Elsevier

Neurocomputing

Volume 74, Issue 10, May 2011, Pages 1673-1680
Neurocomputing

Chaos control and associative memory of a time-delay globally coupled neural network using symmetric map

https://doi.org/10.1016/j.neucom.2011.01.007Get rights and content

Abstract

A chaotic neural network called time-delay globally coupled neural network using symmetric map (TDSG) is proposed for information processing applications. Firstly, its rich dynamic behaviors are exhibited and the output stability is demonstrated by using a parameter modulated control method. Secondly, the associative memory of TDSG is investigated by the control method. It is observed that the stable output sequence only contains stored pattern and its reverse pattern and the TDSG finally converges to the stored pattern which has the smallest Hamming distance to the initial patterns with noise. At last, strong information recovery ability of the TDSG is illustrated by comparative experiments.

Introduction

Many artificial neural network models (ANN) have been proposed to realize flexible intelligent information processing that is related to functions of real neurons and the brain. However, because the conventional artificial neuron model is simply a threshold element that transforms a weighted summation of inputs into the output via a nonlinear output function with a threshold, it is so oversimplified that the chaotic behavior of biological neurons cannot be well represented. In order to exhibit chaotic dynamics better, Aihara and his collaborators proposed chaotic neural network (CNN) models, which are composed of chaotic neurons derived based on the electrophysiological experiments on squid giant axons [1], [2], [3], [4], [5], [6]. Ever since then a lot of chaotic neural network models [7], [8], [9], [10], [11], [12] have been successively presented and investigated. There are several kinds of chaotic neural networks according to the mechanism of chaos. For example, the Inoue network made up of chaotic oscillators [7], the Kaneko model composed of coupled chaotic neurons [8], the network proposed by Aihara and Chen [3], which can be seen as natural extended model generated by introducing certain transformation to Hopfield network and so on. In spite of different mechanisms, they all combine the characteristics of common neural networks and chaos so well that they are applied widely in combinatorial optimization, image processing and pattern recognition and so on.

The associative memory ability is one of the most important characteristics of biological neural systems such as human brain. Through associative memory, intelligent information processing in engineering can also be realized. The associative memory dynamics of CNN is different from that of traditional ANN. For example, equilibrium dynamics is structuralized in an association process of Hopfield neural network corresponding to minimizing the network's Lyapunov function. But nonequilibrium dynamics is exhibited in that of CNN, namely, model's spatiotemporal complexity is not generated by the network structure, rather by the dynamics of each single neuron. From a technical viewpoint, it is important to implement nonequilibrium information processing such as that in the human brain. So a CNN is a promising technique in information processing such as memory recalling or pattern recognition. However, there are also two factors that affect the application of CNNs. On the one hand, the outputs of CNNs wander around all the stored patterns and cannot be stabilized to one of them or a periodic orbit. On the other hand, it is also difficult to judge when to terminate chaotic dynamics. Therefore several chaos control methods have been adopted for CNNs [13], [14], [15], [16], [17]. Different from many previous chaos control methods that the control targets must have been specified at the beginning, the method proposed by G.G.He and his collaborators shows dynamic associative memory of CNN can be realized by controlling the network without the control target in advance during information processing [18], [19], [20]. The existence of time delay is the other most important characteristic of biological neural systems. It also lies in the role of chaos in the engineering realization of neural network. So the study of time-delay network is necessary and important. Although many neural networks with time delay have been proposed and the stability, periodicity, control and synchronization of the solutions have been studied [21], [22], [23], [24] in theory, the applied research is rarely carried on.

The globally coupled map model using symmetric map (S-GCM) is a chaotic neural network with globally coupled chaotic elements, which is designed and investigated by Ishii [25], [26]. It has cluster frozen attractors (CFA), each of which is taken to represent information. Motivated by the two aspects above, we propose a time-delay globally coupled neural network using symmetric map (TDSG) based on S-GCM for information processing applications. Firstly, we show the rich dynamics of TDSG and investigate in detail the chaos control of the TDSG by the parameter modulated control method. Secondly, the associative memory of the controlled TDSG injected the initial patterns with different levels of noise is illustrated by the experiment. It is observed that the stable output sequence only contains stored pattern and its reverse pattern and the TDSG finally converges to the stored pattern which has the smallest Hamming distance to the initial patterns with noise. The results suggest that the TDSG has excellent performance by contrast. The experiment verifies the effectiveness of the application of the TDSG to information processing.

The rest of this paper is arranged as follows. In Section 2, the TDSG model is introduced and its dynamics is exhibited. In Section 3, the principle of chaos control of the TDSG is demonstrated. In Section 4, associative memory of TDSG is illustrated by experiments and the tolerant ability is also compared. Some discussions and conclusions are given in Section 5 finally.

