Neural Processing Letters

, Volume 38, Issue 3, pp 433–444 | Cite as

Stability of Stochastic \(\theta \)-Methods for Stochastic Delay Hopfield Neural Networks Under Regime Switching

  • Feng Jiang
  • Yi Shen


This paper is concerned with the general mean-square (GMS) stability and mean-square (MS) stability of stochastic \(\theta \)-methods for stochastic delay Hopfield neural networks under regime switching. The sufficient conditions to guarantee GMS-stability and MS-stability of stochastic \(\theta \)-methods are given. Finally, an example is used to illustrate the effectiveness of our result.


Stochastic delay Hopfield neural networks Regime switching MS-stability GMS-stability Stochastic \(\theta \)-methods 



The work is supported by the Fundamental Research Funds for the Central Universities, China Postdoctoral Science Foundation funded project under Grant 2012M511615 and the State Key Program of National Natural Science of China under Grant 61134012.


  1. 1.
    Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Nat Acad Sci (Biophysics) 79:2554–2558MathSciNetCrossRefGoogle Scholar
  2. 2.
    Forti M, Tesi A (1995) New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans Circuits Syst. I Fundam Theory Appl 42(7):354–366MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Venetianer P, Roska T (1998) Image compression by delayed CNNs. IEEE Trans Circuits Syst I 45:205–215CrossRefGoogle Scholar
  4. 4.
    Jiang M, Shen Y, Liao X (2006) Boundedness and global exponential stability for generalized Cohen-Grosssberg neural networks with variale delay. Appl Math Comput 172:379–393MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wen S, Zeng Z (2012) Dynamics analysis of a class of memristor-based recurrent networks with time-varying delays in the presence of strong external stimuli. Neural Process Lett 35:47–59CrossRefGoogle Scholar
  6. 6.
    Chen H, Zhang Y, Hu P (2010) Novel delay-dependent robust stability criteria for neutral stochastic delayed neural networks. Neurocomputing 73:2554–2561CrossRefGoogle Scholar
  7. 7.
    Chen G, Shen Y, Zhu S (2011) Non-fragile observer-based \(\text{ H}_\infty \) control for neutral stochastic hybrid systems with time-varying delay. Neural Comput Appl 20:1149–1158CrossRefGoogle Scholar
  8. 8.
    Mahmoud MS, Shi P (2003) Robust stability, stabilization and \(\text{ H}_\infty \) control of time-delay systems with Markovian jump parameters. Int. J. Robust Nonlinear Control 13:755–784MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lou X, Cui B (2009) Stochastic stability analysis for delayed neural networks of neutral type with Markovian jump parameters. Chaos Solitons Fractals 39:2188–2197MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huang H, Ho DWC, Qu Y (2007) Robust stability of stochastic delayed additive neural networks with Markovian switching. Neural Netw 20:799–809CrossRefzbMATHGoogle Scholar
  11. 11.
    Chen W, Xu J, Guan Z (2003) Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays. IEEE Trans Autom Control 48:2270–2277MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhu S, Shen Y, Liu L (2010) Exponential stability of uncertain stochastic neural networks with Markovian switching. Neural Process Lett 32:293–309CrossRefGoogle Scholar
  13. 13.
    Mao X, Yuan C (2006) Stochastic differential equations with Markovian switching. Imperial College Press, LondonCrossRefzbMATHGoogle Scholar
  14. 14.
    Küchler U, Platen E (2000) Strong discrete time approximation of stochastic differential equations with time delay. Math Comput Simul 54:189–205CrossRefGoogle Scholar
  15. 15.
    Higham DJ, Mao X, Stuart AM (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J Numer Anal 40(3):1041–1063MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hu P, Huang C (2011) Stability of stochastic \(\theta \)-methods for stochastic delay integro-differential equations. Int J Comput Math 88:1417–1429MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang F, Shen Y, Hu J (2011) Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching. Commun Nonlinear Sci Numer Simul 16:814–821MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhou S, Wu F (2009) Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching. J Comput Appl Math 229:85–96MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li R, Pang W, Leung P (2010) Exponential stability of numerical solutions to stochastic delay Hopfield neural networks. Neurocomputing 73:920–926CrossRefzbMATHGoogle Scholar
  20. 20.
    Jiang F, Shen Y, (2012) Stability in the numerical simulation of stochastic delayed Hopfield neural networks. Neural Comput Applic. doi: 10.1007/s00521-012-0935-0

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Statistics and MathematicsZhongnan University of Economics and LawWuhanChina
  2. 2.Department of Control Science and EngineeringHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations