Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions

In this letter, without assuming the boundedness of the activation functions, we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions. A relaxed set of sufficient conditions is derived, guaranteeing the existence, uniqueness, and global stability of the equilibrium point. Convergence behaviors for both state and output are discussed. The constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique. We also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high-slope continuous activation functions.

[1]  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.

[2]  Leon O. Chua,et al.  Neural networks for nonlinear programming , 1988 .

[3]  Leon O. Chua,et al.  Linear and nonlinear circuits , 1987 .

[4]  Mark P. Joy,et al.  Results concerning the absolute stability of delayed neural networks , 2000, Neural Networks.

[5]  G. Haddad,et al.  Monotone viable trajectories for functional differential inclusions , 1981 .

[6]  S. Sastry,et al.  A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators , 1987 .

[7]  M. Forti,et al.  Global convergence of neural networks with discontinuous neuron activations , 2003 .

[8]  John S. Spraker.  A Comparison of the Carathéodory and Filippov Solution Sets , 1996 .

[9]  A. F. Filippov.  Classical Solutions of Differential Equations with Multi-Valued Right-Hand Side , 1967 .

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

[11]  Tianping Chen,et al.  Global exponential stability of delayed Hopfield neural networks , 2001, Neural Networks.

[12]  L. Pandolfi,et al.  On stability of cellular neural networks with delay , 1993 .

[13]  Tianping Chen,et al.  Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions , 2005, Neural Networks.

[14]  Eduardo Liz,et al.  Existence theory for first order discontinuous functional differential equations , 2002 .

[15]  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.

[16]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[17]  Qingshan Liu,et al.  A One-layer Recurrent Neural Network with a Unipolar Hard-limiting Activation Function for k-Winners-Take-All Operation , 2007, 2007 International Joint Conference on Neural Networks.

[18]  J. Hopfield,et al.  Computing with neural circuits: a model. , 1986, Science.

[19]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[20]  Xuyang Lou,et al.  Boundedness and Stability for Integrodifferential Equations Modeling Neural Field with Time Delay , 2007, Neural Computation.

[21]  Josef A. Nossek,et al.  An analog implementation of discrete-time cellular neural networks , 1992, IEEE Trans. Neural Networks.

[22]  Francesca Maria Ceragioli,et al.  Discontinuous ordinary differential equations and stabilization , 2000 .

[23]  V. Utkin.  Variable structure systems with sliding modes , 1977 .

[24]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[25]  Tianping Chen,et al.  Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations , 2008, Neural Computation.

[26]  Jun Wang,et al.  Global exponential stability of a general class of recurrent neural networks with time-varying delays , 2003 .

[27]  Zhigang Zeng,et al.  Global Stability of a General Class of Discrete-Time Recurrent Neural Networks , 2005, Neural Processing Letters.