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corrected 10 times according to the template. The required shifts in all frames were normally close to zero after the 10 corrections. Note that although the original imaging acquisition frequency was 30 Hz, the effective acquisition frequency of the motion-corrected data was 10 Hz due to the down-sampling mentioned abo...	{ "page_id": null, "source": 7370, "title": "from dpo" }
in very few cases, were set to zero. The fractional change in fluorescence with respect to baseline (ΔF/F) was calculated as (F(t) - F 0(t)) / F 0(t), similar to what described previously (Low et al., 2014 denotes baseline fluorescence over time, which was estimated as a sum of two components: F 0(t)= F s(t)+ m. F s(t)...	{ "page_id": null, "source": 7370, "title": "from dpo" }
time interval: 0.1 s), during which the mouse’s running speed met or exceeded a speed threshold, which was specifically calculated for each imaging/behavioral session as follows. For each session, the instantaneous running speed of an imaging frame n was calculated by dividing the instantaneous travel distance (i.e., t...	{ "page_id": null, "source": 7370, "title": "from dpo" }
in a 5cm bin was above 50 cm/s, the bin contained no imaging data because the mouse spent less than 0.1 s within the bin. This high speed was frequently observed in well-trained mice running in 1D virtual reality. #### Tetrode recordings Tetrode recordings provided ground truth information for distinguishing cue cells ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
mate with the headstage for these miniature microdrives. 32 input channels were filtered (5 Hz-7.7 kHz), amplified, and multiplexed by the headstage into a single output using a multiplexing amplifier array (Intan Technologies, RHA2132). The multiplexing array used a 1MHz crystal oscillator clock for 32 channels, which...	{ "page_id": null, "source": 7370, "title": "from dpo" }
after surgery and were able to walk and lift their heads. ##### Environment—real 2D arena The real 2D arena was a 0.5 m square environment with black walls (≥30 cm high) made from plastic sheet or foam board. The floor was made of styrofoam. Black vertical construction rails (Thorlabs) defined the corners of the arena....	{ "page_id": null, "source": 7370, "title": "from dpo" }
to regularly explore the arena and forage for chocolate crumbs (Hershey’s milk chocolate) during these training sessions. The animals were then trained to run on the virtual linear tracks. ##### Behavior during recording Animals were first placed in a real 2D area and neural activity was recorded as they foraged for sm...	{ "page_id": null, "source": 7370, "title": "from dpo" }
across all electrodes on the same tetrode. Features of extracted waveforms were calculated. The features included the baseline-to-peak amplitudes of the waveforms on each of the tetrode wires, and the top three principal components calculated from a concatenation of the waveforms from all wires. ##### Cluster separatio...	{ "page_id": null, "source": 7370, "title": "from dpo" }
smoothed spike counts in the bin divided by the time spent in the bin. Firing rate was not defined for bins visited for less than 0.3 s. Spatial firing rate on virtual linear tracks were computed as follows. Virtual tracks were divided into 5cm non-overlapping spatial bins. Spike counts and the total amount of time the...	{ "page_id": null, "source": 7370, "title": "from dpo" }
(see “_Position-related calcium signals_”), were used for the analysis. In- and out-of-field periods were defined by comparing the mean ΔF/F value in each 5 cm bin to that of a random distribution created by 1000 bootstrapped shuffled responses. Each bootstrapped shuffled response was generated by rotating the ΔF/F of ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
cells must have a number of transitions between an in-field and out-of-field period for a track of length L larger than L/(5w), where w is the mean field width of the 1D response. (3) The widest field of the response must be smaller than 5w. (4) At least 30% of the bins must be assigned to either in-field or out-of-fie...	{ "page_id": null, "source": 7370, "title": "from dpo" }
at 5cm spatial bins along the track. Values in most bins of the template were set to 0, whereas the bins containing visual cues were set to 1. This template captured both the locations and the spatial extents of the cues. For tracks with asymmetric cues only cues on the right were used to create the template as we foun...	{ "page_id": null, "source": 7370, "title": "from dpo" }
