zbMATH — the first resource for mathematics

Exponential stability and instability of stochastic neural networks. (English) Zbl 0848.60058
Summary: We discuss stochastic effects to the stability property of a neural network \(\dot u(t) = - Bu(t) + Ag(u(t))\). Suppose the stochastically perturbed neural network is described by an ItĂ´ equation \[ dx(t) = \biggl [-Bx(t) + Ag \bigl( x(t) \bigr) \biggr] dt + \sigma \bigl( x(t) \bigr) dw(t). \] The general theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established. The theory is then applied to investigate the stochastic stabilization and destabilization of the neural network. Several interesting examples are also given for illustration.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
Full Text: DOI
[1] Coben, M.A. and Crosshery, S. 1983.IEEE Trans. on Systems, Man and CyberneticsVol. 13, 815–826.
[2] Denker, J.S., ed. Neural Networks for Computing. Proceedings of the Conference on Neural Networks for Computing. 1986, New York. Snowbird, UT: AIP.
[3] Friedman A., Stochastic Differential Equations and Applications (1975) · Zbl 0323.60056
[4] Hopfield, J.J. 1982. Neural networks and physical systems with emergent collect computational abilities. Proe. Natl. Acad, Sci. 1982, USA. Vol. 79, · Zbl 1369.92007
[5] 1984. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad, Sci. 1984, USA. pp.2554 · Zbl 1371.92015
[6] Hopfield J.J., Model Science pp 3088– (1986)
[7] Liao, X.X. 1992. Stability of a class of nonlinear continuous neural networks. Proceedings of the First World” @Conference on Nonlinear Analysis. 1992. Vol. WC313, · Zbl 0981.34037
[8] Liptser R., 0 (1986)
[9] Mao X., Exponential Stability of Stochastic Differential Equations (1994) · Zbl 0806.60044
[10] Metivier M., Semimartingales (1982) · Zbl 0503.60054
[11] Quezz A., IEEE Trans. on Systems 18 pp 80– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.