Structure of the grid cell code.

A: Neurons are tuned to a hexagonal lattice of positions in 2D space. B: They are grouped into modules: neurons in the same module have translated (but not rotated) receptive fields, and across a module they uniformly sample the phases (translations). C: There are only a handful of modules in one animal, each with its own lattice, and ~ 1000s neurons covering the possible phases. D: For each grid module there is a population of grid cells that are conjunctively tuned to both the underlying grid of the module, and a particular heading direction. E: These conjunctive neurons can implement path-integration by pushing the bump of neural activity around the module (Burak and Fiete), like the ring attractor in the fly central complex (Hulse and Jayaraman), using a shifted connectivity pattern: pure spatial neurons project to conjunctive neurons with the same spatial tuning profile (red connections), which project back to the spatial neurons shifted by their velocity tuning (blue connections). When the rightward neurons are more active than the leftward, this will cause the activity bump to move rightwards on the ring, implementing path-integration.

Path-integration with different codes

A: Path-integrating with a place cell code is easy, current cell plus step uniquely determines next cell, but it is limited by the number of cells. B: Multifield cells improve the coding capacity but make path-integration more challenging, instead resources must be devoted to learning a mapping between unique combinations of cells. C: Within a grid moudle, current cell plus movement again uniquely determines the next cell: no matter which firing field of a grid cell you are in, thanks to the translational symmetry, you always know which cell to activate after a step. As such, grid cells elegantly combine the easy path-integration of place cells, with the higher capacity coding of multifield cells, and the path-integration mechanism generalises across space.

Grid Cells via Bandpass Filtering.

A: A Gaussian place cell code has a covariance whose frequency content is a smoothly-decaying Gaussian, left, but a difference-of-Gaussian code has covariance whose frequency content peaks at a non-zero frequency, figure from Sorscher et al. B: The grid cells that result from nonnegative PCA on difference-of-Gaussian place cells are not translationally symmetric, each population contains grid cells whose axes are rotated relative to one another (for example, the left and rightmost grid cells from dordek have lattices rotated 30° relative to one another), figures from Dordek et al. and Sorscher et al. C: We create a representation, g(x), that contains a single frequency, and plot the conformal loss, eq. (3), as a function of this single frequency for a few σ values. This loss is minimised (dark blue) at an intermediate value of frequency: a bandpass filtering effect. D: Metric encoding also produces a population of grid cells that are rotated relative to one another, figure from (Pettersen et al.).

Successor representation eigenvectors are poor models of grid cells, figure from Stachenfeld, Botvinick, and Gershman.

A Space of Optimal Codes.

We optimise a nonnegative, unit-norm representation of position to minimise a similarity matching objective either linear, eq. (2), or nonlinear, eq. (8), with or without a path-integrating constraint, eq. (5). With more neurons than positions all choices lead to place cells (not shown). With few neurons and no path-integration (left column) we get place cells with a linear objective, and random multifields with a nonlinear objective (see also fig 15C, Dorrell et al.). Adding a path-integration constraint leads to either one grid module for the linear similarity loss, or multiple under the nonlinear loss (for more discussion, see Dorrell et al.).

We optimise a metric encoding loss, eq. (3) with large σ and find the optimal representation is place cells, matching the correspondance with the similarity matching objective, section 3.1.

We use a periodic environment for convenience, hence the multiple patches observed correspond to parts of the same field.