the SMRNN one symbol per cycle. The SMRNN uses separate internal states to store symbol-level context, as well as segment-level context. The symbol-level context is updated for each symbol presented for input. The segment-level context is updated after each segment. The SMRNN is trained using an extended real-time recurrent learning algorithm. We test the performance of SMRNN on the information latching problem, the "two-sequence problem" and the problem of protein secondary structure (PSS) prediction. Our implementation results indicate that SMRNN performs better on long-term dependency problems than conventional RNNs. Besides, we also theoretically analyze how the segmented memory of SMRNN helps learning long-term temporal dependencies and study the impact of the segment length.

Global dissipativity analysis on uncertain neural networks with mixed time-varying delays

In this paper, the problems of global dissipativity and global exponential dissipativity are investigated for uncertain neural networks with discrete time-varying delay and distributed time-varying delay as well as general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing Newton-Leibniz formulation and linear matrix inequality (LMI) technique, several new criteria for checking the global dissipativity and global exponential dissipativity of the addressed neural networks are established in terms of LMI, which can be checked numerically using the effective LMI toolbox in MATLAB. Illustrated examples are given to show the effectiveness and decreased conservatism of the proposed criteria in comparison with some existing results. It is noteworthy that the traditional assumptions on the differentiability of the time-varying delays and the boundedness of its derivative are removed.