Multistability of Delayed Recurrent Neural Networks with Mexican Hat Activation Functions

2017 ◽  
Vol 29 (2) ◽  
pp. 423-457 ◽  
Author(s):  
Peng Liu ◽  
Zhigang Zeng ◽  
Jun Wang
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Activation Function ◽  
Mexican Hat ◽  
Exponentially Stable ◽  
Attraction Basins ◽  
Stability Results

This letter studies the multistability analysis of delayed recurrent neural networks with Mexican hat activation function. Some sufficient conditions are obtained to ensure that an [Formula: see text]-dimensional recurrent neural network can have [Formula: see text] equilibrium points with [Formula: see text], and [Formula: see text] of them are locally exponentially stable. Furthermore, the attraction basins of these stable equilibrium points are estimated. We show that the attraction basins of these stable equilibrium points can be larger than their originally partitioned subsets. The results of this letter improve and extend the existing stability results in the literature. Finally, a numerical example containing different cases is given to illustrate the theoretical results.

DELAY-DEPENDENT ROBUST EXPONENTIAL STABILITY FOR UNCERTAIN RECURRENT NEURAL NETWORKS WITH TIME-VARYING DELAYS

2007 ◽  
Vol 17 (03) ◽  
pp. 207-218 ◽  
Author(s):  
BAOYONG ZHANG ◽  
SHENGYUAN XU ◽  
YONGMIN LI
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Time Varying ◽  
Parameter Uncertainties ◽  
Robust Exponential Stability ◽  
Neuron Activation ◽  
Exponentially Stable

This paper considers the problem of robust exponential stability for a class of recurrent neural networks with time-varying delays and parameter uncertainties. The time delays are not necessarily differentiable and the uncertainties are assumed to be time-varying but norm-bounded. Sufficient conditions, which guarantee that the concerned uncertain delayed neural network is robustly, globally, exponentially stable for all admissible parameter uncertainties, are obtained under a weak assumption on the neuron activation functions. These conditions are dependent on the size of the time delay and expressed in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness and less conservatism of the proposed stability results.

ASYMPTOTIC HYPERSTABILITY OF A CLASS OF NEURAL NETWORKS

1999 ◽  
Vol 09 (02) ◽  
pp. 95-98 ◽  
Author(s):  
ANKE MEYER-BÄSE
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lyapunov Methods ◽  
Network Parameters ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

This paper is concerned with the asymptotic hyperstability of recurrent neural networks. We derive based on the stability results necessary and sufficient conditions for the network parameters. The results we achieve are more general than those based on Lyapunov methods, since they provide milder constraints on the connection weights than the conventional results and do not suppose symmetry of the weights.

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

2008 ◽  
Vol 18 (07) ◽  
pp. 2029-2037
Author(s):  
WEI WU ◽  
BAO TONG CUI ◽  
ZHIGANG ZENG
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Exponential Stability ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Hopfield Neural Network ◽  
Cellular Neural Network ◽  
Distributed Delays ◽  
Globally Exponential Stability ◽  
Exponentially Stable

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.

GUARANTEED ATTRACTIVITY OF EQUILIBRIUM POINTS IN A CLASS OF DELAYED NEURAL NETWORKS

2006 ◽  
Vol 16 (09) ◽  
pp. 2737-2743 ◽  
Author(s):  
XIAOFAN YANG ◽  
XIAOFENG LIAO ◽  
YUANYAN TANG ◽  
DAVID J. EVANS
Keyword(s):  
Neural Networks ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Practical Importance ◽  
Activation Functions ◽  
Qualitative Properties ◽  
Domains Of Attraction ◽  
Delayed Neural Networks ◽  
Exponentially Stable ◽  
First Time

This paper addresses qualitative properties of equilibrium points in a class of delayed neural networks. We derive a sufficient condition for the local exponential stability of equilibrium points, and give an estimate on the domains of attraction of locally exponentially stable equilibrium points. Our condition and estimate are formulated in terms of the network parameters, the neurons' activation functions and the associated equilibrium point; hence, they are easily checkable. Another advantage of our results is that they neither depend on monotonicity of the activation functions nor on symmetry of the interconnection matrix. Our work has practical importance in evaluating the performance of the related associative memory. To our knowledge, this is the first time to present an estimate on the domains of attraction of equilibrium points for delayed neural networks.

Analysis on Cellular Neural Networks with Negative Slope Activation Function

2013 ◽  
Vol 427-429 ◽  
pp. 2493-2496
Author(s):  
Qi Han ◽  
Qian Xiong ◽  
Chao Liu ◽  
Jun Peng ◽  
Le Peng Song ◽  
...  
Keyword(s):  
Neural Networks ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Activation Function ◽  
Cellular Neural Networks ◽  
Negative Slope ◽  
The Relationship ◽  
Inputs And Outputs

In the paper, the region of the number of equilibrium points of every cell in cellular neural networks with negative slope activation function is considered by the relationship between parameters of cellular neural networks. Some conditions are obtained by using the relationship among connection weights. Depending on these sufficient conditions, inputs and outputs of a CNN, the regions of the values of parameters can be obtained. Some numerical simulations are presented to support the effectiveness of the theoretical analysis.

