  Grid cells in the entorhinal cortex fire when animals that are exploring a
certain region of space occupy the vertices of a triangular grid that spans the
environment. Different neurons feature triangular grids that differ in their
properties of periodicity, orientation and ellipticity. Taken together, these
grids allow the animal to maintain an internal, mental representation of
physical space. Experiments show that grid cells are modular, i.e. there are
groups of neurons which have grids with similar periodicity, orientation and
ellipticity. We use statistical physics methods to derive a relation between
variability of the properties of the grids within a module and the range of
space that can be covered completely (i.e. without gaps) by the grid system
with high probability. Larger variability shrinks the range of representation,
providing a functional rationale for the experimentally observed co-modularity
of grid cell periodicity, orientation and ellipticity. We obtain a scaling
relation between the number of neurons and the period of a module, given the
variability and coverage range. Specifically, we predict how many more neurons
are required at smaller grid scales than at larger ones.