Section snippets

Neuron model of TDSG

We introduce time delay into S-GCM and obtain TDSG. According to the characteristics of the system structure, we assume time delay exists in the transmission of neural signals except reflexive signals, then neuron model of TDSG can be described as follows:x(t+1)=(1ɛ)f(x(t)),f(x)=αx3αx+x,x[1,1],y(t+1)=O(x(t+1)),where x(t) is the internal state of the neuron at time t, y(t) is the output of the neuron at time t. f(·) is the symmetric cubic map, α is the bifurcation parameter of f(·). f(·) has

Chaos control of the TDSG

Several methods have been proposed for the chaos control of chaotic neural networks. Roughly speaking, chaos control methods can be divided into two categories. One is that the chaos is controlled by perturbing its parameters based on their sensitive effects on the chaotic dynamics to make the unstable periodic orbit stabilized such as parameter modulated control [25], [26]. This kind of method realizes the transformation from chaotic state to periodic state naturally by changing the value of

Associative memory

Now we investigate the associative memory of the controlled TDSG. We use Hamming distance to measure the difference between the initial pattern and the stored patterns or between the output pattern and the stored patterns. The Hamming distance is a measure used for comparing two binary patterns, which is defined as follows:H=i=1100|xixip|,where xi is the ith state of an output pattern, xip is the ith component of the pth stored pattern. For the pth output pattern, the Hamming distance will be

Discussion and conclusion

Considering the rich chaos dynamic features of S-GCM and time-delay occurrence in the transmission of signals, we investigate chaos control and associative memory of TDSG for the application of information processing. The TDSG exhibits chaotic associative memory dynamics because it is a auto-associative network, to which the memory information is not stored in the synaptic weights but input by iteration of the control parameters αi. The feature is different from the conventional networks and

Acknowledgments

We are grateful to the anonymous reviewers and the editor for their valuable suggestions and comments which have led the improvement of this paper. This work is supported by National High Technology Research and Development Program of China (Grant no. 2008AA01Z148) and Scientific Research Fund of Heilongjiang Provincial Education Department of China (Grant no. 11551140).

Tao Wang received his B.Sc. degree and M.Sc. degree in Mathematics from Harbin Normal University, China, in 2000 and 2003, respectively. He is a Ph.D. candidate at College of Automation in Harbin Engineering University since 2007. Now he is a lecturer at College of Mathematics Science in Harbin Normal University. His current research interests include neural networks and intelligent control.

Cited by (13)

  • Associative memory network and its hardware design

    2015, Neurocomputing
    Citation Excerpt :

    The associative results can be obtained by forward numerical calculation. The stored binary patterns are from references [17–19] shown in Fig. 5. Each stored pattern contains 10×10=100 units.

  • A novel GCM chaotic neural network for information processing

    2012, Communications in Nonlinear Science and Numerical Simulation
    Citation Excerpt :

    In addition, since the iteration function of the neurons is not unique, modified GCM models have been proposed such as GCM model with circle maps [12] and GCM model using symmetric map (S-GCM) proposed by Ishii [13,14] with the objective of information processing. However, except our improvement called time-delay S-GCM (TDSG) [15] recently, no other model based on GCM is proposed by far for information processing. From another point of view, it is important to implement nonequilibrium information processing such as that of nervous system in the human brain.

View all citing articles on Scopus

Tao Wang received his B.Sc. degree and M.Sc. degree in Mathematics from Harbin Normal University, China, in 2000 and 2003, respectively. He is a Ph.D. candidate at College of Automation in Harbin Engineering University since 2007. Now he is a lecturer at College of Mathematics Science in Harbin Normal University. His current research interests include neural networks and intelligent control.

Kejun Wang received his Ph.D. degree in special auxiliary ships and marine equipment and systems from Harbin Engineering University in 1995. From 1996 to 1998, he was a Postdoctoral Research Fellow in Fluid Power Transmission and Control at Harbin Institute of Technology. He is now a professor and doctoral supervisor at College of Automation in Harbin Engineering University. He has held and participated in many projects such as fingerprinting and has published more than 80 refereed journal papers. His current research interests include biological feature identification technology, intelligence system and the applications in pattern recognition and ship motion control.

Nuo Jia received her B.Sc. degree and M.Sc. degree in Mathematics from Harbin Normal University, China, in 2000 and 2003, respectively. She is a Ph.D. candidate at College of Automation in Harbin Engineering University since 2008. Now she is an associate professor at College of Mathematics Science in Harbin Normal University. Her current research interests include complex systems, control theory and applications.

View full text