Compared with the traditional neural network, the memristor-based neural networks can better reflect the variable intensity of synapse, so it can better simulate the neural system of human brain. Under the framework of Filippov solution, the dynamical behavior of a class of stochastic memristor-based neural network is studied by employing appropriate Lyapunov functional, It$\hat{\rm o}$'s differential formula, theories of differential inclusions and set-valued maps. Several sufficient conditions are obtained for ensuring the system to be mean square exponential stability in this paper. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed results.