Hexagons all the way down: Grid cells as a conformal isometric map of space

The brain's ability to navigate is often attributed to spatial cells in the hippocampus and entorhinal cortex. Grid cells, found in the entorhinal cortex, are known for their hexagonal spatial activity patterns and are traditionally believed to be the neural basis for path integration. However, recent studies have cast grid cells as a distance-preserving representation. We further investigate this role in a model of grid cells based on a superposition of plane waves. In a module of such grid cells, we optimise their phases to form a conformal isometry (CI) of two-dimensional flat space. With this setup, we demonstrate that a module of at least seven grid cells can achieve a CI, with phases forming a regular hexagonal arrangement. This pattern persists when increasing the number of cells, significantly diverging from a random uniform distribution. In particular, when optimised for CI, the phase distribution becomes distinctly regular and hexagonal, offering a clear experimentally testable prediction. Moreover, grid modules encoding a CI maintain constant energy expenditure across space, providing a new perspective on the role of energy constraints in normative models of grid cells. Finally, we investigate the minimum number of grid cells required for various spatial encoding tasks, including a unique representation of space, the population activity forming a torus, and achieving a CI, where we find that all three are achieved when the module encodes a CI. Our study not only underscores the versatility of grid cells beyond path integration but also highlights the importance of geometric principles in neural representations of space. ### Competing Interest Statement The authors have declared no competing interest.