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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 above. ##### ROIs, raw fluorescence, ΔF/F, significant transients All regions of interest (ROIs) were identified by the following algorithm, except for pyramidal and stellate cells that were manually annotated and identified in related analyses (Figures 2. Details about the ROI extraction algorithm is described as follows. For each FOV, motion-corrected stacks (≤40 stacks) were used to identify ROIs. The stacks were first down-sampled by a factor of 2 in all three dimensions (time and spatial axes (X and Y)). The whole FOV was then evenly split as nine blocks. For each block, an independent component analysis (ICA) based algorithm (Mukamel et al., 2009, 150 independent components, s.d.threshold= 3). Independent ROIs in the nine blocks corresponding to individual cell bodies in the FOV were manually identified and retained for subsequent analyses. Fluorescence time series of these ROIs were extracted from all motion-corrected stacks. For each cell, a negative offset, which was produced by ScanImage5 and indicated the voltage value of zero in the case of zero signal, was subtracted from its fluorescence time series so that all fluorescence values were non-negative. Remaining negative values, which only existed
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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) denotes the smoothed fluorescence signal, calculated as the 8 th percentile of F(t) within a ± 15 s moving window centered at time _t_. The _m_ is a small, constant value such that, under baseline conditions, F 0(t) matches the local mean of F(t) and ΔF/F has mean zero. _m_ was calculated for each cell as the mean of F(t) - F s(t) over a set of time points (baseline points), during which no transients or other fast fluctuations occurred. The baseline points were mathematically determined as points, the standard deviations of which were smaller than std min+0.05∗(std max-std min). std min and std max were the minimal and maximal standard deviations among all data points within a 20 frame (±1 s) moving window, respectively. The significant calcium transients were identified as transients that exceeded cell-specific amplitude and duration thresholds so that the artifactual fluctuations were expected to account for less than 1% of detected transients, as described previously ([Heys et al., 2014]( ##### Position-related calcium signals Analysis included only motion-corrected data points (effective acquisition frequency: 10 Hz, i.e., frame
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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., track position difference between frame n and frame n-1, 0 if n= 1) by the frame time interval (0.1 s). All positive speeds of the session were binned into 100 bins and the bin edge of the first bin was set as the speed threshold of the session. In our data, the thresholds largely (95.4%) varied from 0.7 to 2.0 cm/s, whereas mean running speeds (the average positive speed of a session) were mostly (94.9%) distributed from 10 to 45 cm/s. To calculate mean ΔF/F as a function of position along the virtual linear track, the track was divided into non-overlapping 5cm bins and the mean ΔF/F within each bin was calculated. Heat plots, which showed ΔF/F as a function of position across multiple traversals along the track, were similarly generated except that the mean ΔF/F within each 5cm bin was calculated for each individual traversal. For display purposes, color values representing the mean ΔF/F were normalized to span the minimum and maximum values for each cell. As explained in Figure 2 within the bin the mouse traversed below the speed threshold, so the imaging data was eliminated from the analysis. (2) Within the bin the mouse traversed at very high speed, so there was no imaging data in the bin. Note that the effective acquisition frequency in motion-corrected data was 10 Hz. When the mouse’s running speed
{ "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 from grid cells (Figures S1 were assembled manually. Tetrodes, which were made of PtIr (18 micron, California Fine Wire), were plated using Platinum Black (Neuralynx) to 100-150kΩ at 1kHz. Shuttles were made of dental acrylic and moved with an embedded drive screw. Each shuttle contained two tetrodes. A reference wire (0.004” coated PtIr, 0.002” uncoated 300μm top) was placed into the brain at a similar depth as the tetrode, but medial to the MEC on both hemispheres. A stainless steel screw or wire above the cerebellum was used as reference ground. ##### Electrode recording system The system was similar to that described previously (Aronov and Tank, 2014, printed (Sunstone Circuits), and assembled at the Princeton Physics Department electronics facility. The headstage design was identical to the one used previously (Aronov and Tank, 2014 to power two LEDs to track animal location and head orientation. A lightweight 9-wire cable (3 m, Omnetics) connected the headstage to the electrode interface board, which was customized to
{ "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 provided a sampling rate of 31.25 kHz to each channel. The multiplexed signals were digitized with a data acquisition card (National Instruments, PCI-6133) and acquired using custom software in MATLAB. ##### Surgery The surgery was performed as described previously (Domnisoru et al., 2013 so that the microdrive could be implanted vertically. Bilateral craniotomies were performed at 3.2mm lateral to the midline and just rostral to the lambdoid suture. A headplate was centered on the skull and affixed with dental cement (Metabond, Parkell). Shuttle screws driving the tetrodes were retracted by 2-4 turns and mineral oil was applied inside the cannula of the microdrive. The microdrive was then mounted on the stereotaxic arm and lowered so that the bottom of the cannula touched the dura of each craniotomy. A fillet of composite (Dentin, Parkell) was made around the cannula to protect the inside tetrodes from the Metabond. The Metabond was then applied to affix the microdrive to the skull. Once the Metabond solidified, the tetrodes were lowered at least 1mm into the brain by slowly turning the screw on each drive by 4-6 turns. Animals woke up within 10 min
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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. A single white cue card was placed on one wall. The arena was positioned near the virtual reality setup for easy transfer of the animal when plugged into the headstage between the two systems, which facilitated the recording of the same cells in both environments. In most cases, a single light was positioned above the area to illuminate the environment. In some cases, the light was turned off so that only the LED light on the headstage was present during the recording. ##### Environment—virtual reality The virtual reality system was similar to that described previously (Domnisoru et al., 2013, and monitor running velocity of the mouse. The animal ran on a cylindrical treadmill, the rotational velocity of which was proportional to mouse velocity. This velocity was measured using sequential sampling of an angular encoder (US Diginal) on each ViRMEn iteration (∼60 iterations per second). ##### Training in real 2D arena and virtual reality At least one week after surgery, animals were introduced to the real 2D arena. They quickly began
