A Group-Theoretic Perspective on Path Integration and The Origin of Grid Cells

We develop a group-theoretic framework predicting that the neural representations supporting path integration reflect the mathematical structure of the space being navigated. Path integration is the process through which an animal estimates its allocentric position by integrating egocentric movements and is thought to be a core computation of the entorhinal–hippocampal system. We formalize path integration as _sequential group composition_ over finite groups, with different groups describing different spaces. Using group Fourier analysis we analytically construct recurrent neural networks that perform the computation exactly. These circuits decompose into modules indexed by irreducible representations (irreps) and achieve asymptotically optimal hidden width. For a group coupling translations and rotations in 2D space, our construction reproduces key features of the entorhinal grid-cell code: hexagonal spatial tuning, modules with distinct spatial scales, conjunctive position–heading tuning, and toroidal population geometry. Extending the construction to 3D navigation predicts a grid-cell code with irregular, yet locally structured, spatial tuning built from irreps of a 3D navigation group. Lastly, we introduce _irrep selectivity_, a group-theoretic generalization of gridness that identifies new units aligned to path integration tasks without requiring strict hexagonal symmetry. Taken together, our results provide simple accounts of detailed grid cell phenomenology and new ways for analyzing grid cell data.