  Certain types of neurons, called "grid cells", have been shown to fire on a
triangular grid when an animal is navigating on a two-dimensional environment,
whereas recent studies suggest that the face-centred-cubic (FCC) lattice is the
good candidate for the same phenomenon in three dimensions. The goal of this
paper is to give new evidences of these phenomena by considering a infinite set
of independent neurons (a module) with Poisson statistics and periodic spread
out Gaussian tuning curves. This question of the existence of an optimal grid
is transformed into a maximization problem among all possible unit density
lattices for a Fisher Information which measures the accuracy of grid-cells
representations in $\mathbb{R}^d$. This Fisher Information has translated
lattice theta functions as building blocks. We first derive asymptotic and
numerical results showing the (non-)maximality of the triangular lattice with
respect to the Gaussian parameter and the size of the firing field. In a
particular case where the size of the firing fields and the lattice spacing
match with experiments, we have numerically checked that it is possible to find
a value for the Gaussian parameter above which the triangular lattice is always
optimal. In the case of a radially symmetric distribution of firing locations,
we also characterize all the lattices that are critical points for the Fisher
Information at fixed scales belonging to an open interval (we call these
lattices "volume stationary"). It allows us to compare the Fisher Information
of a finite number of lattices in dimension 2 and 3 and to give another
evidences of the optimality of the triangular and FCC lattices.
