Several differential equation models have been proposed to explain the formation of patterns characteristic of the grid cell network. Understanding the robustness of these patterns with respect to noise is one of the key open questions in computational neuroscience. In the present work, we analyze a family of stochastic differential systems modelling grid cell networks. Furthermore, the well-posedness of the associated McKean--Vlasov and Fokker--Planck equations, describing the average behavior of the networks, is established. Finally, we rigorously prove the mean field limit of these systems and provide a sharp rate of convergence for their empirical measures.