On the optimality of grid cells

TL;DR

This paper provides theoretical evidence that grid cells' firing patterns are optimized for path integration, with sinusoidal tuning curves maximizing sensitivity to small displacements in one and two dimensions.

Contribution

It demonstrates mathematically that grid-like tuning curves are optimal for path integration, linking neural tuning properties to efficient spatial displacement measurement.

Findings

Sinusoidal tuning curves maximize displacement sensitivity in 1D.

Phase-shifted sinusoid populations enable displacement computation.

Product of two sinusoids yields a grid pattern in 2D.

Abstract

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.