The grid covering the 2D space occupied by the track is binned into squares that are 3 cm × 3 cm. The red line illustrates how we sequenced the bins in the planar and linear reference frames, with m = 1 at the top right square, m = 2 the square below it, etc. The activity vector ui or uj is the firing rate of a place cell given a particular track configuration (i on the left, j on the right, respectively) in the 2D frame of reference and is constituted by the firing rates of the cell in particular bins (ui1, ui2, etc). In the example in panel A, the same spatial bin produces vector components ui,m (in configuration i) and uj,m (in track configuration j). We calculate the correlation coefficient, Ci,j, to understand the relationship between two activity vectors for a given place cell firing in two different track configurations (for the sake of simplicity, we depict only a change of position in the top arm, from i to j; we can do this safely because the place cells we recorded did not have multiple place fields). Note that any given configuration can be arrived at from a number of movements: the top arm could move to position 2 from vertical sections 1 or 3, 7, or 10 (see Figure 1D), so the correlation coefficient for a cell that has a place field on the top arm, Ci,j would be denoted by C1,2, C3,2, C7,2, or C10,2, respectively. As explained in Figure 1D, the greater the numeric difference between i and j, the greater the geometric difference between track configurations 1 and 2. The correlation matrix in the middle graphically depicts correlation coefficients by color. In panel B we show activity vectors vi, vj for linearized trajectories of the animal along the same configurations i and j depicted in 2D in panel A.