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Hexagons all the way down: grid cells as a conformal isometric map of space

Fig 2

Distortions in spatial representation are encoded in the metric tensor.

a) An illustration of a standard football field (top) and a distorted version (bottom), where vertical distances compress as one moves to the right. b) The metric tensor components show how distances in a) are locally deformed in each direction. For example, the component shows that as one goes to the right, distances in the vertical direction will shrink, whereas the component shows that as one goes up or down, distances to the right will expand. c) Phase arrangements of 100 grid cells inside a unit cell (upper left in each set of four figures), along with a three-dimensional UMAP projection of the generated activity coloured by the metric tensor components (indicated by labels). Unit cell plots are coloured by the determinant of the metric tensor. The example on the first row shows a low-dimensional projection of a conformally isometric torus, and trajectories are undistorted. The example on the second row shows a projection of the torus, when the phases are sampled densely around a small region. In this case, trajectories crossing phase-dense regions appear more dense on the manifold.

doi: https://doi.org/10.1371/journal.pcbi.1012804.g002