Abstract
This paper studies the stability of stochastic reaction-diffusion delayed recurrent neural networks with Levy noise. Using key tools such as Ito’s formula for general semimartingales, Lyapunov method, and inequality techniques, we find conditions under which the solutions to the neuron models driven by Levy noise are exponentially stable in the mean square.
Similar content being viewed by others
References
Applebaum D (2004) Levy processes and stochastic calculus. Cambridge University Press, Cambridge
Applebaum D, Siakalli M (2009) Asymptotic stability of stochastic differential equations driven by Levy noise. J Appl Prob 46:1116–1129
Arnold L (1972) Stochastic differential equations: theory and applications. Wiley, New York
Cao J (2001) Global stability conditions for delayed CNNs. IEEE Trans Circuits Syst I 48:1330–1333
Chen T, Amari S (2001) Stability of asymmetric Hopfield networks. IEEE Trans Neural Netw 12:159–163
Chen W, Guan Z, Lu X (2004) Delay-dependent exponential stability of neural networks with variable delays. Phys Lett A 326:355–363
Cohen MA, Grossberg S (1983) Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybernet 13:815–821
Friedman A (1976) Stochastic differential equations and applications. Academic Press, New York
Grossberg S (1988) Nonlinear neural networks: principles, mechanisms, and architectures. Neural Netw 1:17–61
Haykin S (1994) Neural networks. Prentice-Hall, Englewood Cliffs
Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. 2nd edn. Springer, Berlin
Kunita H (2004) Stochastic differential equations based on Levy processes and stochastic flows of diffeomorphisms. In real and stochastic analysis, New Perspectives, Boston
Liang J, Cao J (2003) Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays. Phys Lett A 314:434–442
Lu J (2008) Global exponential stability and periodicity of reaction diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fractals 35:116–125
Mao X (1997) Stochastic differential equations and applications. Ellis Horwood, Chichester
Patel A, Kosko B (2008) Stochastic resonance in continuous and spiking neuron models with Levy noise. IEEE Trans Neural Netw 19:1993–2008
Liu Z, Peng J (2010) Delay-independent stability of stochastic reaction diffusion neural networks with Dirichlet boundary conditions. Neural Comput Appl 19:151–158
Wang L, Xu D (2003) Asymptotic behavior of a class of reaction-diffusion equations. J Math Anal Appl 281:439–453
Wan A, Peng J, Wang M (2006) Exponential stability of a class of generalized neural networks with time-varying delays. Neurocomputing 69:959–963
Xia Y, Cao J, Cheng S (2007) Global exponential stability of delayed cellular neural networks with impulses. Neurocomputing 70:2495–2501
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Peng, J., Liu, Z. Stability analysis of stochastic reaction-diffusion delayed neural networks with Levy noise. Neural Comput & Applic 20, 535–541 (2011). https://doi.org/10.1007/s00521-011-0541-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-011-0541-6