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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 small pieces of chocolate scattered in the arena at random times. Trials generally lasted 10-20 min. Animals were then transferred to the virtual reality setup and ran along a virtual linear track. Most animals were recorded on just a single virtual track. To collect 1D data for module assignment, the animals were recorded when they ran on two different virtual tracks sequentially. ##### Position tracking in the real 2D arena A Neuralynx acquisition system (Digital Lynx) was used for the video tracking (Domnisoru et al., 2013 above the arena. The midpoint between the two LEDs was defined as the animal’s location. ##### Timing Digital timing signals were used to synchronize all computers. These signals were sent and acquired using a NI-DAQ card and transmitted once per iteration in ViRMEN (60Hz). #### General processing of tetrode data Data analysis was performed using custom MATLAB code. Electrophysiology data were first demultiplexed and filtered (500Hz highpass). Spikes were detected using a negative threshold, which was set to be three times the standard deviation of the averaged signal
{ "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 separation Features of the waveforms were plotted with a custom MATLAB GUI and polygons were manually drawn in the feature space to define clusters. All data from one day were grouped and analyzed together. Criteria for eliminating clusters from the dataset were: (1) units with less than 100 spikes (in the real 2D arena or virtual tracks); (2) the minimal spatial firing rate along the virtual track was larger than 10Hz and the maximal firing rate was larger than 50Hz; (3) the tetrode was positioned outside of the MEC based on histology. ##### Spatial firing rates Position data were subsampled at 50Hz and spikes were assigned to the corresponding 0.02 s bins. Velocity at the spike time point was calculated by smoothing the instantaneous velocity with a moving window of 1 s. Only data in which the animal’s velocity exceeded 1cm/sec were used for further analysis. Spatial firing rates in the real 2D arena were calculated as follows. The 2D arenas were divided into 2.5× 2.5cm non-overlapping spatial bins. Spike counts and the total amount of time the animal spent in these bins were convolved with a Gaussian window (5×5 bins, σ= 1 bin). The firing rate in each bin was given by the
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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 animal spent in these bins were convolved with a Gaussian window (3-point, σ=1 bin). The firing rate in each bin was calculated as the smoothed spike count in the bin divided by the smoothed time in the bin. ### Quantification and Statistical Analysis Except the last section “_Tetrode data analysis_,” all the following sections are analyses of calcium imaging data recorded in 1D virtual reality. #### Statistics Values in bar graphs or curves with error bars are reported as mean ± SEM. Statistical analyses are described in figure legends and below in individual sections. Most statistical analyses used standard parametric approaches, such as t tests (for Gaussian-distributed data) and Kolmogorov-Smirnov tests (for non-Gaussian-distributed data). Significance was defined using a p value threshold of 0.05. In figures, we use different numbers of “∗” to represent different ranges of p values: ∗p ≤ 0.05, ∗∗p ≤ 0.01, ∗∗∗p ≤ 0.001. “n.s.” indicates that two groups were not statistically significant when p > 0.05. #### Classifying grid cells based on 1D calcium responses ##### In- and out-of-field periods The in- and out-field periods in the 1D calcium-dependent responses of cells were similarly determined as those in previous studies ([Domnisoru et al., 2013]( [Heys et al., 2014]( [Yoon et al., 2016]( The mean ΔF/F, ΔF/F and significant-transient-only traces, which occurred when the running velocity was greater than the velocity threshold
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(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 the entire recording so that for every time point of the recording, its track position was preserved but its calcium response was changed. The ΔF/F trace was rotated by starting the trace from random sample numbers chosen from the interval 0.05× N samples to 0.95× N samples, where N samples was the number of samples in the ΔF/F trace. A shuffled mean ΔF/F was calculated for each rotation. For each 5 cm bin, a p value equaled the percent of shuffled mean ΔF/F that were higher than the real mean ΔF/F. Therefore, 1-p value equaled the percent of shuffled mean ΔF/F lower than the real mean ΔF/F. In-field-periods were defined as three or more adjacent bins (except at the beginning and end of the track where two adjacent bins were sufficient) whose 1-p value ≥ 0.8 and for which at least 10% of the runs contained significant calcium transients within the period. Out-of-field periods were defined as two or more bins whose 1-p value ≤ 0.25. Bins with intermediate mean ΔF/F remained unassigned. ##### 1D grid cell classifier The 1D classifier used to identify grid cells based on their 1D responses was identical to what was used previously (Yoon et al., 2016 a grid cell must have at least two spatial fields on a track. (2) The 1D response of grid
{ "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-field periods. (5) The mean ΔF/F of in-field periods divided by the mean ΔF/F of out-of-field periods must be larger than 2. ##### Cue cells (Figures S1. In virtual reality, these cells are termed ‘cue cells’ because their firing fields are tied to salient cues on virtual linear tracks (Kinkhabwala et al., 2015, Soc. Neurosci., conference). These firing field patterns can sometimes produce misclassification of cue cells as grid cells in our original grid cell classifier. Therefore, cue cells, which accounted for 7.28 ± 1.3% of cells originally classified as grid cells, were eliminated from the grid cell corpus used in further analyses (Figures S1 were stellate cells and anatomically intermingled with grid cells (Figures S2. The pattern of visual cues was expressed as a cue template, which was calculated
