Medial entorhinal cortex (MEC) grid cells fire at regular spatial intervals and project to the hippocampus, where place cells are active in spatially restricted locations. One feature of the grid population is the increase in grid spatial scale along the dorsal–ventral MEC axis. However, the difficulty in perturbing grid scale without impacting the properties of other functionally defined MEC cell types has obscured how grid scale influences hippocampal coding and spatial memory. Here we use a targeted viral approach to knock out HCN1 channels selectively in MEC, causing the grid scale to expand while leaving other MEC spatial and velocity signals intact. Grid scale expansion resulted in place scale expansion in fields located far from environmental boundaries, reduced long-term place field stability and impaired spatial learning. These observations, combined with simulations of a grid-to-place cell model and position decoding of place cells, illuminate how grid scale impacts place coding and spatial memory.
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We thank J. Dickinson, A.S. Henderson, K. Muench, N.L. Saw and L. Willmore for assistance in gathering behavioral data and A. Borrayo and A.S. Henderson for histology assistance. L.M.G. is a New York Stem Cell Foundation – Robertson Investigator. This work was supported by funding from The New York Stem Cell Foundation, Whitehall Foundation, NIMH grant MH106475, the Simons Foundation, the James S McDonnell Foundation, a Klingenstein-Simons Fellowship awarded to L.M.G. and an NSF Graduate Research Fellowship awarded to C.S.M.
Integrated supplementary information
Histology for MEC-implanted mice. Injected wildtype mice (iWT) shown at top, injected floxed HCN1 (iCre-KO) mice shown at bottom. The mouse ID is shown above each section. For each mouse, left columns: Nissl stained sagittal sections. Black arrow marks the deepest tetrode location. An area was considered infected if the mean fluorescent intensity was greater than two standard deviations (SD) above the mean background intensity. The approximate area of virus infection in MEC is outlined in green. This area was estimated from multiple GFP images covering ± 160 µm from the slice containing the final tetrode location. In the same slices, any GFP detected in the hippocampus was additionally quantified using Image-Pro Plus. A small amount of GFP expression was detected in CA1 of three iWT (mice #34, #48, #79) and one iCre-KO mouse (mice #18) (portion of CA1 infected: iWT = 2–5%, iCre-KO = 3%). Three iWT mice (mice #34, #47, #81) and two iCre-KO (mice #70, #82) mice had minor infection of the dentate gyrus, where HCN1 is only lowly expressed1,2 (portion of dentate gyrus infected: iWT = 12–30%, iCre-KO= 6–11%). For each mouse, right columns: Close up of GFP expression in the area near the final tetrode location (indicated by the white arrowheads). 1. Seo, H., Seol, M.J. & Lee, K. Differential expression of hyperpolarization-activated cyclic nucleotide-gated channel subunits during hippocampal development in the mouse. Molecular brain 8, 13 (2015). 2. Santoro, B., et al. Molecular and functional heterogeneity of hyperpolarization-activated pacemaker channels in the mouse CNS. J Neurosci 20, 5264–5275 (2000).
iWT mice shown on right, iCre-KO mice shown on left. Mouse ID is shown at the top of each brain slice. All scale bars are 1 mm. a) Nissl stained sagittal sections of implant site. Black arrow marks the recording location of CA1 cells. While in some cases tetrodes were lowered below CA1, only cells from CA1 were included in analysis. The morphology of the hippocampus of mouse #106 was distorted during brain extraction, however the site of the lesion was still evident. (b) Overlaid images of DAPI and GFP for the slice containing the tetrode track. Importantly, GFP was not detected in the hippocampus in these slices. Images of DAPI and GFP were taken separately and then combined in Photoshop. The intensities of the blue and green channels were adjusted separately to allow for clear visualization of the GFP expression. White arrowheads denote the end of the tetrode track. (c-d). GFP expression in the left (c) and right/implanted (d) hemispheres. White lines denote the boundary between MEC and neighboring brain structures. In two mice (iCre-KO, #99 and #118) GFP was not detectable in the left/non-implanted hemisphere. GFP expression was greatest in and largely restricted to the MEC. In a few mice a small amount of GFP leaked into the neighboring subiculum. Any GFP observed in hippocampus was further quantified using Image-Pro Plus software. Two iCre-KO mice had minor GFP expression in CA1 (#99: infected area <<1 % right CA1, ~15 cells; #144: infected area <2 % left CA1, <<1 % right CA1). GFP was also detected in the CA1 of two iWT mouse (#100: infected area = ~1 % of right CA1; #119: infected area <4 % left CA1). Two iCre-KO mice had GFP expression in a small portion of dentate gyrus (#87: infected area = ~7 % of bilateral dentate gyrus; #118: infected area <1 % of left dentate gyrus), where HCN1 expression is very low1,2. (e) There was no difference in the mean or maximum field size of place cells recorded in iCre-KO mice with or without minor spread to hippocampus [HPC] (n = 59 place cells from 4 mice with perfect virus restriction, n = 67 place cells from 4 mice with minor virus spread; mean ± SD; mean field size without spread = 455.28 ± 353.99 cm2, with spread = 424.13 ± 323.36 cm2, two-tailed WRS Z = 0.32, p = 0.75; max field size without spread: 556.99 ± 377.99 cm2, with spread: 495.71 ± 329.28 cm2, two-tailed WRS Z = 0.79, p = 0.43). Box depicts the first and third quartiles. The median is depicted by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range (plotted separately). (f) Exclusion of all mice with any hippocampal spread did not affect the primary place cell findings (virus perfectly restricted to MEC: n = 64 cells from 9 iWT mice, 59 cells from 4 iCre-KO mice). (g) boxplot of mean and maximum place field sizes (mean ± SD; mean place field size, iWT = 259.76 ± 186.94 cm2, iCre-KO = 455.28 ± 353.99 cm2, two-tailed WRS Z = −3.66, p = 2.5e-4; max place field size, iWT = 342.19 ± 200.11 cm2, iCre-KO = 556.99 ± 377.99 cm2, two-tailed WRS Z = −3.50, p = 4.6e-4). Box depicts the first and third quartiles. The median is depicted by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range (plotted separately). (h) Scatterplot of distance from the nearest boundary versus max field size (F(3,123) = 7.50, p = 0.0071, ANCOVA). Blue and red lines depict the line of best linear fit for iWT and iCre-KO place cells, respectively. (i) Boxplot showing the difference in size between fields located in the center (> 14 cm; n = 11 iWT cells, 16 iCre-KO cells) versus the boundaries (< 14 cm; n = 53 iWT cells, 43 iCre-KO) of the environment, which was greater for iCre-KO (red) place cells compared to iWT (blue) (two-tailed t-test on bootstrapped distributions of the mean difference, t(19998) = 141.21, p << 0.0001). Box depicts the first and third quartiles. The median is depicted by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range (plotted separately). ***p<0.0001.
