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Influence of noise and delay on reaction-diffusion recurrent neural networks

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Analysis in Theory and Applications

Abstract

In this paper, the influence of the noise and delay upon the stability property of reaction-diffusion recurrent neural networks (RNNs) with the time-varying delay is discussed. The new and easily verifiable conditions to guarantee the mean value exponential stability of an equilibrium solution are derived. The rate of exponential convergence can be estimated by means of a simple computation based on these criteria.

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References

  1. Arik, S., An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks, IEEE Trans. Neural Networks, 13(2002), 1239–1242.

    Article  Google Scholar 

  2. Arik, S., An Improved Global Stability Result for Delayed Cellular Neural Networks, IEEE Trans. Circuits Systems I, 49(2002), 1211–1214.

    Article  MathSciNet  Google Scholar 

  3. Arik, S., Global Robust Stability of Delayed Neural Networks, IEEE Trans. Circuits Systems I, 50(2003), 156–160.

    Article  MathSciNet  Google Scholar 

  4. Arnold, L., Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.

    Google Scholar 

  5. Blythe, S., Mao, X. and Liao, X., Stability of Stochastic Delay Neural Networks, J. Franklin Inst. 338 (2001), 481–495.

    Article  MATH  MathSciNet  Google Scholar 

  6. Buhmann, J. and Schulten, K., Influence of Noise on the Function of A ‘Physiological’ Neural Network, Biol. Cynern. 56 (1987), 313–327.

    Article  MATH  MathSciNet  Google Scholar 

  7. Carpenter, G. A., A Geometric Approach to Singular Perturbation Problems with Application to Nerve Impulse Equation, J. Differential Equations, 23 (1977), 355–367.

    Article  MathSciNet  Google Scholar 

  8. Cao, J., A Set of Stability Criteria for Delayed Cellular Neural Networks,IEEE Trans. Circuits Systems I, 48 (2001), 1330–1333.

    Article  MATH  Google Scholar 

  9. Cao, J. and Wang, J., Global Asymptotic Stability of a General Class of Recurrent Neural Networks with Time-Varying Delays, IEEE Trans. Circuits Systems I, 50(2003), 34–44.

    Article  MathSciNet  Google Scholar 

  10. Cao, J., New Results Concerning Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks, Phys. Lett. A, 307 (2003), 136–147.

    Article  MATH  MathSciNet  Google Scholar 

  11. Cao, J. and Wang, L., Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays, IEEE Trans. Neural Networks, 13 (2002), 457–463.

    Article  Google Scholar 

  12. Chen, T. and Amari, S., Stability of Asymmetric Hopfield Networks, IEEE Trans. Neural Networks, 12(2001), 159–163.

    Article  Google Scholar 

  13. Chen, T. and Amari, S., New Theorems on Global Convergence of Some Dynamical Systems, Neural Networks, 14 (2001), 251–255.

    Article  Google Scholar 

  14. Chen, Y., Global Stability of Neural Networks with Distributed Delays, Neural Networks, 15 (2002), 867–871.

    Article  Google Scholar 

  15. Chen, W., Guan, Z. and Lu, X., Delay-dependent Exponential Stability of Neural Networks with Variable Delays, Phys. Lett. A, 326 (2004), 355–363.

    Article  MathSciNet  Google Scholar 

  16. Chua, Leon O. amd Yang, L., Cellular Neural Networks: Applications, IEEE Trans. Circuits Systems I, 35 (1988), 1273–1290.

    Article  MathSciNet  Google Scholar 

  17. Chua, Leon O. and Yang, L., Cellular Neural Networks: Theory, IEEE Trans. Circuits Systems I, 35 (1988), 1257–1272.

    Article  MATH  MathSciNet  Google Scholar 

  18. Chua, Leon O. and Roska, T., The CNN Paradigm, IEEE Trans. Circuits Systems I, 40 (1993), 147–156.

    Article  MATH  MathSciNet  Google Scholar 

  19. Coben, M. A. and Crosshery, S., Absolute Stability and Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks, IEEE Trans. Syst. Man Cybernet, (1983), 815–826.

  20. Driessche, C. M. and Zou, X., Global Attractivity in Delayed Hopfield Neural Network Models, SIAM J. Appl. Math. 58 (1998) 1878–1890.

    Article  MATH  MathSciNet  Google Scholar 

  21. Friedman, A., Stochastic Differential Equations and Applications, Academic Press. New York, 1976.

    MATH  Google Scholar 

  22. Gopalsamy, K. and He, X., Delay-independent Stability in Bidirectional Associative Memory Networks, IEEE Trans. Neural Networks, 5 (1994), 998–1002.

    Article  Google Scholar 

  23. Haykin, S., Neural Networks, Prentice-Hall, NJ, 1994.

    MATH  Google Scholar 

  24. Hastins, A., Global Stability in Lotka-Volterra System with Diffusion, J. Math. Biol. 6 (1978), 163–168.

    Article  MathSciNet  Google Scholar 

  25. Horn, R. A. and Johnson, C. R., Matrix Analysis, Cambridge Univ. Press, London, 1985.

    MATH  Google Scholar 

  26. Hou, C. and Qian, J., Stability Analysis for Neural Dynamics with Time-varying Delays, IEEE Trans. Neural Networks, 9 (1998), 221–223.

