The Sense of Place: Grid Cells in the Brain and the Transcendental Number e

TL;DR

This paper presents a theoretical model explaining the organization of grid cells in the brain, predicting their scale ratios and lattice structure based on minimizing neuronal resources for spatial encoding.

Contribution

It introduces a mathematical framework linking grid cell scales to the transcendental number e, predicting geometric progression and lattice arrangements in spatial navigation.

Findings

Grid scales follow a geometric progression with ratios near √e.

The ratio between grid scale and field width is consistent within and across animals.

Grid fields are arranged on a triangular lattice.

Abstract

Grid cells in the brain respond when an animal occupies a periodic lattice of "grid fields" during spatial navigation. The grid scale varies along the dorso-ventral axis of the entorhinal cortex. We propose that the grid system minimizes the number of neurons required to encode location with a given resolution. We derive several predictions that match recent experiments: (i) grid scales follow a geometric progression, (ii) the ratio between adjacent grid scales is the square root of e for idealized neurons, and robustly lies in the range 1.4-1.7 for realistic neurons, (iii) the scale ratio varies modestly within and between animals, (iv) the ratio between grid scale and individual grid field widths at that scale also lies in this range, (v) grid fields lie on a triangular lattice. The theory also predicts the optimal grids in one and three dimensions, and the total number of discrete scales.