Supplementary Figure 3 Quantification of virus expression for mice performing the DMP and behavioral analysis.
(a) Flat maps indicating the extent of virus expression across the MEC in iWT and iCre-KO mice undergoing behavioral tests. The mouse number is indicated at top of each plot. Left and right hemispheres are shown on the left and right, respectively. Sections are arranged from lateral MEC (left) to medial MEC (right). Each map was constructed by analyzing the GFP expression of six 40 μm-thick brain slices, selected at evenly-spaced intervals spanning 960 μm across the medial–lateral MEC axis. For each slice the area of virus expression in MEC was calculated using Image Pro Plus software. A region was considered infected if the average fluorescence intensity was greater than two standard deviations above background. After identifying an infected region, the dorsal–ventral range of expression was determined. As HCN1 reduction in the hippocampus has been linked to improved performance on the Morris water maze3, upon histological examination, mice with any virus leakage into the hippocampus (n = 4 iWT, 5 iCre-KO mice) were excluded from all behavioral analyses and not considered further. (b) The percentage of MEC infected did not differ between iWT and iCre-KO mice (mean ± SD; iWT = 52.7 ± 24.2 %, iCre-KO = 59.4 ± 23.9 %; two-tailed t-test, t(19) = 0.54, p = 0.51; n = 8 iWT mice, 13 iCre-KO mice). Box depicts the first and third quartiles. The median is depicted by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range (plotted separately). Individual data points are overlaid. (c) There was no correlation between the amount of virus expression and distance savings on the DMP (iWT: r(6) = −0.11, p = 0.79; iCre-KO: r(11) = 0.24, p = 0.43, Pearson’s correlation; n = 8 iWT mice, 13 iCre-KO mice). Dash line indicates zero savings. (d) Latency to find the hidden platform during the standard water maze, in which the location of the platform was held constant across days, did not differ between iWT and iCre-KO mice (main effect of group: F(1,19) = 0.050, p = 0.83, group × day interaction: F(4,76) = 0.40, p = 0.81; repeated measures ANOVA with day as a within subject factor and group as a between subject factor; n = 8 iWT mice, 13 iCre-KO mice). (e) Rapid place learning deficits were most pronounced for distance traveled on the delayed-match-to-place (DMP) task. Similar trends were observed for latency, (effect of trial on latency: F(3,57) = 3.51, p = 0.021, group × trial interaction: F(3,57) = 1.61, p = 0.20; repeated measures ANOVA with trial as a within subject factor and group as a between subject factor), however the tendency of iWT mice to swim more slowly on later trials potentially obfuscated further improvements in latency to locate the platform (group × trial interaction on velocity: (F(3,57) = 4.30, p = 0.0084, effect of trial on velocity: iWT F(3,21) = 11.74, p = 9.9e-5, iCre-KO F(3,36) = 0.21, p = 0.89; n = 8 iWT mice, 13 iCre-KO mice; see Fig. 6j). (f) Though the effects were greater for distance traveled, the average savings for latency to find the platform (difference between late and early trials) was significantly larger in iWT (blue) compared to iCre-KO (red) mice (mean ± SEM; iWT = 7.16 ± 1.84, iCre-KO = 1.39 ± 1.73 s, t(19) = 2.19, p = 0.041, unpaired two-tailed t-test; n = 8 iWT mice, 13 iCre-KO mice). (g) The time taken to find the platform on the cued version of the DMP task did not differ between iWT (blue) and iCre-KO (red) mice (main effect of group: F(1,19) = 0.20, p = 0.66, group × trial interaction: F(3,57) = 1.56, p = 0.21, repeated measures ANOVA with trial as within subject factor and group as between subject factor; n = 8 iWT mice, 13 iCre-KO mice). (h) While the loss of hippocampal HCN1 channels results in anxiolytic behaviors4 we found similar levels of anxiety in iWT (n = 8) and iCre-KO (n = 9) mice, as measured by time spent in the open portion of an arena (mean ± SD; iWT = 32.91 ± 15.67, iCre-KO = 42.28 ± 28.61 s; t(16) = 0.82, p = 0.42, unpaired two-tailed t-test). Data are expressed as mean (bar height) ± SEM (solid line), with all individual data points overlaid. (i-j) The total number of entries (i) and percent alternation (j) did not differ between groups on a Y-maze test of working memory (mean ± SD; entries: iWT = 35.13 ± 11.61, iCre-KO = 40.46 ± 15.61, t(19) = 0.83, p = 0.42; alternation: iWT = 58.91 ± 11.75 %, iCre-KO = 58.52 ± 6.45 %, t(19) = 0.099, p = 0.92; unpaired two-tailed t-tests; n = 8 iWT mice, 13 iCre-KO mice). The dashed line represents chance. All plots in (d–j) show the mean ± SEM with individual data points from each mouse overlaid. 3. Nolan, M.F., et al. A behavioral role for dendritic integration: HCN1 channels constrain spatial memory and plasticity at inputs to distal dendrites of CA1 pyramidal neurons. Cell 119, 719–732 (2004). 4. Kim, C.S., Chang, P.Y. & Johnston, D. Enhancement of dorsal hippocampal activity by knockdown by HCN1 channels leads to anxiolytic- and antidepressent-like behaviors. Neuron, 503–516 (2012).
