Grid cells in rat medial entorhinal cortex discharge when the animal moves through particular regions of the external world. For each grid cell, these regions form a hexagonal lattice whose spatial phase, orientation and scale vary from cell to cell. As the animal crosses one of these firing fields, spikes tend to occur at successively earlier theta phases of the local field potential, a phenomenon called phase precession. To better understand these different salient features of grid cells, we re-analyzed published data from Hafting et al. (Nature 2008) and Sargolini et al. (Science 2006). We find, in particular, that spike phases provide about twice as much spatial information as firing rates, phase precession is stronger at the single-run level than suggested by pooled data, and that the multiple firing fields of a grid cell operate as independent encoders of physical space.

Using mathematical theory we prove that optimizing spatial resolution predicts that grid scales are discrete with constant scaling ratio, that the spatial resolution of such grid codes dwarfs place codes, and that optimal grids for isotropic space are hexagonal in 2D and face-centered-cubic in 3D. These results provide an explanation for why grid codes exist and make quantitative predictions that can be tested experimentally and compared with the predictions of alternative theoretical frameworks. Finally, I will discuss how a simple population vector of grid cell activity across multiple scales can be used for navigation.

These results have been obtain in collaboration with Alexander Mathis (Harvard), Johannes Nagele and Martin Stemmler (Bernstein Center Munich), and Eric Reifenstein, Richard Kempter and Susanne Schreiber (Bernstein Center Berlin). Some of the findings have been published previously; see, e.g., E Reifenstein et al. PNAS 2012, A Mathis et al. Neural Comp 2012 and PRL 2012. We thank Edvard Moser for making data publicly available.