{ "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 found the MEC cells on the left hemisphere, where we always imaged, mostly responded to cues on the right, but rarely to those on the left (data not shown). To calculate the cue score, the cross-correlation between the cue template and the 1D response was first calculated under spatial shifts (lags) up to 300 cm in both directions. The peak in the cross-correlation with the smallest displacement from zero was chosen as the best correlation of the firing rate to the cue template. The lag at which this peak occurred was then used to displace the cue template to best align it with the 1D response. Having found the best lag to align the activity with the cue template, we then selected a subset of the track close to and including the visual cues (termed ‘cue bins’) so that we could calculate the correlation between the activity and the cue as follows. For each visual cue represented by the cue template, the cue-bins were defined as those within the extent of the cue and those extending past the cue in both directions by the half-width of the cue. Cues located near the start or the end of the track were removed from the analysis if their cue bins exceeded the range of the track. A local correlation of the template and 1D response within the cue-bins was calculated. The local correlations
{ "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 permuting the ΔF/F, identical to what was described for the p value analysis. The mean ΔF/F and cue score of each shuffled response was calculated. The distribution of all shuffled cue cell scores combined (203400 shuffles for 1017 imaged cells) was used to calculate a threshold at the 95 th percentile. The cells with cue scores above the threshold were identified as cue cells and removed from the grid cell population. Based on our previous study, this method effectively distinguished grid cells from cue cells, which generally did not correlate well with visual cues at consistent locations (Kinkhabwala et al., 2015, Soc. Neurosci., conference). #### Assigning grid modules based on 1D calcium responses Grid modules are typically determined by the clustering of spatial response scales in 2D environments ([Stensola et al., 2012]( Previous work showed that the 1D response of a grid cell was well approximated as a slice through a 2D triangular lattice; the 2D field width, spacing, and orientation of the cell were estimated as the width and spacing of the lattice, and the angle of the slice, respectively ([Yoon et al., 2016]( Since the 2D grid fields are typically grazed by 1D slices, and slices can skip one or more adjacent fields depending on angle, we reasoned that the best estimates of the 2D field width and spacing are the largest 1D field width and smallest 1D field spacing, respectively. Furthermore, if a group of
{ "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 trials on the same track in four-dimensional space, and assigned cells to different modules according to this clustering (Figure 3 and separately estimated using clustering of the 1D field widths and spacings in the two 1D tracks. The detailed procedures for the assignment are described below. The analysis for the ground truth tetrode data is described in “Tetrode data analysis.” ##### Modified p value As described above, a set of p value thresholds and 5cm bin widths, which were similar to those used previously (Domnisoru et al., 2013 and drifts in their spatial tuning, the same set of parameters could not always be used to faithfully identify every field of every cell in every dataset. Since the module assignments required precise identification of spatial fields for the measurements of field widths and spacings, we modified the parameters for different imaging sessions. Based on manual inspection of cells in each dataset, the p value thresholds for in-field periods were set from 0.75 to 0.95. A 2.5cm bin
{ "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 drifted across many runs. These drift responses mostly occurred at the beginning of the behavioral session when the mouse was just exposed to the VR. To more precisely identify in-field periods for these cells, we computed a stability score, as described previously (Yoon et al., 2016 of all pairs of single-run responses within a consecutive n-run block [i, i+n-1]. n is generally 10-30 depending on the length of the recording and the consistency of the run by run activity. We repeated this process for different starting positions i in steps of 1 run, starting from i= 1 up to i= N-n+1. We finally selected the n-run block with the largest mean stability score for all cells in the same imaging session and analyzed all cells within this block. After these modifications, we discarded cells whose identified fields did not match manual inspection. ##### 1D spacing and 1D field width of a cell ([Figure 3]( Adjacent field spacing was measured as the distance between two adjacent field centers. The 1D field spacing of a cell was then determined as the shortest spacing among all the adjacent field spacings. Field width was measured as the spatial width a field spanned. The 1D field width of a cell was determined as the largest width among all field widths. ##### Assigning cells to modules The 1D field
{ "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 width on track 1 (or trial 1), 1D field spacing on track 2 (or trial 2), and 1D field width on track 2 (or trial 2). The clustering was performed by k-means and the optimal number of clusters was obtained using the MATLAB function “evalclusters” under the Calinski-Harabasz clustering evaluation. One to six clusters were evaluated. The clustering of cells was then used to assign their module identities. In rare cases, the ratio of mean 1D spacings of cells in two different clusters were equal or smaller than 1.1. We considered that these cells belonged to the same module. #### Phase distances of co-modular grid cells The phase distances of simultaneously imaged co-modular grid cells were calculated by fitting their 1D calcium dependent responses on the same VR track to parallel slices in a 2D triangular lattice. The accuracy of this method was tested by applying the analysis to tetrode-recordings of grid cells in both real 2D and virtual 1D environments (Figures S3 and then separately estimated using the best fit parallel slices of their 1D responses in the same 2D triangular lattice. The details for identifying phase distances from 1D data are as follows. The analysis for ground truth tetrode data is described in “Tetrode data analysis” ##### Brute-force database In order to fit cells’ 1D responses to the best slice of a [2D lattice]( "Learn more about