    Article  Google Scholar 

  27. Huang, H., Cao, J. and Wang, J., Global Exponential Stability and Periodic Solutions of Recurrent Neural Networks with Delays, Phys. Lett. A, 298 (2002), 393–404.

    MATH  MathSciNet  Google Scholar 

  28. Kosko, B., Bi-directional Associative Memories, IEEE Trans. Syst. Man Cybern. 18 (1988), 49–60.

    Article  MathSciNet  Google Scholar 

  29. Liang, J. and Cao, J., Global Exponential Stability of Reaction-diffusion Recurrent Neural Networks with Time-varying Delays, Phys. Lett. A, 314 (2003), 434–442.

    Article  MATH  MathSciNet  Google Scholar 

  30. Liao, X., Chen, G. and Sanchez, E.N., Delay-dependent Exponential Stability Analysis of Delayed Neural Networks: An LMI Approach, Neural Networks, 15 (2002), 855–866.

    Article  Google Scholar 

  31. Liao, X. and Li, J., Stability in Gilpin-Ayala Competition Models with Diffusion, Nonlinear Analysis, 28 (1997), 1751–1758.

    Article  MATH  MathSciNet  Google Scholar 

  32. Liao, X. and Mao, X., Exponential Stability and Instability of Stochastic Neural Networks, Stochast. Anal. Appl. 14 (1996), 165–185.

    MATH  MathSciNet  Google Scholar 

  33. Liao, X. and Mao, X., Stability of Stochastic Neural Networks, Neural, Parallel. Sci Comput. 4 (1996), 205–224.

    MATH  MathSciNet  Google Scholar 

  34. Mao, X., Stochastic Differential Equations and Applications, Horwood Publishing, 1997.

  35. Mohamad, S. and Gopalsamy, K., Dynamics of A Class of Discrete-time Neural Networks and Their Continuous-time Counterparts, Math. Comput. Simulat. 53 (2001), 1–39.

    Article  MathSciNet  Google Scholar 

  36. Roska, T., Wu, C. W., Basli, M. and Chua, L. O., Stability and Dynamics of Delay-Type General and Cellular Neural Networks IEEE Trans. Circuits Systems I, 39 (1992), 487–490.

    Article  MATH  Google Scholar 

  37. Roska, T., Wu, C. W. and Chua, L. O., Stability of Cellular Neural Networks with Dominant Nonlinear and Delay-Type Templates, IEEE Trans. Circuits Systems I, 40 (1993), 270–272.

    Article  MATH  Google Scholar 

  38. Rothe, F., Convergence to the Equilibrium State in the Volterra-Lotka Diffusion Equations, J. Math. Biol. 3 (1976), 319–324.

    MATH  MathSciNet  Google Scholar 

  39. Sree Hari Rao, V. and Phaneendra Bh R, M., Global Dynamics of Bidirectional Associative Memory Neural Networks Involving Transmission Delays and Dead Zones, Neural Networks, 12 (1999), 445–465.

    Article  Google Scholar 

  40. Sun, C., Zhang, K., Fei, S. and Feng, C., On Ecponential Stability of Delayed Neural Networks with A General Class of Activation Functions, Phys. Lett. A, 298 (2002), 122–132.

    Article  MATH  MathSciNet  Google Scholar 

  41. Wang, L. S. and Xu, D. Y., Global Exponential Stability of Hopfield Reaction-diffusion Neural Networks with Time-varying Delays, Science in China (Series F), 46 (2003), 466–474.

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang, L.S. and Xu, D. Y., Asymptotic Behavior of A Class of Reaction-diffusion Equations with Delays, J.Math. Anal. Appl. 281 (2003), 439–453.

    Article  MATH  MathSciNet  Google Scholar 

  43. Zhang, J. and Jin, X., Global Stability Analysis in Delayed Hopfield Neural Network Models, Nerual Networks, 13 (2000), 745–753.

    Article  Google Scholar 

  44. Zhang, Q., Ma, R., Wang, C. and Xu, J., On the Global Stability of Delayed Neural Networks, IEEE Trans. Automatic Control, 48 (2003), 794–797.

    Article  MathSciNet  Google Scholar 

  45. Zhang, Q., Wei, X. and Xu, J., Global Exponential Stability of Hopfield Neural Networks with Continuously Distributed Delays, Phys. Lett. A, 315 (2003), 431–436.

    Article  MATH  MathSciNet  Google Scholar 

  46. Zhang, Q., Wei, X. and Xu, J., An Analysis on the Global Asymptotic Stability for Neural Networks with Variable Delays Phys, Lett. A, 328 (2004), 163–169.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Basic Courses, Nanjing Institute of Technology, Nanjing, 211167, P.R. China

    Li Wu

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Wu, L. Influence of noise and delay on reaction-diffusion recurrent neural networks. Analys in Theo Applic 22, 283–300 (2006). https://doi.org/10.1007/s10496-006-0283-y

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  • DOI: https://doi.org/10.1007/s10496-006-0283-y

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