Efficient and flexible representation of higher-dimensional cognitive variables with grid cells
(a) If the encoded variable is 3D (here, the animal leaves the 2D plane), simple projection down to the 2D phase is ambiguous and consistent with multiple locations in the z-direction. (b) There are two sources of ambiguity, the periodicity of the grid code within the xy-plane, and the ambiguity in the z-direction. (c) Two different 2D phases for two modules are set by two distinct projections (red and blue) of the 3D value onto a plane. Together they are able to simultaneously resolve both sources of ambiguity. (d) Estimates of the value of the encoded 3D variable obtained by combining the ambiguous estimates of 1, 2, 3, and 4 modules as in (c). Given the cell responses we compute a probability estimate and show only areas that exceed a fixed threshold (blue blobs). The spacing between the blobs defines the coding range, that is, the range over which the code is unique. With an increasing number of modules the range quickly grows larger than the individual periods.