of all cues were similarly calculated and their mean was defined as the “cue score.” Cue cells in imaging data were identified by comparing their cue scores with the cue scores of shuffled data (Kinkhabwala et al., 2015, Soc. Neurosci., conference). 200 shuffled 1D responses were generated for each cell by circularly p...	{ "page_id": null, "source": 7370, "title": "from dpo" }
grid cells cluster into more than one module, their 1D spacings and widths should also form separate clusters corresponding to the modules. To obtain a robust module assignment, we clustered the 1D field spacings and widths of the same set of grid cells measured on two different virtual linear tracks or two different t...	{ "page_id": null, "source": 7370, "title": "from dpo" }
width was used to better separate out firing fields, especially ones that were spatially close. The same parameters were applied to all cells in a given FOV. We also observed that while most cells exhibited good run-by-run consistency of their responses, some imaging sessions contained cells in which spatial tuning dri...	{ "page_id": null, "source": 7370, "title": "from dpo" }
spacings and widths of simultaneously imaged cells on two VR tracks (or two trials of the same track) were obtained based on their 1D responses on the two tracks (or two trials on the same track), as described above. The cells were clustered based on four parameters: 1D field spacing on track1 (or trial1), 1D field wid...	{ "page_id": null, "source": 7370, "title": "from dpo" }
2D lattice from ScienceDirect's AI-generated Topic Pages"), we first generated a large, ‘brute-force database’ (database 1) containing a large number of modeled 1D responses of grid cells along slices though 2D lattices, which were regular triangular lattices and the widths of all fields (local Gaussian peaks) were equ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
the 1D spacings of the simultaneously imaged co-modular grid cells. The density distribution of 1D spacings of all cells was first estimated based on a normal kernel function (using MATLAB function “ksdensity”). The spacing with the highest distribution probability was used as the main spacing (MS) of these grid cells....	{ "page_id": null, "source": 7370, "title": "from dpo" }
in the section “Assigning grid cells to modules based on their 1D responses”). Second, in some imaging sessions, we observed that in more than half of the cells, prominent fields existed at the beginning and the end of the track, where the rewards were delivered. These fields were generally narrow and precisely aligned...	{ "page_id": null, "source": 7370, "title": "from dpo" }
of the slice fit of real data in the brute-force database 1 and obtaining the spacing of the best fit lattice and the angle of the best fit slices in the lattice (s1 and a1, as described above), the slice fit was further refined in two steps: (1) If the spacing of the best fit lattice was smaller than 1 m, we further o...	{ "page_id": null, "source": 7370, "title": "from dpo" }
the lattice were mathematically identical. Therefore, to calculate the phase distances of two cells (cells a and b), we fixed the starting point of cell a and placed the starting point of cell b at the nine mathematically identical locations around nine fields. The phase distance of a and b was calculated as the shorte...	{ "page_id": null, "source": 7370, "title": "from dpo" }
same procedure was applied with X and Y replaced by vectors of pairwise distances for a single collection of cells in the same FOV. The plots for the pairwise physical distances versus the percentage of cell pairs within a certain range of phase distances (Figures 4 and pairwise phase distances (vector Y) described abo...	{ "page_id": null, "source": 7370, "title": "from dpo" }
plot depending on the purpose of different analyses. #### Grid score of phase clusters ##### Score calculation (Figure S4 only for those cell pairs whose pairwise phase distance was below the _a_ th percentile of all phase distances. This collection of 2D offsets (phase plot) was then binned using bin width _b_ to gene...	{ "page_id": null, "source": 7370, "title": "from dpo" }
about annular region from ScienceDirect's AI-generated Topic Pages") with inner radius R i and outer radius R o, and rotated the autocorrelation map in steps of 60∘ to compute the Pearson correlation between the rotated and the original map. The grid score was defined by the minimum difference between crests and trough...	{ "page_id": null, "source": 7370, "title": "from dpo" }
score with those of shuffled data. The shuffled data were created by preserving the anatomical locations of all cells while randomly permuting their phases. The phase distances, phase plot, and grid scores were generated as for real cells. The significance of the grid score of real cells was given by its percentile amo...	{ "page_id": null, "source": 7370, "title": "from dpo" }
