Supplementary Figure 15: Increasing grid spacing reduces long-term place stability in a winner-take-all model of grid-to-place cell formation.

(a) In addition to varying the number of grid and unstable spatial inputs received by each place cell (i.e. “conditions” in Fig. 4), we also varied the scale of the unstable spatial inputs (small: 1000–3000 cm², in the main text and large: 3000–5000 cm², shown here). Fig. 4 presents results from simulations using the smallest scale of unstable spatial inputs. Far left: average place field size increased with the spacing of the smallest grid module in all simulation conditions and larger scales of unstable spatial input. Each line depicts the mean ± SEM for 10 iterations of the simulation. The color of the line represents the condition. For statistical analysis, for each of the 9 conditions we computed the Pearson’s correlation between the spacing of the smallest grid and the mean field size for all iterations (all r(78) > 0.67, p < 8.2e-12). Middle left: the correlation coefficients for place maps across days declined with increasing grid scale in all 9 conditions (the average of 10 iterations per condition is indicated by a different colored line), all r(78) < −0.60, p < 2.1e-09 (Pearson’s correlations). Middle right: for each of the 9 simulation conditions (colored dots), the p-value for the Pearson’s correlation between the spacing of the smallest grid module and the correlation coefficient for place maps across days is shown. The dashed line represents p = 0.001. Far right: the slope between the spacing of the smallest grid module and the correlation coefficient for place maps across days is plotted for each of the 9 simulation conditions (colored dots). Mean ± SEM of 10 iterations is shown for each condition. (b) In the model, increasing grid scale will change both the spacing (distance between grid nodes) and grid field size. To dissociate the separate effects of grid spacing and field size on place cell re-mapping, we implemented a winner-take-all model in which field size was held constant while spacing was varied. For this simulation, we set the radius of all fields for grid cells to be 15 cm, and varied the spacing of the grid inputs to the network in the same manner as our other winner-take-all simulations. In these ‘fixed field size’ simulations, we still see a decrease in place cell stability as grid spacing increases (r(78) = −0.88, p = 5.54e-27, Pearson’s correlation, n = 10 iterations, 8 grid scales). Plot shows mean ± SEM from 10 iterations. (c) Schematic illustrating how grid spacing affects the peak value of the summed activity of multiple grid inputs across space. For simplicity, the cartoon depicts one-dimensional slices through a two-dimensional grid cell map. The phases of each grid cell were selected randomly. Inputs from a small grid module (spacing = 30 cm) and those from a larger module (spacing = 60 cm) are shown on the bottom left and right, respectively. Field sizes are drawn to reflect the experimentally observed relationship between grid spacing and field size. The summed input from each set of grid cell inputs is shown above in gray, with the peak values indicated for the small (blue) and large-scale (red) grid inputs indicated by the dashed lines. (d) Fig. 4f shows that the peak grid input declines with increasing grid scale. Here, we performed a simulation to show that this results from reduced overlap of grid fields across physical space. In each simulation, 600 grid cells were randomly selected from a pool of 3,000 grid cells evenly spanning 5 modules, between which grid spacing increased by a factor of the square root of two. The environment was binned evenly, and the number of grid field centers (region with 80% of the peak firing rate) overlapping in each of the 30 bins counted for each place cell. The entire process was repeated a total of 2000 times each (for 2000 place cells) for 8 different scales of grid input, in which the size of the smallest module increased from 25 cm (black) to 60 cm (red). The average amount of grid field overlap decreased significantly as spacing of the grid input increased (r(15998) = −0.32, p << 0.0001; Pearson’s correlation). The figure shows a histogram (averaged across place cells) plotting the probability of field overlap for different scales of grid input.