{ "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 equal. The spacings of the 2D lattices were evenly sampled numbers from 10 to 1000cm at 10cm intervals. The field widths of the lattices were determined by the standard deviations (σ) of the Gaussian peaks. For each spacing, we generated 2D lattices whose standard deviations were 10 evenly spaced numbers between σmin and σmax, which generated the minimal and maximal field widths (narrowest and widest Gaussian peaks), respectively:σmin=0.03 σ max=S p a c i n g 2×2×ln(50))The maximal lengths of the slices were 1800cm. The angles of slices were 30 integer angles from 0 to 29 degrees. The starting points of the slices were 121 points evenly distributed in a unit-rhombus, which is the smallest repeating unit of the triangular lattice. In summary, the database contained 3630000 modeled grid cell responses on a 1D track, the maximal length of which is 1800cm. ##### Fitting simultaneously imaged co-modular cells to slices of a 2D triangular lattice The 1D mean ΔF/F traces of simultaneously imaged co-modular grid cells were fit using parallel slices of the same 2D triangular lattice. In particular, for each group of co-modular cells, a set of candidate 2D triangular lattices with a range of spacings in the brute-force database was first selected as follows. The minimal spacing of the lattices was 10cm, which was the minimal spacing of lattices the brute-force database. The maximal spacing of the lattices was determined based on
{ "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. The maximal spacing of candidate triangular lattices was calculated as:maximal spacing=2×MS 3 This maximal spacing above could be any number, whereas the spacings of lattices in the brute-force database were discontinuous numbers. Therefore, the maximal spacing of the candidate lattices in the database were further determined as the smallest sampled spacing of the database that was larger than the maximal spacing. The mean ΔF/F of co-modular grid cells were compared with modeled 1D activity patterns, which were generated from parallel slices in the candidate lattices. The similarity of real 1D activity patterns with each simulated 1D slice was quantified by the Pearson’s linear correlation coefficient between them (fit score). For each set of parallel slices under a specific angle in a 2D lattice (note that these slices have different starting points in the rhombus), we found the best fit slice for each cell and averaged the fit scores for all the simultaneously recorded co-modular cells. The 2D lattice with the best average score was then considered as the best fit lattice for these cells, and the spacing of this lattice and the angle of the best fit slices in this lattice were set as s1 and a1, respectively. There were at least two complications affecting the accuracy of the slice fitting. First, the 1D spatial tuning could drift throughout multiple traversals. Therefore, we only used the activity in the subset of traversals, which had highest stability score (as described above
{ "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 to the reward locations. These fields were likely triggered by the rewards, rather than grid activity pattern. In datasets for which this was the case, we only considered the activity between the two reward locations for the purposes of slice fitting. ##### Significance of the slice fitting The significance of the fitting of co-modular cells was evaluated by comparing the fitting results to 100 shuffled datasets. The shuffled activity of a cell was generated by randomly shifting its in-field periods along the track. Individual data points of the out-of-field periods and unassigned periods were randomly placed in gaps between shuffled in-field periods. If only a part of the original 1D response was used for the slice fitting, as mentioned above, the shuffled activity was generated from the same part of the 1D response. The shuffles generated by this method still contained the original fields (albeit rearranged) and allowed us to test whether the distributions of these fields followed the 1D slice pattern through a 2D triangular lattice. Each set of shuffled data, which contained shuffles for each co-modular cell, was similarly fit using parallel slices in triangular lattices in the brute-force database 1, as described above for the real data. The significance of the fit of the co-modular cells was calculated as the percentile of its fit score among the fit scores of 100 shuffled datasets. ##### Optimization of the slice fitting After validating the statistical significance
{ "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 optimized the spacing and angle by fitting the co-modular cells in database 2, which contained lattices under finer sampled spacings from 40cm to 1 m with an interval of 0.25cm. After this step, we obtained a new best fit lattice with spacing s2 with slices at angle a2. We did not perform this step if the spacing of the best fit lattice of cells was larger than 1 m. For these cells, s2= s1, a2= a1. (2) After obtaining s2 and a2, we further optimized the starting points in the unit-rhombus as follows: the 1D responses were fitted in a database 3, which contained a single lattice with spacing s2. Parallel slices at the angle a2 were created under 961 evenly distributed starting points (note that there were only 121 starting points sampled in previous database 1, so database 3 provided finer sampling of starting points of slices) in the unit-rhombus. The new set of optimized starting points led to the higher fit score in database 3. ##### Phase distances of grid cells ([Figure S3]( After optimizing the slice fit, the phases of grid cell 1D responses were determined as the starting points of their slice fits in the unit-rhombus. The phase distance of a pair of co-modular grid cells was therefore calculated as the distance of their starting points in the same unit-rhombus. Due to the symmetry of the triangular lattice, all the connecting points of
{ "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 shortest distance between the starting point of cell a and the nine starting points of cell b. #### Pairwise phase distances versus physical distances Plots of phase distance versus physical distance (Figures 4 were generated as follows. For Figures 4, the corresponding phase distances in Y were grouped and the mean and standard errors of each group were calculated and plotted against the centers of X bins, producing a “binned phase distance versus physical distance curve.” We generated shuffled “binned phase distance versus physical distance curves” by randomly permuting the elements of Y with respect to X and applying the same procedure as above. We generated 100 such shuffled (binned) curves and took their average to produce a “mean shuffled curve.” Finally, the value of the curve computed from real data in a given bin was determined to be significantly different from the overall mean phase distance when it was further (in absolute value) from the value of the mean shuffled curve in that bin than 95% of the shuffled curves. For [Figures 4]( and [S4]( the
