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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 Itô 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.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93E15 Stochastic stability in control theory
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##### References:
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