Table 1.

Axioms from which grid cell firing emerges as the most parsimonious solution to provide a code for trajectories in 2D space.

	Axiom	Remark/Example
Axiom 1	Cell sequences code for trajectories in 2D space.	This axiom states that the sequential activity of two cells, e.g., i → j, can unambiguously be interpreted by a downstream reader mechanism as a a code for one and only one trajectory, e.g., moving from place A to place B.
Axiom 2	The reverse cell sequence codes for the reverse trajectory.	If the cell sequence i → j codes for moving from place A to B, then the reverse cell sequence j → i codes for moving from place B to place A.
Axiom 3	Each cell’s firing field is surrounded symmetrically by a maximal number of other cell’s firing fields so that the angular resolution is constant and maximal across all directions.	This axiom means that a sequence code should represent each direction equally with no “gaps” in the representation (Figures 3D–E).
Axiom 4	The number of cells is finite.	Grid cells are densest in layer II of the medial entorhinal cortex (Sargolini et al., 2006), and this layer has been estimated to contain 24,000 and 58,000 neurons in mice and rats, respectively (Gatome et al., 2010).