Histological analysis of 4 hemispheres revealed only minor leakage of virus into the LEC (5.0 ± 3.5% of LEC). Images of DAPI and GFP were taken separately and then combined in Photoshop. The intensities of the blue and green channels were adjusted separately to allow for clear visualization of the GFP expression. (a) Sagittal brain sections showing virus expression across the medial–lateral axis (left to right: lateral to medial). Images of Cre-GFP (green) are overlaid on images of DAPI (blue). The white dotted lines depict the dorsal MEC border (top), the MEC–LEC border (middle), and the ventral LEC border (bottom). Each row shows slices from a different mouse (n = 3). (b) Horizontal brain sections from an individual mouse showing virus expression. Dorsal to ventral sections are shown from left to right. The white dotted lines depict the anterior LEC border (left), MEC-LEC border (middle), and MEC-parasubiculum border (right). Scale bars: 1 mm.
Supplementary Figure 5 Whole-cell patch-clamp recordings in layer II MEC stellate cells confirm that Cre-mediated knockout of HCN1 reduced Ih.
Whole-cell recordings from n = 13 cells from 8 iWT mice and 13 cells from 8 iCre-KO mice. (a) Image of a representative virus-infected stellate neuron targeted for whole-cell patch clamp recording. Top: GFP expression is visible in the MEC; scale bar: 1 mm. Bottom: higher magnification view; scale bar: 0.1 mm. A red arrow marks the recorded neuron. (b-c) In MEC, the absence of HCN1 leads to a higher input resistance (Rin) and a more hyperpolarized resting membrane potential5. Consistent with a decrease in I(h), infected neurons in iCre-KO mice had a significantly higher input resistance (b) and lower resting membrane potential (c) compared to infected neurons in iWT mice (mean ± standard deviation [SD]; input resistance: iWT = 47.02 ± 15.81 mΩ, iCre-KO = 69.05 ± 17.89 mΩ, unpaired two-tailed t-test, t(24) = 3.33, p = 0.0028; resting membrane potential iWT = −64.23 ± 2.35 mV, iCre-KO = −69.23 ± 7.67 mV, t(24) = 2.25, p = 0.041). Colored bars indicate the mean ± SEM and individual data points are shown in black. *p<0.05, **p<0.01, ***p<0.001. (d) We held the membrane potential at −70 mV and applied 1 s long hyperpolarizing current steps that, in the presence of I(h), result in a slow depolarizing shift in the membrane potential (sag)6. Examples of sag potential from iWT (blue) and iCre-KO (red) infected stellate neurons (top) in response to current injections (bottom; black). (e) Cre-mediated knockout of HCN1 significantly decreased the sag ratio (mean ± SD; iWT = 1.44 ± 0.41, iCre-KO = 1.13 ± 0.07, unpaired two-tailed t-test t(24) = 7.17, p = 2.1e-7). Colored bars indicate the mean ± SEM and individual data points are shown in black. *p<0.05, **p<0.01, ***p<0.001. 5. Nolan, M.F., Dudman, J.T., Dodson, P.D. & Santoro, B. HCN1 channels control resting and active integrative properties of stellate cells from layer II of the entorhinal cortex. J Neurosci 27, 12440–12451 (2007). 6. Dickson, C.T., et al. Properties and role of I(h) in the pacing of subthreshold oscillations in entorhinal cortex layer II neurons. J Neurophysiol 83, 2562–2579 (2000).
(a-b) Shuffled thresholds for all cell types in MEC (a) and hippocampus (b). Top rows show the distribution for values observed (n = 527 iWT MEC cells, 600 iCre-KO MEC cells, n = 323 iWT hippocampus cells, 303 iCre-KO hippocampus cells). Bottoms rows show the distribution of scores generated from randomly shuffled datasets (n = 52,700 iWT MEC shuffles, 60,000 iCre-KO MEC shuffles, 32,300 iWT hippocampus shuffles, 30,300 iCre-KO hippocampus shuffles [100 shuffles per cell]). Green lines indicate the 95th percentile significance level for each shuffled distribution.
(a) Examples of grid cells recorded in iWT and iCre-KO mice. Rate maps (top rows) and autocorrelation maps (bottom rows) are color coded for minimum (blue) and maximum (red) values. Peak firing rates (left) and grid spacing (right) are marked at the top of each plot. (b-c) Scatterplot demonstrating grid spacing (b) and grid field size (c) in iWT and iCre-KO mice along the dorsal–ventral (DV) MEC axis (n = 37 iWT cells, 62 iCre-KO cells). As determined by an analysis of covariance, grid spacing in both groups increased along the dorsal-ventral axis (F(1,96) = 4.94, p = 0.029). The Y intercept in iCre-KO mice was significantly higher compared to iWT, reflecting a ~6 cm shift in grid spacing (F(1, 96) = 10.19, p = 0.0019, η2 = 0.096; Y intercept iWT = 31.97 cm, iCre-KO = 37.48 cm). This expansion in iCre-KO grid spacing occurred across the DV extent examined, as the slope of the grid spacing gradient was not different between groups (F(1, 95) = 0.079, p = 0.80, η2 = 0.001). The increase in grid field size across the dorsal-ventral axis did not reach significance (F(1, 95) = 2.34, p = 0.13, η2 = 0.024), however iCre-KO grid cells fields were significantly larger than those of iWT (two-tailed WRS Z = 2.20, p = 0.028, two sample KS D = 0.29, p = 0.035), and in both groups grid spacing and grid field size were highly correlated (iWT r(35) = 0.79, p = 8.6e-9, iCre-KO r(60) = 0.76, p = 9.5e-13, Pearson’s correlation). Blue and red lines depict the lines of best linear fit for iWT and iCre-KO grid cells, respectively. (d) Flat maps showing the MEC dorsal–ventral and medial–lateral position of each grid cell recorded in iWT (left; n = 37 cells) and iCre-KO (right; n = 62 cells) mice. Each gray bar represents a single medial–lateral position within MEC (spaced at 200 µm), and bars are organized from most lateral (left) to most medial (right). The position of the grid cell along the dorsal–ventral axis is shown as distance from the dorsal entorhinal border. Neither the dorsal–ventral nor the medial–lateral positions differed significantly between the two groups (mean ± SD; dorsal–ventral: iWT = 386 ± 222 µm, iCre-KO = 327 ± 148 µm, two-tailed WRS Z = 1.67, p = 0.095; medial–lateral (distance from midline): iWT = 3.05 ± 0.26 mm, iCre-KO = 2.97 ± 0.18 mm, two-tailed WRS Z = 1.65, p = 0.10). (e) For each individual iWT (n = 11; blue) or iCre-KO (n = 15; red) mouse, the grid cell recorded with smallest spacing is