Efficient and flexible representation of higher-dimensional cognitive variables with grid cells
(a) Change of coding range while increasing the dimensionality of the input (N) and keeping the projection fixed (per trial), illustrating the flexibility of the scheme (10 trials are shown). (b) Left: The conventional grid code encodes only 2D variables (x, y, black arrows; and not z, red); movements in x, y result in phase changes across modules, but changes in z do not. These phase changes evolve along only one direction in the coding space (here, showing 2 modular phases), governed by the ratios of the grid periods. Middle: The disjoint grid module code: Modules are partitioned into disjoint groups, one set coding x, y, the other coding z. Changes in x, y update one set of phases, changes in z another; the red circle cannot be reached from the black by only changing x, y, without changing z. Right: Updates in module activity are decoupled as for the disjoint code but each module participates in the representation of all input dimensions. The periodicity of the code makes it possible to reach (and thus use) the white coding state from the black in two ways: by moving along x, y, or along z (red and black arrows).