Global Exponential Stability of Multitime Scale Competitive Neural Networks With Nonsmooth Functions

In this paper, we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions, which models a literally inhibited neural network with unsupervised Hebbian learning. The network has two types of state variables, one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights. Based on the nonsmooth analysis techniques, we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network. Numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

[1]  S. Grossberg Competition, Decision, and Consensus , 1978 .

[2]  Roman Bek,et al.  Discourse on one way in which a quantum-mechanics language on the classical logical base can be built up , 1978, Kybernetika.

[3]  B. Pourciau Hadamard's theorem for locally Lipschitzian maps , 1982 .

[4]  Shun-ichi Amari,et al.  Competitive and Cooperative Aspects in Dynamics of Neural Excitation and Self-Organization , 1982 .

[5]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[7]  Shun-ichi Amari,et al.  Field theory of self-organizing neural nets , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Allon Guez,et al.  On the stability, storage capacity, and design of nonlinear continuous neural networks , 1988, IEEE Trans. Syst. Man Cybern..

[10]  Morris W. Hirsch,et al.  Convergent activation dynamics in continuous time networks , 1989, Neural Networks.

[11]  B. V. K. Vijaya Kumar,et al.  Emulating the dynamics for a class of laterally inhibited neural networks , 1989, Neural Networks.

[12]  D. Kelly,et al.  Stability in contractive nonlinear neural networks , 1990, IEEE Transactions on Biomedical Engineering.

[13]  Kiyotoshi Matsuoka,et al.  Stability conditions for nonlinear continuous neural networks with asymmetric connection weights , 1992, Neural Networks.

[14]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[15]  M. Forti,et al.  Necessary and sufficient condition for absolute stability of neural networks , 1994 .

[16]  A. Tesi,et al.  New conditions for global stability of neural networks with application to linear and quadratic programming problems , 1995 .

[17]  Anke Meyer-Bäse,et al.  Singular Perturbation Analysis of Competitive Neural Networks with Different Time Scales , 1996, Neural Computation.

[18]  N. Swindale The development of topography in the visual cortex: a review of models. , 1996, Network.

[19]  Christoph von der Malsburg,et al.  Self-organization and the brain , 1998 .

[20]  Herbert Witte,et al.  Learning continuous trajectories in recurrent neural networks with time-dependent weights , 1999, IEEE Trans. Neural Networks.

[21]  Liang Jin,et al.  Stable dynamic backpropagation learning in recurrent neural networks , 1999, IEEE Trans. Neural Networks.

[22]  Xue-Bin Liang A comment on "On equilibria, stability, and instability of Hopfield neural networks" [and reply] , 2000, IEEE Trans. Neural Networks Learn. Syst..

[23]  S. Arik,et al.  A sufficient condition for absolute stability of a larger class of dynamical neural networks , 2000 .

[24]  Johan A. K. Suykens,et al.  Robust local stability of multilayer recurrent neural networks , 2000, IEEE Trans. Neural Networks Learn. Syst..

[25]  X. B. Liwang A comment on "On equilibria, stability, and instability of Hopfield neural networks". , 2000, IEEE transactions on neural networks.

[26]  Anke Meyer-Bäse,et al.  Global exponential stability of competitive neural networks with different time scales , 2003, IEEE Trans. Neural Networks.

[27]  S. Grossberg,et al.  Adaptive pattern classification and universal recoding: I. Parallel development and coding of neural feature detectors , 1976, Biological Cybernetics.

[28]  Xiaoqi Yang,et al.  Deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis , 2004, IEEE Transactions on Neural Networks.

[29]  C. Malsburg Self-organization of orientation sensitive cells in the striate cortex , 2004, Kybernetik.

[30]  Sabri Arik,et al.  Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays , 2005, IEEE Transactions on Neural Networks.

[31]  Zhidong Teng,et al.  Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays , 2005, IEEE Transactions on Neural Networks.

[32]  Duccio Papini,et al.  Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain , 2005, IEEE Transactions on Neural Networks.

[33]  Xinzhi Liu,et al.  Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays , 2005, IEEE Trans. Neural Networks.