plotted. Black bars indicate the medians. The smallest grid cells recorded in each mouse were significantly larger in the iCre-KO mice compared to the iWT mice (median: iWT = 26.6 cm, iCre-KO = 36.7 cm; two-tailed WRS, Z = −2.08, p = 0.038). *p<0.05. (f). Top: an estimate of the probability mass function of the iWT grid spacing, based on the observed data and smoothed using a kernel density function (blue), with the observed iCre-KO grid spacing data overlaid (red). Bottom: The likelihood that the iCre-KO data comes from the iWT distribution, LLH KO , is computed, where ), where is the spacing of grid cell , is the probability of observing a cell with a given spacing according to the iWT distribution, and N is the number of iCre-KO grid cells that fall within the iWT distribution (red dashed line). This value is then compared a null distribution of LLH values, which were computed in the same way as LLH KO , but with N randomly sampled iWT values (n = 10,000 shuffles). Using this approach, we found it highly unlikely that the iCre-KO data was generated from a similar distribution as the iCre-WT (p = 0.00010, meaning that the observed iCre-KO was significantly different from what would be expected if the values were sampled from the iWT distribution).
Supplementary Figure 8 The impact of HCN1 loss on spatially modulated cells is restricted to grid cells.
P-values are provided in Table 1. (a) Examples of border cells. Rate maps (top) are coded from maximum firing rates (red) to minimum firing rates (blue). Peak firing rate (left) and border width (right) are indicated at the top of each plot. (b) The width of border cells was not significantly different between iWT (blue) and iCre-KO (red) mice. Inset: Schematic showing border cell width calculation. Left: rate map of a border cell with black dots indicating the center of mass of each detected firing field. Right: fields detected from the rate map are shown in black. The width of each field was first calculated by averaging the distance at each point along the field’s inner edge from the wall associated with that field (the wall over which the field spanned the greatest length). A border cell’s width was then defined as the average width of all of its fields, depicted in this schematic as red lines. (c) The firing rates of iWT and iCre-KO border cells similarly decrease as distance from the nearest arena wall increases. For each border cell (n = 86 iWT cells, n = 71 iCre-KO cells), the average firing rate was calculated at increasing distances from the nearest arena wall (for example, the average firing rate was calculated for all bins located 1 cm away from the nearest wall, then 3 cm away from the nearest wall, then 5 cm away from the nearest wall, and so on). For comparison across cells, firing rates were normalized by dividing the firing rate at each distance by the peak rate at any distance. As expected for border cells, firing rates decrease with distance from the nearest wall (repeated measures ANOVA; main effect of distance: F(17, 2635) = 279.16, p = 1.0e-13). However, there was no main effect of group (F(1,155) = 0.029, p = 0.87), nor was there an interaction between group and distance from the nearest arena wall (F(17,2635) = 0.31, p = 0.81). Plot shows the mean and SEM for all iWT (blue) or iCre-KO (red) cells. (d) Examples of spatially stable cells. Black dots indicate firing fields. Peak firing rate and percentage of the environment covered by a firing field are indicate at the top left and right, respectively. (e) The size of the average field is similar between spatially stable cells recorded in iWT and iCre-KO mice. (f) The percentage of the environment covered by a field is similar between spatially stable cells recorded in iWT and iCre-KO mice. (g) Examples of interneurons recorded in iWT and iCre-KO mice. Color coding of rate maps as in (a). Peak firing rate is indicated at top. Right: Spike time autocorrelation diagrams. (h) Box plot showing that the interspike-interval latency of theta-modulated interneurons (n = 13 iWT cells, 16 iCre-KO cells) does not differ between iWT and iCre-KO. Box shows first and third quartiles, and whiskers show the maximum and minimum values without outliers (plotted separately). A solid line indicates the median. (i) Box plot (depicted as in [h]) showing that bursting frequency of interneurons (n = 15 iWT cells, 22 iCre-KO cells) does not differ between iWT and iCre-KO.
Data from iWT are shown in blue, iCre-KO in red. Boxplots indicate the first and third quartiles by a box, and the median by a solid black line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are indicated by plus symbols. Individual data points are plotted at the left of each plot. (a-b) Examples of layer II/III (a) and layer V (b) head direction cells in iWT and iCre-KO mice. Rate maps (top) are coded from maximum firing rates (red) to minimum firing rates (blue). The polar plot (bottom) indicates firing rate as a function of head direction. Maximum firing rates (left) and tuning width (right) are indicated at the top of each plot. (c-d) Cumulative frequency plots showing that the tuning width of head direction cells in layer II/III (c) and layer V (d) are not significantly different between iWT (blue) and iCre-KO (red) mice. (e) Examples of tuning curves in iWT and iCre-KO speed cells. (f-g) The score (f) and slope between firing rate and running speed (g) of speed cells do not differ between groups. (h-i) Examples of MEC (h) and hippocampal (i) LFP theta frequency relative to running speed in four individual animals. (j-k) Theta frequency does not differ significantly between iWT and iCre-KO mice in either MEC (j) or hippocampus (k). (l-m) The slope of running speed modulation is not significantly different between the groups in either MEC (l) or hippocampus (m). (n-o) Theta power is not significantly different between the groups in either MEC (n) or hippocampus (o). (p-q) Neither fast nor slow gamma power differ significantly between the groups in MEC (p) or hippocampus (q). For j-q, n = 12 iWT and 16 Cre-KO mice with tetrodes in layer II/III MEC, and n = 11 iWT and 8 iCre-KO mice with tetrodes in hippocampus.