length at least 1. If a lattice is a regular hexagon, the angle of v2 should be 60 degrees, so its _x_-coordinate should be 0.5, and its length should be 1. In our results, we observed that the _x_-coordinate of all the v2 vectors were between 0 and 0.5, suggesting a bias in the set of lattice deformations occurring in...	{ "page_id": null, "source": 7370, "title": "from dpo" }
unit-rhombus, folding score Cells in a cell group were assigned new locations in phase space based on their spatial tuning phases as follows. The center cell was assigned a phase location equal to its original phase. The remaining cells were assigned phase locations given by the phase offsets between their original pha...	{ "page_id": null, "source": 7370, "title": "from dpo" }
view. Only the 29 sets of co-modular cells whose autocorrelation of phase cluster lattice showed at least five local maxima (Figure S4 in L. These vectors determine the exact alignment of the unit phase rhombus with the autocorrelation vertices (or hypothetical brain lattice). 1 In this section, we will often speak dir...	{ "page_id": null, "source": 7370, "title": "from dpo" }
Equation 1As described in “Grid score of phase clusters,” for each set of co-modular cells in a FOV, we first computed an autocorrelogram from the modified, binned phase plot _H_, using the percentile and bin width parameters _a_ and _b_ which yield the highest grid score (and using the same parameter ranges as describ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
2∈P, we constructed the 8 shortest vectors in the lattice L p 1,p 2 generated by p 1 and p 2 (here, L p 1,p 2 is the set m p 1+n p 2 for all integers m and n). To do this we first constructed a pair of shortest generating vectorsp˜1,p˜2 for L p 1,p 2. The 8 shortest vectors E then consist of p˜1,p˜2,p˜1+p˜2,p˜1−p˜2 and...	{ "page_id": null, "source": 7370, "title": "from dpo" }
in this analysis we created shuffles which preserved some amount of the local phase structure of the data. Rather than shuffle all the phases at random, we first clustered the phases in the unit rhombus, and then shuffled the clusters. More specifically: * 1)For each set of co-modular cells, we clustered the phases in ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
regularity criteria. Specifically, we rejected a shuffle if its best fit vectors v 1,v 2 had: a b s c o s a n g l e v 1,v 2>0.9,\|v 1−v 2\|/m e a n\|v 1\|,\|v 2\|<0.5, or if the minimum norm of v 1 and v 2 for the shuffle was less than 0.8 times the minimum norm of v 1 and v 2 for the data, and similarly for 1.2 times the ma...	{ "page_id": null, "source": 7370, "title": "from dpo" }
was r. For each set of co-modular cells, we constructed 10 sets of noisy phases under six values of r, evenly spaced between 0 and 3/4, as well as 10 sets with completely random phases, for which the noise vectors were drawn uniformly from the rhombus. We then created a fit curve (shown in the Figure S5 Sequential grid...	{ "page_id": null, "source": 7370, "title": "from dpo" }
above. The track location with the maximal mean ΔF/F within a field was termed as the “field location’. For each pair of cells, if the field location distance of their two, closely-occurring fields (one field from each cell) was shorter than S, these two fields were called sequential fields. S was defined as follows: t...	{ "page_id": null, "source": 7370, "title": "from dpo" }
of vector directions: the longer the resultant vector length was, the more consistent vector directions were. The resultant vector length was calculated using a MATLAB toolbox “CircStat” (Berens, 2009 of Figure 6 within the bin were called “co-active cells” in the bin. A single 2D matrix with the same number of pixels ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
The autocorrelograms were similarly generated for all the bins and averaged as a single autocorrelogram (AA), which reflected the general pattern of co-active cells in the FOV. ##### Correlation of the autocorrelograms of co-active cells and phase cluster lattice To quantify whether the co-active cells formed a similar...	{ "page_id": null, "source": 7370, "title": "from dpo" }
phase clusters”), and y was determined as 0.8× (the shortest spacing of the phase cluster autocorrelogram) (see “Spacing of phase clusters” for the calculation of spacings of the phase cluster autocorrelogram). (2) For other sets of co-modular cells, x and y were set to 15 (i.e., the 15 th percentile) and 70μm, respect...	{ "page_id": null, "source": 7370, "title": "from dpo" }
did not have phase clusters either because they did not have cells with similar phases, or because none of their cluster sets met the criteria above. We then kept unique sets of phase clusters to perform the missed-field analysis below. ##### Missed-field pattern of a cell (Figures S7 The missed-field pattern of a cell...	{ "page_id": null, "source": 7370, "title": "from dpo" }
