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

Fig 5

Impact of the Number of Cells on Spatial Encoding.

a) One dimensional tuning curves for populations of 2, 3, and 10 neurons modelled by Eq 1. b) Population activity vectors corresponding to the waves from (a) form an isotropic ring with radius , embedded in a plane. Increasing the number of cells expands this ring, intersecting with more unique voxels (distinct population states) and enhancing spatial resolution by enabling finer distinctions between encoded positions. c) The conformal scale () from Eq 2 plotted against the firing rate (A) and number of cells (N), with light blue contour lines representing the square grid’s scale (). d) Population vector norm, or energy, shown relative to the firing rate A and cell count N, illustrating how energy scales with module size. e) A schematic of the unit cell with overlaid meshes of different granularities to depict spatial resolution. The left side, with a coarser mesh, represents a module with fewer cells, while the right side, with a finer mesh, represents a module with more cells, highlighting the resolution enhancement with increased cell count. f) The average geodesic distance between initial and optimised phase positions, plotted against module size (N = 7 , 14 , 21 , … , 133). g) Illustration of how increasing cell count leads to larger toroidal structures and extended neural trajectories, highlighting spatial and topological expansion in larger modules.

doi: https://doi.org/10.1371/journal.pcbi.1012804.g005