Supplementary Figure 10 Capturing cells that encode position (P), head direction (H) or speed (S) in medial entorhinal cortex using an LN model framework.
The majority of cells classified as grid, head direction, or speed cells according to shuffled criteria were found to significantly encode either P, H, or S, respectively, using the LN model. Furthermore, the percentage of each MEC cell type captured by both methods did not differ between iWT and iCre-KO mice (P encoding: grid cells in iWT n = 33/37, iCre-KO 60/62, Z = −1.53, p = 0.13; border cells in iWT = 77/86, iCre-KO = 66/71, Z = −0.75, p = 0.45; H encoding: head direction cells in iWT = 208/225, iCre-KO = 229/240, Z = −1.35, p = 0.18; S encoding: speed cells in iWT = 52/69, iCre-KO = 69/89, Z = −0.32, p = 0.75; all two-tailed binomial tests). (a-b) Response profiles for cells that significantly encode position (a) and speed (b). Examples on the left show individual MEC cells that were classified as a grid, border (a, two leftmost examples) or speed (b, two leftmost examples) cell using the shuffled criterion. Examples on the right show cells that significantly encode position (a) or speed (b) but have tuning curve shapes that, while unconventional, still carry significant information regarding the position or running speed of the animal. Note that response profiles are model derived but qualitatively similar to tuning curves and plotted using similar units and were computed as in Hardcastle et al., 20177. (c) Histogram illustrating the fraction of neurons identified as significantly encoding one or more variable using the LN model approach (PHS: iWT = 33.4%, iCre-KO = 36.2%, PH: iWT = 32.1%, iCre-KO = 35.3%, PS: iWT = 8.7%, iCre-KO = 7.3%, HS: iWT = 0.2%, iCre-KO = 0.2%, P: iWT = 13.9%, iCre-KO = 11.8%, H: iWT = 0.2%, iCre-KO = 0.2%, S: iWT = 0.8%, iCre-KO = 0.3%). Similar to what was reported in Hardcastle et al., 20177, a large portion of superficial MEC neurons encode more than one navigational variable (exhibit mixed-selectivity) and, as a population, MEC neurons tend to encode position at a higher frequency than head direction or speed. (d) Boxplots of model fit for cells that significantly encoded one or more variables. Boxplots indicate the first and third quartiles by a box, and the median by a solid black line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are indicated by plus symbols. In essence, model fit indicates how well the spiking of a single-cell can be predicted by the variable, or set of variables, of interest (e.g. position or position and head direction). We did not find any significant differences in model fit for any cell-type (Model fit mean ± SD: PHS iWT = 0.34 ± 0.34, n = 176, iCre-KO = 0.31 ± 0.32, n = 217, two-tailed WRS Z = 0.51, p = 0.61; PS iWT = 0.30 ± 0.30, n = 169, iCre-KO = 0.31 ± 0.31, n = 212, two-tailed WRS Z = 0.29, p = 0.77; PHS iWT = 0.12 ± 0.13, n = 46, iCre-KO = 0.13 ± 0.13, n = 44, Z = −0.38, p = 0.70; P iWT = 0.10 ± 0.13, n = 73, iCre-KO = 0.083 ± 0.096, n = 71, Z = 0.31, p = 0.76; H iWT = 0.043 ± 0.027, n = 5, iCre-KO = 0.41 ± 0.55, n = 4, t(7) = −1.55, p = 0.17; S iWT = 0.050 ± 0.049, n = 4, iCre-KO = 0.023 ± 0.010, n = 2, t(4) = 0.72, p = 0.51). Boxplots are not shown for HS cells, as the LN model detected significant HS encoding in one 1 iWT and 1 iCre-KO neuron (Model fit iWT = 0.018, iCre-KO = 0.020). (e) The majority of grid cells identified using shuffled criterion also significantly encoded P but a few (n = 4 iWT grid cells and 2 iCre-KO grid cells) only passed based on the shuffled criterion. Even so, we found a significant increase in spacing in iCre-KO mice when only considering the grid cells that passed the shuffled criterion and significantly encoded position (mean ± SD; iWT = 37.24 ± 9.30 cm, iCre-KO = 42.31 ± 10.67 cm, two-tailed WRS Z = −2.45, p = 0.014; n = 33 iWT grid cells and 60 iCre-KO grid cells). 7. Hardcastle, K., Maheswaranathan, N., Ganguli, S. & Giocomo, L.M. A multiplexed, heterogeneous, and adaptive code for navigation in medial entorhinal cortex. Neuron 94, 375–387 (2017).
