  The global asymptotic behavior of a stochastic Hopfield neural network model
(HNNM) with delays is explored by studying the existence and structure of
random attractors. It is first proved that the trajectory field of the
stochastic delayed HNNM admits an almost sure continuous version, which is
compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction
based on the random semiflow generated by the diffusion term. Then, this
version is shown to generate a random dynamical system (RDS) by piece-wise
linear approximation, after which the existence of a random absorbing set is
obtained by a careful uniform apriori estimate of the solutions. Subsequently,
the pullback asymptotic compactness of the RDS generated by the stochastic
delayed HNNM is proved and hence the existence of random attractors is
obtained. Moreover, sufficient conditions under which the attractors turn out
to be an exponential attracting stationary solution are given. Numerical
simulations are also conducted at last to illustrate the effectiveness of the
established results.
