The firing fields of grid cells display their striking hexagonal lattice pattern in 2d environments. However, the structure of grid cell fields on a linear track has not been adequately investigated or explained. We use a straightforward but powerful Fourier analysis method to analyze 1d grid fields, showing that they do indeed arise as "slices" of an underlying 2d hexagonal grid. Furthermore, the fields of anatomically nearby cells can be realized as parallel slices of the same grid, with different phase shifts. Using new data from the Tank lab, we show that these 1d shifts are related to the shifts seen in 2d between the same cells.
Our underlying motivation is to understand the mechanism responsible for grid cell firing patterns. A grid cell model must be able to produce hexagonal fields, but can a more detailed study of grid cell dynamics in 1d actually provide constraints on which classes of models are feasible? For example, can 1d dynamics be used to support or refute the hypothesis that grid cells form a 2d attractor network? Combined with some new modeling work, our Fourier method, along with the view of 1d fields as slices, can be used to make and test new predictions which distinguish between different classes of network models, and hopefully yield new insights into the explanation of grid cell behavior.