Multistability in a class of stochastic delayed Hopfield neural networks

In this paper, multistability analysis for a class of stochastic delayed Hopfield neural networks is investigated. By considering the geometrical configuration of activation functions, the state space is divided into 2n+12n+1 regions in which 2n2n regions are unbounded rectangles. By applying Schauder’s fixed-point theorem and some novel stochastic analysis techniques, it is shown that under some conditions, the 2n2n rectangular regions are positively invariant with probability one, and each of them possesses a unique equilibrium. Then by applying Lyapunov function and functional approach, two multistability criteria are established for ensuring these equilibria to be locally exponentially stable in mean square. The first multistability criterion is suitable to the case where the information on delay derivative is unknown, while the second criterion requires that the delay derivative be strictly less than one. For the constant delay case, the second multistability criterion is less conservative than the first one. Finally, an illustrative example is presented to show the effectiveness of the derived results.