  Grid cells, discovered more than a decade ago [5], are neurons in the brain
of mammals that fire when the animal is located near certain specific points in
its familiar terrain. Intriguingly, these points form, for a single cell, a
two-dimensional triangular grid, not unlike our Figure 3. Grid cells are widely
believed to be involved in path integration, that is, the maintenance of a
location state through the summation of small displacements. We provide
theoretical evidence for this assertion by showing that cells with grid-like
tuning curves are indeed well adapted for the path integration task. In
particular we prove that, in one dimension under Gaussian noise, the
sensitivity of measuring small displacements is maximized by a population of
neurons whose tuning curves are near-sinusoids -- that is to say, with peaks
forming a one-dimensional grid. We also show that effective computation of the
displacement is possible through a second population of cells whose sinusoid
tuning curves are in phase difference from the first. In two dimensions, under
additional assumptions it can be shown that measurement sensitivity is
optimized by the product of two sinusoids, again yielding a grid-like pattern.
We discuss the connection of our results to the triangular grid pattern
observed in animals.
