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Stability analysis of stochastic reaction–diffusion neural networks with Markovian switching and time delays in the leakage terms

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Abstract

This paper investigates a class of stochastic reaction–diffusion neural networks with both Markovian jumping parameters and time delays in the leakage terms. By using the Lyapunov functional method, linear matrix inequality approach and stochastic analysis technique, a novel sufficient condition is derived to ensure the stochastic stability of the networks in the mean square sense. The proposed results, which do not require the differentiability and monotonicity of the activation functions, can be easily checked via Matlab LMI Toolbox. Moreover, they indicate that the stability behavior of neural networks is very sensitive to the time delay in the leakage term. Finally, two numerical examples are given to demonstrate the effectiveness of our theoretical results.

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References

  1. Chen W, Lu X (2003) Mean square exponential stability of uncertain stochastic delayed neural networks. Phys Lett A 372:1061–1069

    Article  Google Scholar 

  2. Gopalsamy K (2007) Leakage delays in BAM. J Math Anal Appl 325:1117–1132

    Article  MATH  MathSciNet  Google Scholar 

  3. Gopalsamy K (1992) Stability and oscillations in delay differential equations of population dynamics. Kluwer Academic Publishers, Dordrecht

  4. Gu K (2000) An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE conference decision and control, Sydney, pp 2805–2810

  5. Huang G-B, Wang D-H, Lan Y (2011) Extreme learning machines: a survey. Int J Machine Learn Cybern 2(2):107–122

    Article  Google Scholar 

  6. Li X, Cao J (2007) Delay-independent exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays and reaction–diffusion terms. Nonlinear Dyn 50:363–371

    Article  MATH  MathSciNet  Google Scholar 

  7. Li X, Fu X, Balasubramaniamc P, Rakkiyappan R (2010) Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations. Nonlinear Anal Real World Appl 11:4092–4108

    Article  MATH  MathSciNet  Google Scholar 

  8. Li Z, Xu R (2012) Global asymptotic stability of stochastic reaction–diffusion neural networks with time delays in the leakage terms. Commun Nonlinear Sci Numer Simulat 17:1681–1689

    Article  MATH  Google Scholar 

  9. Mathiyalagan K, Sakthivel R, Anthoni SM (2012) New robust passivity criteria for stochastic fuzzy BAM neural networks with time-varying delays. Commun Nonlinear Sci Numer Simulat 17:1392–1407

    Article  MATH  MathSciNet  Google Scholar 

  10. Poznyak AS, Sanchez EN (1995) Nonlinear systems approximation by neural networks: error stability analysis. Intell Autom Soft Compt Int J 1:247–258

    Google Scholar 

  11. Sakthivel R, Arunkumar A, Mathiyalagan K, Anthoni SM (2011) Robust passivity analysis of fuzzy Cohen-Grossberg BAM neural networks with time-varying delays. Appl Math Comput 218:3799–3809

    Article  MATH  MathSciNet  Google Scholar 

  12. Sakthivel R, Mathiyalagan K, Anthoni SM (2012) Robust H\(_\infty\) control for uncertain discrete-time stochastic neural networks with time-varying delays. IET Control Theory Appl 6:1220–1228

    Google Scholar 

  13. Raja R, Sakthivel R, Anthoni SM (2011) Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays. Int J Appl Math Comput Sci 21(1):127–135

    Article  MATH  MathSciNet  Google Scholar 

  14. Sakthivel R, Mathiyalagan K, Anthoni SM (2012) Delay dependent robust stabilization and H\(_\infty\) control for neural networks with various activation functions. Phys Scr 85:045801

    Google Scholar 

  15. Sakthivel R, Raja R, Anthoni SM (2011) Exponential stability for delayed stochastic bidirectional associative memory neural networks with Markovian jumping and impulses. J Optim Theory Appl 150:166–187

    Article  MATH  MathSciNet  Google Scholar 

  16. Sakthivel R, Samidurai R, Anthoni SM (2010) New exponential stability criteria for stochastic BAM neural networks with impulses. Phys Scr 82:045802

    Article  Google Scholar 

  17. Sarlin P (2012) Visual tracking of the millennium development goals with a fuzzified self-organizing neural network. Int J Mach Learn Cybern 3(3):233–245

    Article  Google Scholar 

  18. Song Q, Cao J, Zhao Z (2006) Periodic solutions and its exponential stability of reaction–diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal Real World Appl 7:65–80

    Article  MATH  MathSciNet  Google Scholar 

  19. Syed Ali M. (2012) Robust stability of stochastic uncertain recurrent neural networks with Markovian jumping parameters and time-varying delays. Int J Machine Learn Cybern. doi:10.1007/s13042-012-0124-6

  20. Tang Y, Wang Z, Gao H, Swift S, Kurths J (2012) A constrained evolutionary computation method for detecting controlling regions of cortical networks. IEEE/ACM Trans Comput Biol Bioinforma 9:1569–1581

    Article  Google Scholar 

  21. Tang Y, Wong WK (2013) Distributed synchronization of coupled neural networks via randomly occurring control. IEEE Trans Neural Netw Learn Syst 24(3):435–447

    Article  Google Scholar 

  22. Wang L, Zhang Z, Wang Y (2008) Stochastic exponential stability of the delayed reaction–diffusion recurrent neural networks with Markovian jumping parameters. Phys Lett A 372:3201–3209

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang L, Zhou Q (2008) Exponential stability of stochastic reaction–diffusion Cohen-Grossberg neural networks with delays. Appl Math Comput 206:818–824

    Article  MathSciNet  Google Scholar 

  24. Wang X, Chen A, Feng H (2011) Upper integral network with extreme learning mechanism. Neurocomputing 74(16):2520–2525

    Article  Google Scholar 

  25. Xu X, Zhang J, Zhang W (2011) Mean square exponential stability of stochastic neural networks with reaction–diffusion terms and delays. Appl Math Lett 24:5–11

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang W, Tang Y, Fang J, Wu X (2012) Stability of delayed neural networks with time-varying impulses. Neural Netw 36:59–63

    Article  MATH  Google Scholar 

  27. Zheng H, Wang H (2012) Improving pattern discovery and visualisation with self-adaptive neural networks through data transformations. Int J Mach Learn Cybern 3(3):173–182

    Article  Google Scholar 

  28. Zhu Q, Cao J (2011) Exponential stability analysis of stochastic reaction–diffusion Cohen-Grossberg neural networks with mixed delays. Neurocomputing 74:3084–3091

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Science, Dalian Jiaotong University, Dalian, 116028, People’s Republic of China

    Cheng-De Zheng & Yue Zhang

  2. School of Information Science and Engineering, Northeastern University, Shenyang, 110004, People’s Republic of China

    Zhanshan Wang

Authors
  1. Cheng-De Zheng
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  2. Yue Zhang
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  3. Zhanshan Wang
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Corresponding author

Correspondence to Cheng-De Zheng.

Additional information

This work was supported by the National Natural Science Foundation of China under Grants 61074073, 61034005, 61273022, Program for New Century Excellent Talents in University of China (NCET-10-0306), and the Fundamental Research Funds for the Central Universities under Grants N110504001.

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Zheng, CD., Zhang, Y. & Wang, Z. Stability analysis of stochastic reaction–diffusion neural networks with Markovian switching and time delays in the leakage terms. Int. J. Mach. Learn. & Cyber. 5, 3–12 (2014). https://doi.org/10.1007/s13042-013-0165-5

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  • DOI: https://doi.org/10.1007/s13042-013-0165-5

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