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

Grid cells in the entorhinal cortex are known for their hexagonal spatial activity patterns and are thought to provide a neural metric for space, and support path integration. In this study, we further investigate grid cells as a metric of space by optimising them for a conformal isometric (CI) map of space using a model based on a superposition of plane waves. By optimising the phases within a single grid cell module, we find that the module can form a CI of two-dimensional flat space with phases arranging into a regular hexagonal pattern, supporting an accurate spatial metric. Additionally, we find that experimentally recorded grid cells exhibit CI properties, with one example module showing a phase arrangement similar to the hexagonal pattern observed in our model. These findings provide computational and preliminary experimental support for grid cells as a CI-based spatial representation. We also examine other properties that emerge in CI-optimised modules, including consistent energy expenditure across space and the minimal cell count required to support unique representation of space and maximally topologically persistent toroidal population activity. Altogether, our results suggest that grid cells are well-suited to form a CI map, with several beneficial properties arising from this organisation.Author summary: Grid cells in the brain’s entorhinal cortex are neurons that fire in hexagonal patterns as an animal moves through space, effectively creating an internal map of the environment. These cells are believed to perform two main functions: path integration—the process by which animals determine their position based on self-movement—and serving as a spatial metric, accurately representing distances and angles within their surroundings. In our study, we focus on the latter function: how grid cells act as a metric of space. Using a mathematical model based on the superposition of plane waves, we investigate how and when grid cells naturally preserve distances and angles in flat two-dimensional space. Our findings demonstrate that grid cells can inherently form a conformal isometric (CI) map, maintaining the geometric properties essential for accurate spatial representation. This understanding suggests that grid cells are not only crucial for navigation but also for constructing a precise metric of space, naturally preserving distances and angles. By elucidating the conditions under which grid cells form a CI map, our work contributes to the broader knowledge of spatial cognition and may have implications for developing artificial navigation systems that mimic biological processes.