Robust and efficient coding with grid cells
Fig 2
Interference depends on the choice of the scales.
(a) Interference with rational scale ratio. Left: Representative posteriors (P(x|s)) for two modules with scale 1 and α = 3/2. Encoding becomes ambiguous at distance 3 from the origin where perfect interference occurs (3 = 2α). Right: Phase plot of the two modules, with the colour (red to blue) encoding the distance from the origin (see the coloured line below the left panel). Perfect interference occurs when the phase-curve overlaps with itself. (b) Interference with α = 1.76 …, which is close to 7/4 and therefore leads to strong interference at distance 7. Right: Interference occurs when the distance between two neighbouring segments of the phase curve becomes smaller than the limit set by the neuronal noise (grey squares of side δ around the origin, see inset). Note, that both grids are around phase 0.3 at the distance 2.3 without interference. (c) Interference with α = σ ≈ 1.618, which is the golden ratio. Interference still becomes stronger at larger distances, (e.g. at distance 5, since ). Interference in grid codes is related to the approximation of irrationals with rational numbers having small denominators (see text for further details). Right: Interference is inevitable since the phase space has a limited volume.