{ "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 above. The elements of X were binned using a bin width of 15 μm (X bins). Only elements in X bins ≤ 120 μm were used in the analysis. In each of these bins, N represented the total number of elements, and n represented the number of elements, the phase distances of which were within a particular range (e.g., above 90 th percentile of all elements in Y). The n/N (as a percentage) was plotted against the center of its X bin. #### Phase plot (Figure S3. For a given reference cell _c_, we shifted the physical positions of all (simultaneously fitted, co-modular) cells so that _c_ was at (0,0), and colored the other cells according to their phase distance from _c_. This was repeated for each choice of reference cell in the given set, and the individual plots were then overlaid to form a summary plot across all cell pairs. In this final plot, colors of dots, which indicated phase distances between a cell and a centered reference cell, were all renormalized to the same color scale. Cells within a certain range of phase distances could be further eliminated or kept in the
{ "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 generate a 2D histogram _H_. _H_ can be thought of as the density of 2D offsets between cells with similar phases. In what follows, we refer to the 2D autocorrelogram of _H_ as “the autocorrelogram,” which we used to study the pattern of grid cells with similar phases (phase clusters). The vertices (local maxima) around the center of the autocorrelogram typically produced a distorted hexagonal grid. We calculated grid scores of the autocorrelogram using a previously described method, which allows both isotropic (equilateral triangle) and anisotropic (squeezed or stretched) grids to assign high scores despite elliptical distortions in a hexagonal grid (Yoon et al., 2013, fitted those vertices with an ellipse, and transformed the autocorrelogram to a regular triangular lattice by applying a linear transformation so that the long and short axes of the ellipse became equal. Given the transformed autocorrelogram, we defined an [annular region]( "Learn more
{ "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 troughs in rotated correlations.Gridscore(R i,R o)=min ρ i,o 60∘,ρ i,o 120∘−max ρ i,o 30∘,ρ i,o 90∘,ρ i,o 150∘where ρ i,o(ϕ) was the correlation value when one map was rotated by angle ϕ relative to another over the annular region defined by R i,R o. We repeated this by letting R i change from 0.5 r to r and R o change from R i+1 cm to 1.5 r (or to the maximum allowed value based on the autocorrelogram), each independently and in steps of 1cm. r was the mean distance to the nearest neighboring vertices of the central lattice in the transformed autocorrelogram. The grid score of the cell was then defined as the maximum score over these various annular regions. Finally, the above procedure was repeated over a fixed range of values for parameters _a_ and _b_, and the highest grid score so obtained was defined to be the “grid score of phase clusters” for that set of cells. The percentile _a_ could take values 5, 10, 15, 20, 25, 30. The bin width _b_ was the same in the X and Y directions and could take values 10, 15, 20, and 25 (units= μm). ##### Significance of the score The significance of the grid score of phase clusters for a set of cells was determined by comparing the
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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 among grid scores of 200 shuffled datasets. #### Spacing of phase clusters Here and below, unless otherwise indicated, “the autocorrelogram” now refers to the autocorrelogram computed with the optimal parameter values, yielding the best grid score as describe above (and therefore defining the “grid score of phase clusters”). The “spacing of phase clusters” was measured using the untransformed version of this autocorrelogram. Its “vertices” were defined to be its six local maxima nearest to the origin. The spacing in Figure S4 of the autocorrelogram nearest to the origin, excluding reflections; denote the corresponding vectors (pointing from the origin to the vertices) by v1 and v2, with v1 the shortest and v2 the second shortest. We now apply the following transformations to v1 and v2 together, in sequence (1) the unique rotation and scaling transformations sending v1 to (1,0); (2) the transformation v2 → -v2, if the _y_-coordinate of v2 is negative (i.e., v2 is below the x axis); (3) v2 → v2+ (1,0), if the _x-_ coordinate of v2 is negative. After these transformations, v2 now has _x_-coordinate between 0 and 0.5, and
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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 the phase maps. #### Folding triangle analysis (Figures S3 We applied a method that we called a “folding triangle analysis” to reveal how the physical arrangement of grid phases in the brain neighborhood matched their phase arrangement in the unit-rhombus. Only the 29 sets of co-modular cells with at least five local maxima in the autocorrelation of phase cluster lattice (Figure S4 in a set of co-modular cells, the cells within a certain distance (r) surrounding it were identified. The center cell and its surrounding cells were called a “cell group.” The parameter r was determined based on the half minimal spacing (s/2) of the spatial autocorrelogram of phase clusters of the set of co-modular cells. If s/2 was shorter than 50μm, r was set to s/2. Otherwise, r was set to 50μm. Only cell groups with at least four cells were used to calculate folding scores. Non-overlapping triangles (brain triangles) were drawn between cells in the same cell group using the Delaunay triangulation algorithm. The resulting assignment of cells to triangles was held fixed in the computation of a folding score from cell phases, as follows. ##### Triangulation in phase
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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 phases and the center cell phase: i.e., their phase locations could be shifted in order to ensure the shortest distances to the center cell phase (as mentioned above in “Phase distances of grid cells” and Figure S3 using the same assignment as in real brain space. The folding score of a cell group was calculated based on the phase triangles. It was the ratio of the area of the smallest polygon that contains all the triangles to the sum of the areas of all the individual triangles. If the phase arrangement matches the cell arrangement in the brain, the folding score should be close to 1 because the phase triangles are relatively non-overlapping. Otherwise, the folding score should be much smaller than 1. The folding score of a set of co-modular cells was the mean folding scores of all cell groups in the set. Many cells were involved in more than one cell group. ##### Comparing to shuffles One shuffle of a set of co-modular cells was generated by randomly permuting the phases within each cell group while maintaining their physical locations. The folding score of this shuffle was the mean folding score of all permuted cell groups in the set. 100 shuffles were generated for each set to obtain 100 shuffled folding scores. #### Global lattice fit analysis ([Figures S4]( The global lattice analysis was designed to quantify the global, continuous layout of the 2D grid phases across each field of
