Spatial periodicity in grid cell firing is explained by a neural sequence code of 2D trajectories

Spatial periodicity in grid cell firing has been interpreted as a neural metric for space providing animals with a coordinate system in navigating physical and mental spaces. However, the specific computational problem being solved by grid cells has remained elusive. We here provide mathematical proof that spatial periodicity in grid cell firing is the only possible solution to a neural sequence code of 2D trajectories and that the hexagonal firing pattern of grid cells is the most parsimonious solution to such a sequence code. We thereby provide a teleological cause for the existence of grid cells and reveal the underlying nature of the global geometrical organization in grid maps as a direct consequence of a simple local sequence code using a minimal number of neurons. A sequence code by grid cells provides intuitive explanations for many previously puzzling experimental observations and may transform our thinking about grid cells.