(a) Field size on the linear track was larger in iCre-KO mice (mean ± SD; iWT = 22.98 ± 11.56, iCre-KO = 30.39 ± 14.92, two-tailed WRS Z = −2.26, p = 0.023; n = 42 fields from 30 cells from 3 iWT mice, 38 fields from 28 cells from 4 iCre-KO mice). *p<0.05. (b) As an alternative approach, we also estimated field width from the population vector cross-correlation matrices for iWT and iCre-KO place cells (n = 30 iWT, 28 iCre-KO place cells)8, 9. Decorrelation curves, at right, show the mean correlation ± SEM for all possible population vector pair distances between 0 and 15 bins (2.5 cm each). Field width was estimated as the mean distance from the diagonal matrix to the population vectors with a mean correlation of 0.2 (iWT = 28.8 ± 8.1 cm, iCre-KO = 34.2 ± 8.5 cm, two-tailed WRS Z = −2.12, p = 0.034) (c) Histograms of the slopes of regression lines fit between normalized field position and spike phase in iWT (left, n = 26) and iCre-KO (right, n = 23) fields. Filled bars represent significant slopes (p < 0.05). The gray region delimits negative slopes. There was no difference in the proportion of fields with negative slopes (iWT = 17/26, 0.65, iCre-KO = 18/23, 0.78, two-tailed binomial test, Z = −1.00, p = 0.32) or with significant slopes (iWT = 6/26, 0.23, iCre-KO = 6/23, 0.26, two tailed binomial test, Z = −0.24, p = 0.81). Insets: mean place-phase firing maps for all fields with negative slopes. Warm colors indicate the position-phase bins with the highest instantaneous firing rates, cool colors indicate those with lowest instantaneous firing rates. In no bin did the difference between the two group- averaged maps exceed the 95% confidence interval (see Methods). (d) Boxplot for position-phase slopes for all iWT and iCre-KO fields (open boxes; n = 26 iWT fields, 23 iCre-KO fields), and significant slopes (filled boxes; n = 6 iWT fields, 6 iCre-KO fields). Individual data points for significant slopes are overlaid (black circles). The slopes did not differ between iWT and iCre-KO mice (mean ± SD; significant slopes: iWT = −270.61 ± 298.48°/field, iCre-KO = −321.56 ± 236.45°/field, two-tailed WRS p = 0.82; all slopes: iWT = −116.86 ± 345.60°/field, iCre-KO = −190.19 ± 254.16°/field, two-tailed WRS Z = 0.51, p = 0.61). (e) Individual examples of iWT (left) and iCre-KO (right) fields displaying significant phase precession. Left: Theta phase and location of spikes emitted while the mouse was running through the place field indicated at right. The peak of the theta oscillation is set to 360 degrees, and corresponds to the most positive portion of the cycle recorded in the EEG of the CA1 pyramidal layer. The phase of each spike is plotted in a second cycle to aid visualization. Right: the place field recorded on the linear track. Histogram of firing rates across the track, with green lines indicating the boundaries of the place field. The direction of travel for the field is indicated by the arrow. A 2D heat map is shown below, with unexplored bins depicted in black. The corresponding place field recorded in the open field arena is shown at top, with black dots indicating the center of mass of each field detected. For all heat maps, warm colors denote higher firing rates, cool colors denote lower firing rates. (f) Left: Boxplot showing running speed along the analyzed portion of the linear track (central 70 cm). There was no difference in running speed between groups (mean ± SD; iWT = 9.03 ± 4.55 cm/s, iCre-KO = 7.20 ± 2.74 cm/s, two-tailed WRS Z = 1.32, p = 0.19, n = 22 sessions from 3 iWT mice, 20 sessions from 4 iCre-KO mice). Right: Running speed versus track position for iWT (top) and iCre-KO (bottom) mice. Solid line depicts the mean across sessions (n = 22 sessions from 3 iWT mice, 20 sessions from 4 iCre-KO mice), gray region denotes the standard deviation of the mean. (g-i) Analysis of sharp wave ripples (SWRs) during linear track sessions when place cells were recorded (n = 16 sessions from 3 iWT mice, 13 sessions from 4 iCre-KO mice). (g) The frequency of SWRs during periods of awake rest did not differ significantly between iWT and iCre-KO mice (mean ± SD; iWT = 0.11 ± 0.12 SWRs/s, iCre-KO = 0.16 ± 0.16 SWRs/s, two-tailed WRS Z = −1.25, p = 0.21). (h) The number of spikes per cell per SWR event did not differ significantly between iWT and iCre-KO place cells (mean ± SD; iWT = 0.61 ± 0.44 spikes/SWR, iCre-KO = 0.51 ± 0.39 spikes/SWR, WRS Z = 0.94, p = 0.35). The number of spikes per cell per second of awake rest also did not differ between groups (mean ± SD; iWT = 0.073 spikes/s, iCre-KO = 0.089 spikes/s, two-tailed WRS Z = −0.22, p = 0.83). These analyses include all place cells that fired at least one spike during at least one SWR event (n = 21 iWT cells, 24 iCre-KO cells). (i) The percentage of SWR events for which a cell fired at least one spike was not significantly different between iWT and iCre-KO place cells (mean ± SD; iWT = 27.0 ± 30.0%, iCre-KO = 26.0 ± 19.0%, two-tailed WRS Z = −0.47, p = 0.64; n = 30 iWT place cells, 28 iCre-KO place cells). Data in panels (a, d, f, g-i) are represented as boxplots: the first and third quartiles are depicted by a box, and the median by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are denoted by plus symbols. 8. Maurer, A.P., Vanrhoads, S.R., Sutherland, G.R., Lipa, P. & McNaughton, B.L. Self-motion and the origin of differential spatial scaling along the septo-temporal axis of the hippocampus. Hippocampus 15, 841–852 (2005). 9. Ormond, J. & McNaughton, B.L. Place field expansion after focal MEC inactivations is consistent with loss of Fourier components and path integrator gain reduction. Proc Natl Acad Sci U S A 112, 4116–4121 (2015).
Supplementary Figure 12 Increased field size of iCre-KO place cells does not result from reduced short-term stability or drift in a field’s center of mass over the course of a recording session.