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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 directly of the vertices (local maxima) in the autocorrelogram as vectors; we mean the 2D vectors pointing from the origin to the vertices. More specifically, recall that the 2D phases in the unit-rhombus can be described as vectors H⋅ρ, for elements ρ∈0,1 2 (the unit square) and H=[1 1 2 0 3 2]. Then a choice of vectors v 1,v 2, along with a shift s∈0,1 2, determines a linear lattice fit as follows. Define the 2×2 matrix V=[v 1 v 2]. Then for each cell location x in brain coordinates, the vector(Equation 1)H⋅m o d(V−1⋅x+s,1)defines an anatomical phase of a cell in the unit-rhombus. We quantified the quality of the mapping in
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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 described in that section). However, to improve robustness, we now additionally rejected parameter values, regardless of grid score, if the number of autocorrelogram vertices produced by those parameter values was less than the maximum number of vertices produced across all parameter values for the same set of cells (usually, six vertices). Finally, at the end of this process, we had an autocorrelogram with some number of vertices (local maxima). Note that by design, the autocorrelograms coming from the real data each had six vertices; however, by chance, some shuffled autocorrelograms (see “Comparing to shuffles preserving local structure of phase map”) could have fewer vertices. As long as there are at least four vertices, one can perform the following analyses. In our analysis, since the autocorrelograms of shuffles were very similar to that of real data (locality preserved shuffles, see “Comparing to shuffles preserving local structure of phase map”), we did not observe any shuffle with less than four vertices. * 2)Let P denote the set of vectors corresponding to the vertices of the autocorrelogram from 1). The set P is initially symmetric about 0 (i.e., it consists of pairs p,−p), so we removed one vector from each pair p,−p from P before the next step (if not, the next step would result in redundant fits). * 3)For each choice of 2 distinct vertices p 1,p
{ "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 their negatives. Finally, the candidate vector pairs to define _V_ in Equation 1 consisted of all pairs v 1,v 2 of vectors in E such that L p 1,p 2=L v 1,v 2 (technically, these are the pairs v 1,v 2∈E such that the matrix [v 1 v 2]−1⋅[p˜1 p˜2] has determinant= ±1; there are 40 such pairs). * 4)For every candidate lattice vector pair v 1,v 2, we searched over a set of shifts, parameterized by the vector s in the unit square, with the coordinates of s ranging independently over [0,0.2,0.4,0.6,0.8]. For every complete choice of parameters v 1,v 2,s, we compared the proposed phase map in the brain (“anatomical phases”) to the true phases in the phase rhombus (“spatial tuning phases”) as described in Equation (1) In order to test for global structure in the phase pattern, above and beyond the structure of local phase clustering,
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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 the unit-rhombus into Ndisjoint sets using MATLAB’s built-in hierarchical clustering, with the distance between phases given by their distance in the unit-rhombus, and the distance between clusters given by the average pairwise distance between cluster elements. We took N=5 for all co-modular cell sets. * 2)For each phase cluster t constructed in 1), we defined its center c to be the center of mass of the phases in that cluster in the unit-rhombus (which is well-defined when the phases are sufficiently close to each other). A shuffle consisted of a permutation of the N clusters. To permute cluster t 1 to cluster t 2, we translated the phases in t 1 by the unique translation of the rhombus sending cluster center c 1 to c 2 (Figure S4For some shuffles, the non-robustness of the vertex finding algorithm led to vertices which corresponded to highly skewed, unrealistic lattices. To avoid this phenomenon, we rejected shuffles for which the best fit lattice failed to pass a number of
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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 maximum norm (here, |x| denotes the Euclidean norm for a 2d vector _x_). All these criteria were meant to ensure that the shuffles were not too different than the data (recall that the shuffle test was meant to preserve the autocorrelation as much as possible). #### Estimating the noise of phase lattice map ##### Modeling phase lattice map with noise (Figure S5 and their lengths ranged between 0 and 3r/2 so that the average vector length
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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 cell pairs should have spatial fields that occur one after another; this sequential activation can repeat many times as multiple fields are crossed for each cell. Specifically, spatial fields were identified based on the p value, as described
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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: the density distribution of 1D spacings of all co-modular grid 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 cells. To ensure that two fields were truly sequential fields elicited within a short distance on the track, we set S= 0.4× MS. Only cell pairs with at least two pairs of sequential fields were considered to be potential sequential cell pairs and used in the following analysis. For these cell pairs, if most fields of cell A occurred before those of cell B, we defined cell A as an upstream cell of cell B, and vice versa, cell B as a downstream cell of cell A. ##### Consistency of vector directions of sequential cell pairs As illustrated in the top panel of Figure 6, short vectors were generated between sequential cell pairs within the anatomical distance D, which was the minimal length of lattice vectors used to generate the best fit anatomical phase lattice for these cells (as described in “Global lattice fit analysis”). The length of the vector between a sequential cell pair was set to 1 and its direction was set from the upstream cell to the downstream cell. The resultant vector length of all these vectors was used to quantify the consistency
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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 as the imaging FOV was generated so that the areas of these co-active cells were set to 1 and other pixels were set to 0. An autocorrelogram of this matrix was generated using the MATLAB function “normxcorr2.”
