Shifting the phase vector by 
 xmod (

) should produce a shift by x in where the rat thinks it is.(Where the rat thinks it is could be assayed easily by whether therat freezes or displays other unconditioned aversive responses, ifthe shifted location corresponds to where the rat has receivedmultiple foot shocks or some other aversive stimulus.) The ex-periment to shift the dMEC phases requires activation of a spe-cific phase vector in dMEC

Suppose that physiology experiments locate neurons in therodent brain whose responses are explicitly metric functions ofposition over large distances and are independent of the takenpath. Such neurons, if they exist, are likely to be the explicitmetric readouts of the dMEC phase code. The existence of theseneurons is likely to be predictive of behavioral abilities in rats toestimate the distance or angle toward home over large excursions

Recent work has shown that thiscoherence between grid cells is maintained only within and not between grid modules (Stensolaet al. 2012)

The increase in the relative proportion of small place fields over time (Fig. 7 and Extended Data Fig. 7d) may also be achieved as a result of the overall decrease of firing rate in CA1 with familiarity41,43.

We found that the size of the hyperbolic representation was larger when the animal was more familiar with the environment, using the above topological method. The size increased with the logarithm of time that the animal had to explore it

Therefore, higher speed and, thus, less familiarity with the nearby environment produced larger place fields.

One can quantify the additional information provided by the changes in trajectory of the track by computing the probability distribution of the change in angle δθ between segments and using the standard formula for entropy31. With this definition, straight segments have zero additional entropy, with larger values for more curved and varied segments. We found that animal speed was inversely related to the additional information (Fig. 3f), with lower speed in more informative segments.

. We present analytical methods and numerical refinements for extracting the best slice and grid parameters from the Fourier transform of a general 1D spatial tuning curve. We then apply these methods to their 1D responses of grid cells with stable 1D and 2D fields and show that the resulting slices yield excellent fits to their 1D spatial responses.