Efficient Continuous-Time Asymmetric Hopfield Networks for Memory Retrieval

2010 ◽  
Vol 22 (6) ◽  
pp. 1597-1614 ◽  
Author(s):  
Pengsheng Zheng ◽  
Wansheng Tang ◽  
Jianxiong Zhang
Keyword(s):  
Neural Networks ◽  
Continuous Time ◽  
Memory Retrieval ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Hopfield Networks ◽  
Energy Functions ◽  
Asymmetric Networks

A novel m energy functions method is adopted to analyze the retrieval property of continuous-time asymmetric Hopfield neural networks. Sufficient conditions for the local and global asymptotic stability of the network are proposed. Moreover, an efficient systematic procedure for designing asymmetric networks is proposed, and a given set of states can be assigned as locally asymptotically stable equilibrium points. Simulation examples show that the asymmetric network can act as an efficient associative memory, and it is almost free from spurious memory problem.

scholarly journals A perspective on graph theory-based stability analysis of impulsive stochastic recurrent neural networks with time-varying delays

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
M. Iswarya ◽  
R. Raja ◽  
G. Rajchakit ◽  
J. Cao ◽  
J. Alzabut ◽  
...  
Keyword(s):  
Neural Networks ◽  
Graph Theory ◽  
Exponential Stability ◽  
Recurrent Neural Networks ◽  
Lyapunov Functional ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Theoretic Approach ◽  
Discrete Time Delay

AbstractIn this work, the exponential stability problem of impulsive recurrent neural networks is investigated; discrete time delay, continuously distributed delay and stochastic noise are simultaneously taken into consideration. In order to guarantee the exponential stability of our considered recurrent neural networks, two distinct types of sufficient conditions are derived on the basis of the Lyapunov functional and coefficient of our given system and also to construct a Lyapunov function for a large scale system a novel graph-theoretic approach is considered, which is derived by utilizing the Lyapunov functional as well as graph theory. In this approach a global Lyapunov functional is constructed which is more related to the topological structure of the given system. We present a numerical example and simulation figures to show the effectiveness of our proposed work.

Recurrent Neural Networks with Small Weights Implement Definite Memory Machines

2003 ◽  
Vol 15 (8) ◽  
pp. 1897-1929 ◽  
Author(s):  
Barbara Hammer ◽  
Peter Tiňo
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Experimental Studies ◽  
Activation Function ◽  
Transition Function ◽  
Recurrent Network ◽  
Point Of View ◽  
Recurrent Networks ◽  
Contraction Parameter ◽  
Arbitrary Precision

Recent experimental studies indicate that recurrent neural networks initialized with “small” weights are inherently biased toward definite memory machines (Tiňno, Čerňanský, & Beňušková, 2002a, 2002b). This article establishes a theoretical counterpart: transition function of recurrent network with small weights and squashing activation function is a contraction. We prove that recurrent networks with contractive transition function can be approximated arbitrarily well on input sequences of unbounded length by a definite memory machine. Conversely, every definite memory machine can be simulated by a recurrent network with contractive transition function. Hence, initialization with small weights induces an architectural bias into learning with recurrent neural networks. This bias might have benefits from the point of view of statistical learning theory: it emphasizes one possible region of the weight space where generalization ability can be formally proved. It is well known that standard recurrent neural networks are not distribution independent learnable in the probably approximately correct (PAC) sense if arbitrary precision and inputs are considered. We prove that recurrent networks with contractive transition function with a fixed contraction parameter fulfill the so-called distribution independent uniform convergence of empirical distances property and hence, unlike general recurrent networks, are distribution independent PAC learnable.

Analysis and Design of Associative Memories Based on Recurrent Neural Networks with Linear Saturation Activation Functions and Time-Varying Delays

2007 ◽  
Vol 19 (8) ◽  
pp. 2149-2182 ◽  
Author(s):  
Zhigang Zeng ◽  
Jun Wang
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Activation Functions ◽  
Analysis And Design ◽  
Design Procedures ◽  
The Neural Network ◽  
Saturation Region ◽  
The Neural Networks

In this letter, some sufficient conditions are obtained to guarantee recurrent neural networks with linear saturation activation functions, and time-varying delays have multiequilibria located in the saturation region and the boundaries of the saturation region. These results on pattern characterization are used to analyze and design autoassociative memories, which are directly based on the parameters of the neural networks. Moreover, a formula for the numbers of spurious equilibria is also derived. Four design procedures for recurrent neural networks with linear saturation activation functions and time-varying delays are developed based on stability results. Two of these procedures allow the neural network to be capable of learning and forgetting. Finally, simulation results demonstrate the validity and characteristics of the proposed approach.

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