on 2D responses The unbiased autocorrelation of the 2D firing rate in the real arena was calculated (Hafting et al., 2005, an inner radius was defined as the smallest of the following three radii: (1) where the radial autocorrelation was at a local minimum; (2) where the autocorrelation was negative; (3) 10cm. Multiple...	{ "page_id": null, "source": 7370, "title": "from dpo" }
used to calculate a grid score threshold at the 95 th percentile. ##### Grid modules based on 2D responses (Figures S2 estimate of log grid spacings (using MATLAB function “ksdensity”) of cells in the same animal. We set the bandwidth parameter σ for the ksdensity function according to the following formula:σ=a n+b,whe...	{ "page_id": null, "source": 7370, "title": "from dpo" }
was calculated. The combined distribution of all shuffled cue scores was used to calculate a threshold at the 95 th percentile. The tetrode-recorded cells whose cue scores exceeded this threshold were identified as cue cells. ##### In- and out-of field periods for 1D firing rates The in- and out-of-field periods for 1D...	{ "page_id": null, "source": 7370, "title": "from dpo" }
##### 1D spacing and field width The 1D spacing and field width for tetrode-recordings were identified by the same method used for calcium responses, described above. The adjacent field spacings for a cell were defined to be the distances between field centers for all pairs of adjacent fields, and the cell’s 1D spacing...	{ "page_id": null, "source": 7370, "title": "from dpo" }
ROI extraction methods. K.Y. provided phase analysis advice and MATLAB code for calculating grid scores. ### Declaration of Interests The authors declare no competing interests. Recommended articles References ---------- 1. Aronov and Tank, 2014, pp. 442-456 View PDF, pp. 1197-1207 View PDF, 10.18637/jss.v031.i10, pp. ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
D.S. Kim, K. Svoboda Thy1-GCaMP6 transgenic mice for neuronal population imaging in vivo PLoS ONE, 9 (2014), p. e108697 Crossref, pp. 83-92 View in Scopus, pp. 1433-1440 Crossref, pp. 199-204 Crossref, pp. 194-208 View PDF, pp. 4266-4276 View in Scopus, 10.1101/339564, pp. 231-240 View in Scopus, pp. 801-806 [Crossref]...	{ "page_id": null, "source": 7370, "title": "from dpo" }
et al., 2012]( Harvey, P. Coen, D.W. Tank Choice-specific sequences in parietal cortex during a virtual-navigation decision task Nature, 484 (2012), pp. 62-68 Crossref, pp. 1079-1090 View PDF, pp. 521-523 Crossref. Visual cue-related activity of MEC cells during navigation in virtual reality. In Society for Neuroscienc...	{ "page_id": null, "source": 7370, "title": "from dpo" }
in Scopus]( Scholar]( 26. Madisen et al., 2015, pp. 942-958 View PDF, pp. 663-678 Crossref, pp. 507-521 View in Scopus, pp. 747-760 View PDF, pp. 783-806 Crossref, pp. 141-154 View PDF, p. 13 View in Scopus, Sinauer Associates (2001) [Google Scholar]( 34. [Ray et al., 2014]( Ray, R. Naumann, A. Burgalossi, Q. Tang, H. ...	{ "page_id": null, "source": 7370, "title": "from dpo" }
M. Brecht Grid-layout and theta-modulation of layer 2 pyramidal neurons in medial entorhinal cortex Science, 343 (2014), pp. 891-896 Crossref, pp. 758-762 Crossref, pp. 469-475 Crossref, pp. 671-675 Crossref, pp. 186-191 Crossref, pp. 72-78 Crossref, pp. 9466-9471 Crossref, pp. 1040-1053 View PDF, pp. 471-480 [Crossref...	{ "page_id": null, "source": 7370, "title": "from dpo" }
Tang et al., 2014, pp. 1191-1197 View PDF, pp. 12346-12354 View in Scopus, pp. 1313-1324 View PDF, pp. 902-904 Crossref, pp. 49-58 View PDF, pp. 481-495 View PDF, pp. 1110-1116 [View PDF]( article]( in Scopus]( Scholar]( 50. [Yoon et al., 2013]( Yoon, M.A. Buice, C. Barry, R. Hayman, N. Burgess, I.R. Fiete	{ "page_id": null, "source": 7370, "title": "from dpo" }
Specific evidence of low-dimensional continuous attractor dynamics in grid cells Nat. Neurosci., 16 (2013), pp. 1077-1084 Crossref, pp. 1086-1099 View PDF ------------- * ### A unifying perspective on neural manifolds and circuits for cognition Neuron, Volume 96, Issue 2, 2017, pp. 490-504.e5 Mark E.J.Sheffield, …, Dan...	{ "page_id": null, "source": 7370, "title": "from dpo" }
Awake Replay]( "Direct Medial Entorhinal Cortex Input to Hippocampal CA1 Is Crucial for Extended Quiet Awake Replay") Neuron, Volume 96, Issue 1, 2017, pp. 217-227.e4 Jun Yamamoto, Susumu Tonegawa View PDF iScience, Volume 27, Issue 3, 2024, Article 109035 Kyle Puhger, …, Brian J.Wiltgen View PDF Cell, Volume 183, Issu...	{ "page_id": null, "source": 7370, "title": "from dpo" }
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