Kunin, D., Kymn, C., Acosta, F., Marchetti, G., & Miolane, N.
Abstract
Medial entorhinal grid cells are widely viewed as an essential ingredient of mammalian navigation in physical and conceptual spaces [1–4], yet the computations they implement and the mechanisms driving their emergence remain debated. Specific questions include why grid cells organize into discrete modules at consistent scale ratios [5], what environmental factors influence the geometry of their firing patterns [6], and why grid cells emerge abruptly in development [7]. We provide theoretical insights into these questions with an algebraic perspective, studying how simple recurrent neural networks (RNNs) solve a group composition task. Concretely, given a finite group G and an encoding vector x, the task is to map a sequence of encoded group elements g1· x, . . . , gk· x to their cumulative product g1· · · gk· x. This general formulation unifies tasks like path integration and conceptual navigation in a mathematically rigorous way. We prove—and verify empirically—that an RNN learns the task one irreducible representation (irrep) of the group G at a time and that irreps are learned in order of their contribution to the target g1· · · gk·x. For the group of 2D translations, these irreps are precisely the 2D Fourier modes needed to generate hexagonal activation patterns [8]. Strikingly, we find that hidden neurons abruptly specialize to individual irreps and self-organize into independent “modules”, each aligned to a distinct irrep, within which phases distribute approximately uniformly to form regular polygons in phase space. Overall, our findings provide a tractable route toward a general theory of grid cells.
Citation
Kunin, D., Kymn, C., Acosta, F., Marchetti, G., & Miolane, N. A group theoretic perspective on path integration and the emergence of grid cells (2026).