(a) We analyzed short term stability of place cells in two ways. First, we did not find any significant difference in the correlation between smoothed firing rate maps generated from each half of the session (mean ± SD, iWT = 0.50 ± 0.26, iCre-KO = 0.56 ± 0.24, two-tailed WRS Z = −1.68, p = 0.093; n = 113 iWT place cells, 126 iCre-KO place cells). Second, we divided the session into alternating even and odd periods (each 60 s) and computed the correlation between smoothed firing rate maps obtained from the combined even and off periods. We did not find any significant differences between groups in this measure (mean ± SD; iWT = 0.56 ± 0.24, iCre-KO = 0.60 ± 0.23, two-tailed WRS Z = −1.24, p = 0.21). Boxplots indicate the first and third quartiles by a box, and the median by a solid black line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are indicated by plus symbols. (b) We examined the shape of place fields, reasoning that if a field’s center of mass drifted in a consistent direction across a session, the shape of place field would be elliptical (skewed in the direction of the drift). An ellipticity score was calculated for the largest field from each place cell by fitting an ellipse to the central portion of the autocorrelation of the smoothed rate map (top 20%), and computing the ratio between the longer and shorter ellipse axes. 109/113 iWT, and 126/126 iCre-KO place fields were well-fit by an ellipse. Ellipticity was inversely correlated with distance from the nearest boundary, reflecting the fact that fields located near the boundary may be cut off (iWT r(107) = −0.36, p = 9.9e-5, iCre-KO r(124) = −0.36, p = 2.7e-5). However, this relationship did not differ between for iWT and iCre-KO cells (group × distance from boundary interaction: F(1,231) = 0.72, p = 0.40, ANCOVA). There was no difference in the ellipticity score for all iWT and iCre-KO place cells, nor was there a difference when considering only place cells with fields located in the arena’s center (mean ± SD; all fields: iWT = 1.83 ± 0.93, iCre-KO = 2.01 ± 1.31, two-tailed WRS Z = −0.71, p = 0.48, n = 109 iWT fields, 126 iCre-KO fields; central fields: iWT = 1.38 ± 0.30, iCre-KO = 1.49 ± 0.43, two-tailed WRS Z = −0.54, p = 0.56, n = 22 central iWT fields, 35 central iCre-KO fields). This analysis suggests there was no consistent drift over the course of a single session. Boxplots as in (a). (c) Example autocorrelations color coded for minimum (blue) and maximum (red) values. The ellipticity score is listed at the top left of each image. (d) The possibility remained that drift over the session occurred homogeneously across multiple directions, resulting in larger, circular-shaped place fields. To examine this, we computed the standard deviation in the center of mass (COM) across all individual passes through the field. We examined this metric for both place cells recorded on the linear track (top), and for place cells recorded in the open field (bottom). To qualify as a pass through a field in the open field, the animal was required to enter a zone surrounding the field, pass through the center of the field, and then exit the zone surrounding the field. The zone surrounding the field was defined by a circle, centered on the field’s COM and with a radius computed from the field size plus 2 additional bins (5 cm). The central zone was also a circle centered of the field’s COM, but with a radius of 2.5 bins (6.25 cm). To ensure that position over the field was evenly sampled across these passes, for the open field analysis we required that the animal’s average center of mass across trials fall within the center of the place field. In both open field and linear track recordings, we analyzed only fields for which there were at least 10 passes. The standard deviation in the COM across passes did not differ between iWT and iCre-KO fields recorded on either the linear track, or in the open field (mean ± SD: linear track iWT = 4.81 ± 3.03 cm, iCre-KO = 4.96 ± 3.00 cm, two-tailed WRS Z = −0.069, p = 0.95, n = 34 iWT fields, 25 iCre-KO fields; open field, iWT = 6.17 ± 1.81 cm, iCre-KO = 6.68 ± 1.83 cm, two-tailed WRS Z = −1.68, p = 0.092, n = 64 iWT fields, 84 iCre-KO fields). Boxplots as in (a). (e) Examples of iWT (top) and iCre-KO (bottom) place cells exampled in the open field for single pass variability. The animal’s trajectory is shown in light gray, with darker gray regions indicating positions that fell within the place field boundary. Each colored dot represents that COM from spikes emitted on a single pass through the field. The average COM is marked by a black dot. Top left: place cell field size. Top right: standard deviation in the COM across passes. Scale bars are 10 cm. (f) Examples of iWT (top) and iCre-KO (bottom) place cells recorded on the linear track. Rate maps are shown at top, with raster plots at bottom. Green lines denote the boundaries of the field. In the raster plots, the lap number is plotted on the y-axis, and all spikes emitted by the cell are depicted by a black line. On each lap, the center of mass for the spikes from the highlighted place cell is indicated by a red asterisk. The center of mass for a given place cell on a given lap was computed from all spikes falling within 2 bins (5 cm) on either side of the place field boundary. The center of mass was calculated only for laps in which at least 2 spikes were emitted within the place field zone.
(a) Examples of cells co-recorded across two days. Different colors indicate cells recorded in different animals. Shades of blue indicate cells from iWT mice and shades of red highlight cells from iCre-KO mice. The tetrode number and recording day (day 1 = D1, day 2 = D2) are indicated above scatterplots showing the relationship between peak to trough amplitudes for all signals recorded on each pair of electrodes (e) on a given tetrode (from top left to bottom right: e1 vs e2, e1 vs e3, e1 vs e4, e2 vs e3, e2 vs e4, e3 vs e4). Each dot represents a single sampled spike. Spikes associated with each isolated place cell are shown in a different color. For clarify, only clusters containing spikes emitted by place cells recorded on both D1 and D2 are shown (top panels). For each place cell, the average waveform on each channel and each day is shown (bottom right panels). Smoothed firing rate maps (bottom left panels) are color coded for maximum (red) and minimum (blue) values. Maps are shown for each day (map D1 and map D2), as well as the cross-correlation between the two maps (X-corr). (b) Histogram of cluster center-of-mass (COM) shifts for all iWT and iCre-KO place cells recorded across days (n = 25 iWT shifts, 36 iCre-KO shifts). Cluster center-of-mass shifts across days were smaller for iCre-KO cells (mean ± SD: iWT = 0.16 ± 0.067, iCre-KO = 0.11 ± 0.057, two-tailed WRS, Z = 2.49, p = 0.010), indicating that improper identification of cells across days did not lead to the decreased place stability we observed in iCre-KO mice. (c) There was not a significant relationship between the cluster center-of-mass shift and the place cell shift across days in either group (Pearson’s correlation, iWT: r (23) = −0.39, p = 0.051; iCre-KO: r(34) = 0.0075, p = 0.97; n = 25 iWT cells, 36 iCre-KO cells).
Supplementary Figure 14 Place field instability in iCre-KO mice extends beyond 2 days, and does not result from reduced grid or head direction stability across days.
