Several differential equation models have been proposed to explain the formation of patterns characteristic of the grid cell network. Understanding the effect of noise on these models is one of the key open questions in computational neuroscience. In the present work, we continue the analysis of an SDE system commonly proposed for this aim, which we initiated in a previous paper. We show that the fluctuations of the empirical measure associated to the particle system around its mean field limit converge to the solution of a Langevin SPDE. The interaction between different columns of neurons along the cortex prescribes a peculiar scaling regime.