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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 lattice to that predicted by the autocorrelogram of phase clusters (as defined in “Grid score of phase clusters”), a correlation coefficient was calculated between the averaged autocorrelogram of co-active cells (AA) and the autocorrelogram of phase clusters. The AA was binned using the same bin width used for the autocorrelogram of phase clusters. ##### Comparing to shuffles The shuffles were generated by randomly permuting calcium responses among all co-modular cells in the FOV. A new averaged autocorrelogram of co-active cells was similarly generated and its correlation coefficient with the autocorrelogram of phase clusters was calculated. 50 shuffles were used for each group of co-modular cells in a FOV. #### Missed-field pattern of cells in phase clusters ##### Identifying cells in different phase clusters The identification of cells in anatomically-separated phase clusters was determined by two parameters: x and y. The parameter x was a percentile of phases distances of all analyzed co-modular cells in the same FOV and served as the phase similarity threshold for defining phase clusters in this analysis. The parameter y was a physical distance, and it served as the distance threshold to separate cells into different phase clusters. The values of x and y were determined as follows: (1) for a set of co-modular cells, the phase cluster autocorrelogram of which contained at least five local maxima and therefore the grid score of the autocorrelogram was calculated, x was set equal to the phase distance percentile parameter “a” that generated the highest grid score (see “Grid score of
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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, respectively. The phase clusters were identified on a cell-by-cell basis as follows. For a cell A, we first identified the n cells with their phase distances to A no longer than x percentile of all phase distances (so that the total number of cells in the group was n+1, including cell A). The COMs of these cells were then clustered n-1 times using the k-means algorithm, as the number of clusters was varied from 2 to n, producing a sequence of n-1 sets of clusters for a cell group containing n+1 cells. For each set of clusters, we calculated two parameters: (1) p: the maximal pairwise physical distances between cells in the same clusters, and (2) c: the minimal pairwise physical distances between the COMs of the clusters. We only kept the sets of clusters, for which p ≤ y (cells in the same clusters were close enough) and c ≥ y (cells in different clusters were far enough). Among the sets of clusters satisfying these criteria, we determined the optimal set of clusters using the MATLAB function “evalclusters” with the Calinski-Harabasz clustering criterion. In this way, the group of n+1 cells was assigned to N anatomically separated clusters, for which N ≤ n. This procedure was repeated for each cell so that each cell could select one set of phase clusters under the criteria above. Some cells
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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 was calculated as the mean of missed-field patterns across the cell’s spatial fields, which were identified based on the p value of the ΔF/F, as described above. In turn, the missed-field pattern of an individual spatial field was defined to be a binary vector with one entry per run, calculated as follows: first, run-by-run significant transients were identified at 1% false positive rate across the whole session, as described previously. The coordinate of a given spatial field’s missed-field pattern for a given run was set to 1 if that run had a significant transient with its peak located within the field; otherwise it was set to 0. This missed-field patterns, which were vectors composed of 0 s and 1 s, were similarly generated for all spatial fields. These vectors were further averaged as the missed-field pattern of the cell. ##### Missed-field correlation of cells in phase clusters The missed-field correlation of a pair of cells was calculated as the Pearson’s correlation coefficient of their missed-field patterns. The correlations of cell pairs in the same phase cluster (intra-cluster correlation) and in different phase clusters (inter-cluster correlation) were calculated. Particularly, in [Figure 7]( the averaged correlations of cell pairs in the same and different phase cluster were termed as the “inter” and “intra” cluster correlations, respectively. #### Tetrode data analysis ##### Classification of grid cells based
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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 outer radii were used from the inner radius+4 bins to the size of the autocorrelation-4 bins in steps of 1 bin. For each of these outer radii, an annulus was defined between the inner and the outer radius. We then computed the Person’s correlation between each of these annuli and its rotation in 30 degree intervals from 30 to 150 degrees. For each annulus we calculated the difference between the maximum of the 60 and 120 degree rotation correlation, and the minimum of the 30, 90, and 150 degree correlations. The grid score was defined as the maximum of all of these differences across all annuli. A grid cell was defined to be a cell with a grid score exceeding a certain threshold, which was generated from shuffled grid scores as follows. For each cell, its shuffles were generated with spike time circularly permuted by a random amount of time chosen between 0.05× N samples and 0.95× N samples, where N samples was the number of samples in the whole recording. Grid score was similarly calculated for each shuffle. 400 shuffles were generated for each cell and the distribution of shuffled grid scores of all cells combined were
