Quantified symmetry for entorhinal spatial maps

General navigation requires a spatial map that is not anchored to one environment. The firing fields of the “grid cells” found in the rat dorsolateral medial entorhinal cortex (dMEC) could be such a map. Our work provides an explanation for how the context-independent properties of “grid cell” firing arise. We use computational means to analyze and validate the geometric and algebraic invariant properties of the firing fields, leading to a context-independent spatial map. Our method computes the specific symmetry group implicitly associated with the spatial map, and quantifies the regularity of the firing fields to achieve a symmetry-based clustering into two different types of “grid cells.” This quantified regularity makes spatial mapping more computationally efficient and suggests a way to use the dMEC firing patterns to decode the rat’s position in the room. Finally, the highly invariant lattice structure of a “grid cell” firing field encodes the rat’s position with sufficient redundancy to remain the same under changes in the shape of the room. Thus we show formally how the context-independent properties of “grid cells” can arise from their invariance under transformation.