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

Fig 1

Unique representation and toroidal encoding in a grid-cell module with varying phases.

a) Illustration of grid cells modelled as a superposition of three plane waves. b) Visualization of three distinct grid cells from the same module, each with their respective unit cells superimposed, demonstrating the periodic patterns. The right-hand image shows the phases of the three grid cells within a shared unit cell. c) Population vector correlation for 1, 2, and 3 cells’ activity (left to right) relative to the activity at the red cross. White lines and dots highlight ambiguous points, defined as locations where the Euclidean distance between the activity vector at each location and that at the red cross is less than . For populations of 1 and 2 cells, ambiguities are present within the unit cell, while with 3 cells these ambiguities are resolved. The unit cell border colours (yellow, green, and pink) correspond to the phases from panel b) included in the population. d) Grid modules with varying numbers of cells (n = 1 , 2 , 3 , and 15) and randomly distributed phases, shown within a unit cell. e) Persistence barcodes of the grid module from d), indicating the lifetime of zero-, one-, and two-dimensional holes in the population representation. With a larger cell number (here n = 15), we observe one persistent 0D, two 1D, and one 2D bar, suggesting a toroidal topology. f) Population representation of the grid cells from d). The first three plots display the population activity of the grid cells along their firing rate axes. The final plot presents the UMAP projection of the activity of 15 grid cells in three dimensions, resembling a torus. Colours represent the population vector correlation relative to the centre of the unit cell.

doi: https://doi.org/10.1371/journal.pcbi.1012804.g001