< Back to Article

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

Fig 3

Optimal phases for seven cells (a-h) and 100 cells (i-m).

a) Learned phase positions for seven grid cells within the unit cell (blue) and their inferred hexagonal arrangement (orange). b) Three-dimensional UMAP projection of the seven-cell population activity, colour-coded by Voronoi diagram (shown in d). c) A persistence diagram illustrating the activity topology from a). d) Voronoi diagram showing spatial partitioning of phases across an extended grid, with the primitive (rhombus) cell superimposed. e) L2 norm of population activity across the unit cell. Colour range spans from 0 to , the maximum norm for seven cells. f) Conformal isometry grid search when varying the angle and magnitude of the hexagonal phase solution in a). Colours are log ⁡ C + 1 where C is the conformal isometry metric defined in Eq 5. g) Common phase translation grid search, showing invariance to phase shifts (colour scale from f). h) Grid search focusing on varying just one (marked as a red dot) of the phases, using the same colour scale as f). i) Ripley’s H-function analysis comparing the dispersion of optimised (blue) and random (orange) phases. The analysis was conducted on 20 evenly spaced radial distances up to but not including the unit cell radius (r = 2 ∕ 3). Stars indicate significant differences (p < 0 . 01), as determined by a permutation test (details in methods). j) Grid score analysis of learned versus random phase distributions, inferred through Gaussian kernel density estimation, plotted against varying kernel bandwidths in the range of (not including) zero to half the radius length of the unit cell r ∕ 2 = 1 ∕ 3. j) Grid score comparisons of optimised versus random phase distributions across different kernel bandwidths for Gaussian kernel density estimation (KDE). Bandwidths range up to half the unit cell radius (r ∕ 2 = 1 ∕ 3), highlighting the hexagonality of the optimised phase arrangement. k) Kernel density estimate (KDE) of the 15 replications of the phase solution from panel a), each with an additional random phase shift. Bandwidth for KDE is set at 0.1, with the resulting grid score shown in the title. The KDE is normalized within the unit cell to integrate to unity. l) (top) KDE visualisation of optimised phase distributions overlaid with phase locations, with titles indicating grid scores. Four bandwidths were selected: r ∕ 20, 5r ∕ 20, 10r ∕ 20, and 15r ∕ 20. Standard deviations for the KDEs are , reflecting the KDE colour variability across bandwidths. All KDEs are normalized to have a unit integral. (bottom) Autocorrelograms of the KDEs to confirm hexagonal spatial periodicity. m) KDEs for random phase distributions at the same bandwidths as panel l), providing a baseline comparison. KDE standard deviations are .

doi: https://doi.org/10.1371/journal.pcbi.1012804.g003