{ "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,where _n_ is the number of grid cells, _a_= 10, and _b_= 105. The number of modules _p_ was defined to be the number of peaks in the KSD estimate. The assignment of cells to grid modules was then determined via k-means clustering of the grid spacings, with the number of clusters (modules) set to _p_, and the number of iterations for the k-means algorithm set to 300. ##### 2D phase distances The ground truth datasets for the comparison of 1D and 2D grid phases were taken from a previous study (Yoon et al., 2013 distance between the phases of the two cells. ##### Cue cells based on 1D firing rates ([Figures S1]( The cue cells among tetrode-recorded cells were identified by their cue scores, which were calculated as for imaged cells, with firing rates replacing calcium response traces. To calculate the cue score threshold, 400 shuffles were generated for each cell by circularly permuting their spikes, using the same algorithm as for the p value analysis. The mean firing rate and cue score of each shuffled response
{ "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 firing rates were computed as in (Domnisoru et al., 2013 was calculated (see ‘In- and out-of-field periods’ for calcium response analysis). Out-of-field periods were defined as two or more adjacent bins for which 1-p value ≤ 0.05. Candidate in-field periods were defined as three or more adjacent bins (except at the beginning and end of the track, where we only required two or more adjacent bins) all satisfying 1-p value ≥ 0.85. The candidate in-field periods were then extended, by no more than one bin on each side, if those adjacent bins satisfied 1-p value ≥ 0.70. Finally, firing fields were defined as the candidate in-field periods with spikes on at least 20% of all runs (the remaining candidate in-field periods were left unassigned). As with the calcium data, the noise level and drift in spatial tuning varied across datasets. Therefore, slightly different 1-p value threshold for in-field-periods and subsets of the traversals with higher run-by-run consistency were selected in different datasets based on their stability scores so that the identified spatial field matched manual inspection. The same 1-p value threshold was used for all cells in the same animal recorded on the same virtual track. We discard cells, whose identified fields did not match manual inspection.
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##### 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 was defined to be the minimum adjacent field spacing. The 1D field widths of a cell were defined to be the maximum field width across all its spatial fields. ##### Grid modules based on 1D responses The assignments (Figures S2 were performed as for imaged grid cells, based on the clustering of four parameters (1D field width and spacing on two different tracks, as described above) via k-means. ##### 1D phase distances of co-modular grid cells The 1D phase distances of tetrode-recorded co-modular grid cells (Figures S3 and the Simons Foundation (Simons Collaboration on the Global Brain [SCGB]). ### Author Contributions Y.G. and D.W.T. conceived the project and wrote the manuscript with contributions from I.R.F., C.D., and S.L. Y.G. performed two-photon experiments. Y.G. and S.L. applied the folding triangle analysis and global lattice fit developed by S.L. Y.G. performed all other analyses. A.A.K. performed tetrode experiments. Y.G. and C.D. developed the brute-force grid cell fitter. J.L.G. developed
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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. 987-994 Crossref, p. e1000291 Crossref, pp. 2727-2733 View in Scopus, pp. 318-324 [Crossref]( in Scopus]( Scholar]( 8. [Dana et al., 2014]( Dana, T.W. Chen, A. Hu, B.C. Shields, C. Guo, L.L. Looger,
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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]( in Scopus]( Scholar]( 17. [Harvey
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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, …, Daniel A.Dombeck [View PDF]( * ### [Direct Medial Entorhinal Cortex Input to Hippocampal CA1 Is Crucial for Extended Quiet
{ "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, Issue 6, 2020, pp. 1586-1599.e10 Nick T.M.Robinson, …, Michael Häusser View PDF Cell Reports, Volume 43, Issue 7, 2024, Article 114470 Olivier Dubanet, Michael J.Higley View PDF Neuron, Volume 109, Issue 18, 2021, pp. 2967-2980.e11 Isabel I.C.Low, …, Lisa M.Giocomo [View PDF]( Show 3 more articles Article Metrics --------------- ### Citations * Citation Indexes 60 ### Captures * Mendeley Readers 343 ### Social Media * Shares, Likes & Comments 57 ![Image 26: PlumX Metrics Logo]( details]( [![Image 27: Elsevier logo with wordmark]( * [About ScienceDirect]( * [Remote access]( * [Advertise]( * [Contact and support]( * [Terms and conditions]( * [Privacy policy]( Cookies are used by this site. Cookie Settings All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply. [![Image 28: RELX group home page]( ![Image 29: Company Logo](
{ "page_id": null, "source": 7370, "title": "from dpo" }
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{ "page_id": null, "source": 7370, "title": "from dpo" }