Hexagons all the way down: grid cells as a conformal isometric map of space
Fig 4
Signs of conformal mapping and phase hexagonality in a module of 105 experimental grid cells.
a) Euclidean distance between the grid cell population vector at the red cross and population vectors at all other spatial locations. b) Metric tensor components for the experimental data, with values represented on a shared colour scale shown in the colour bar. c) Neural-physical distance plot comparing experimental data, phase-clustered data, and spatially shuffled neural distances, indicating how neural distances correspond to physical distances. d) 1D histogram of the metric tensor components from (b), with distributions from spatially shuffled ratemaps included as a comparative baseline. e) Same as (c), but restricted to short physical distances, defined as less than 25% of the maximum physical distance in (c). Linear regression analyses are conducted for subsets corresponding to 5%, 10%, 15%, 20%, and 25% of the total physical distance range. The legend includes the Pearson correlation coefficients (r-values) from the regressions, illustrating the strength of the linear relationship between neural and physical distances for shorter and longer distance ranges. f) Ripley’s H-function on the experimental phases, with random uniform sampling as a baseline. Shaded areas represent twice the standard deviation from 100 resampling trials, and significant deviations (two standard deviations from the baseline) are marked with red stars. A CI+noise module (105 phases total, derived from 15 noise-induced copies of a 7-cell CI solution) provides an additional comparison, with its radius set at 1.4 times that of the data radius. g) Unit cells and kernel density estimates (KDEs) from the grid modules in (e) with phases superimposed. The colour scale is shared across each KDE.