Convergence Dynamics of Stochastic Reaction-diffusion Recurrent Neural Networks with Delays

Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.

[1]  Jin Xu,et al.  On the global stability of delayed neural networks , 2003, IEEE Trans. Autom. Control..

[2]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[3]  Sabri Arik,et al.  An analysis of global asymptotic stability of delayed cellular neural networks , 2002, IEEE Trans. Neural Networks.

[4]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[5]  Jinde Cao,et al.  Global asymptotic stability of a general class of recurrent neural networks with time-varying delays , 2003 .

[6]  Jinde Cao,et al.  Global exponential stability and periodic solutions of recurrent neural networks with delays , 2002 .

[7]  Jinde Cao.  Global stability conditions for delayed CNNs , 2001 .

[8]  Shun-ichi Amari,et al.  New theorems on global convergence of some dynamical systems , 2001, Neural Networks.

[9]  Xuerong Mao,et al.  Stability of stochastic delay neural networks , 2001, J. Frankl. Inst..

[10]  S. Arik.  An improved global stability result for delayed cellular neural networks , 2002 .

[11]  S. Arik.  Global robust stability of delayed neural networks , 2003 .

[12]  J. Buhmann,et al.  Influence of noise on the function of a “physiological” neural network , 1987, Biological Cybernetics.

[13]  Qiang Zhang,et al.  Global exponential stability of Hopfield neural networks with continuously distributed delays , 2003 .

[14]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[15]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[16]  Xuerong Mao,et al.  Exponential stability and instability of stochastic neural networks 1 , 1996 .

[17]  L.O. Chua,et al.  Cellular neural networks , 1993, 1988., IEEE International Symposium on Circuits and Systems.

[18]  Changyin Sun,et al.  On exponential stability of delayed neural networks with a general class of activation functions , 2002 .

[19]  Jinde Cao,et al.  Global exponential stability of reaction–diffusion recurrent neural networks with time-varying delays , 2003 .

[21]  Yuming Chen,et al.  Global stability of neural networks with distributed delays , 2002, Neural Networks.

[22]  Xuesong Jin,et al.  Global stability analysis in delayed Hopfield neural network models , 2000, Neural Networks.

[23]  Jia Li,et al.  Stability in Gilpin-Ayala competition models with diffusion , 1997 .

[24]  K. Gopalsamy,et al.  Dynamics of a class of discete-time neural networks and their comtinuous-time counterparts , 2000 .

[25]  E. Jenkins.  Light quark spin-flavor symmetry for baryons containing a heavy quark in large N QCD , 1993 .

[26]  Daoyi Xu,et al.  Asymptotic behavior of a class of reaction-diffusion equations with delays ✩ , 2003 .

[27]  Shun-ichi Amari,et al.  Stability of asymmetric Hopfield networks , 2001, IEEE Trans. Neural Networks.

[28]  A. Hastings.  Global stability in Lotka-Volterra systems with diffusion , 1978 .

[29]  Jinde Cao.  New results concerning exponential stability and periodic solutions of delayed cellular neural networks , 2003 .

[30]  Daoyi Xu,et al.  Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays , 2003, Science in China Series F: Information Sciences.

[31]  F. Rothe.  Convergence to the equilibrium state in the Volterra-Lotka diffusion equations , 1976, Journal of mathematical biology.

[32]  Xiaofeng Liao,et al.  (Corr. to) Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach , 2002, Neural Networks.

[33]  Jinde Cao.  A set of stability criteria for delayed cellular neural networks , 2001 .

[34]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.