(a) Examples of place cells recorded in iWT (left) and iCre-KO (right) mice across multiple days. Firing rate maps are color coded for minimum (blue) and maximum (red) values. Black dotes indicated firing fields. The Pearson’s correlation coefficient for maps from the first two days is shown at the top of the left-most rate map, and the Pearson’s correlation coefficient for maps from the latter two days is shown at the top of the right-most rate map. All examples, except for those inside the black box, show rate maps from place cells recorded over three consecutive days. While iWT place cells were observed to typically remain stable across all three days, iCre-KO place cells often shifted between at least one pair of consecutive recordings sessions. Black box: in the top example, the cell was recorded on days 1, 2, and 9; in the bottom example the cell was recorded on days 1, 2, and 5. In each of these examples, the fields remained stable over the first two days, but shifted substantially by the last day. (b-c) Pearson’s correlation coefficients (b) and shift values (c) for place cells recorded over three consecutive days. N = 11 iWT cells (blue), n = 18 iCre-KO cells (red). Boxplots indicate the first and third quartiles by a box, and the median by a solid line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are depicted by plus signs. Friedman’s tests, conducted separately on iWT and iCre-KO cells, showed there to be no significant differences between the three comparisons (Day1-Day2, Day2-Day3, Day1-Day3) for either the Pearson’s correlation coefficients or the magnitude of field shift (Pearson’s correlation coefficients: iWT: χ2(2) = 5.64, p = 0.060; iCre-KO: χ2(2) = 3.11, p = 0.22; shift magnitude: iWT: χ2(2) = 3.94, p = 0.14, iCre-KO: χ2(2) = 0.20, p = 0.91). The average of the three Pearson’s correlation coefficients was significantly higher for iWT cells (iWT median = 0.53, iCre-KO median = 0.25, two-tailed WRS, Z = 2.94, p = 0.0032), and the average of the three shift values was significantly lower for iWT cells (iWT median = 3.33 cm, iCre-KO median = 15.71 cm, two-tailed WRS Z = −2.40, p = 0.016). Taken together, these results suggest that iWT place fields tend to remain stable across at least several days, whereas those of iCre-KOs continue to shift. (d) Co-recorded iCre-KO place cells did not drift coherently across days. Scatterplot showing the mean difference in shift direction versus the mean difference in shift magnitude for co-recorded cells. Colors correspond to the plot at left. Both the differences in shift direction and shift magnitude were significantly greater than zero (mean ± SD; directional difference = 100.81 ± 35.68°, t(7) = 3.87, p = 0.0030; magnitude difference = 17.98 ± 13.14 cm, t(7) = 7.99, p = 4.6e-5, one-tailed t-tests, n = 8 sets of simultaneously recorded iCre-KO place cells). Inset: Arrows depict the direction and magnitude of shift across days for all iCre-KO place cells that were co-recorded with at least one other place cell. Sets of co-recorded cells are shown in different colors. (e) Previous studies have linked place cell remapping to shifts in the orientation of grid cells10, 11, but see also12. In grid cells recorded over consecutive days (n = 8 iWT cells, 13 iCre-KO cells), we did not find evidence that grid cells in iCre-KO mice are unstable across time. Rate maps show grid cells recorded across two days in iWT (left) and iCre-KO (right) mice. (f) Both the Pearson’s correlation coefficients (left) and the shift magnitude (right) of grid cells recorded over two days were similar between iWT and iCre-KO (Pearson’s correlation coefficients: iWT median = 0.36, iCre-KO median = 0.41, two-tailed WRS Z = −0.54, p = 0.59; Shift magnitude: iWT median = 7.50 cm, iCre-KO median = 3.54 cm, two-tailed WRS Z = 0.66, p = 0.51). In addition, the normalized mean and peak firing rate difference indexes across days were similar between iWT and iCre-KO grid cells (mean ± SD; mean firing rate difference index: iWT = 0.32 ± 0.21, iCre-KO = 0.41 ± 0.21, two-tailed WRS Z = −0.62, p = 0.54; peak firing rate difference index: iWT = 0.32 ± 0.20, iCre-KO = 0.35 ± 0.24, two-tailed WRS Z = −0.11, p = 0.91). Boxplots are presented as in (b), with individual data points plotted to the left (blue circles = iWT cells, red circles = iCre-KO cells). (g) The preferred firing direction of head direction cells recorded in WT (left) and iCre-KO (right) mice over multiple days remained stable. The head direction score and preferred firing angle are shown at the top of each polar plot. The spatial firing rate map color coded for minimum (blue) and maximum (red) values associated with each head direction cell is shown at right. (h) Boxplot of angular differences across days in the preferred firing of iWT (n = 18) and iCre-KO (n = 18) HD cells. The differences were similar between groups (mean angular difference ± SD: iWT: 14.86 ± 15.05°, iCre-KO: 19.57 ± 25.50°, two-tailed WRS Z = −0.35, p = 0.73). (i) Boxplot of the correlations between the spatial firing rate maps for head direction cells recorded on multiple days. The correlations did not differ between iWT and iCre-KO mice (mean ± SD; iWT = 0.23 ± 0.16, iCre-KO = 0.20 ± 16; two-tailed WRS Z = 0.090, p = 0.93). Boxplots indicate the first and third quartiles by a box, and the median by a solid black line. Whiskers indicate the range, except for data falling above the third quartile or below the first quartile by at least 1.5 times the interquartile range. Outliers are indicated by plus symbols. 10. Fyhn, M., Hafting, T., Treves, A., Moser, M.B. & Moser, E.I. Hippocampal remapping and grid realignment in entorhinal cortex. Nature 446, 190–194 (2007). 11. Monaco, J.D. & Abbott, L.F. Modular realignment of entorhinal grid cell activity as a basis for hippocampal remapping. J Neurosci 31, 9414–9425 (2011). 12. Brandon, M.P., Koenig, J., Leutgeb, J.K. & Leutgeb, S. New and distinct hippocampal place codes are generated in a new environment during septal inactivation. Neuron 82, 789–796 (2014).
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.