Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Pearson’s correlations between a representative set of movement kinematics measured with the 2-D and 3-D motion capture system. The Pearson’s correlations were strong to very strong for crossing height left toe (r = 0.873, p < 0.001), left foot placement after crossing (r = 0.996, p < 0.001), crossing velocity left toe (r = 0.979, p < 0.001), and single support time left foot (r = 0.851, p < 0.001). All subjects are included in the plots (n = 47).

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques:

Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Pearson’s correlations (r) between the kinematic outcomes of the 2-D and 3-D motion capture system.

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques:

Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Bland-Altman plots that show the agreement between a representative set of movement kinematics measured with the 2-D and 3-D motion capture system. All subjects are included in the plots (n = 47). In each graph, the solid black line represents the mean bias and the dotted lines represent the 95% limits of agreement.

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques:

Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Bland-Altman analysis between the kinematic outcomes of the 2-D and 3-D motion capture system.

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques:

Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Test-retest reliability of the kinematic outcomes of the 2-D and 3-D motion capture system.

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques:

Journal: Scientific Reports

Article Title: A low-cost 2-D video system can accurately and reliably assess adaptive gait kinematics in healthy and low vision subjects

doi: 10.1038/s41598-019-54913-5

Figure Lengend Snippet: Schematic representation of the experimental setup. The grey markers on the shoes represent the reflective markers of the 3-D motion capture system. The markers of the 2-D motion capture system are red (right shoe) and green (left shoe). Note, the 3-D marker on the left toe is obstructed by the green 2-D marker.

Article Snippet: The walkway was surrounded by eight 3-D motion capture cameras (Vicon, Oxford Metrics Ltd).

Techniques: Marker

Journal: bioRxiv

Article Title: Toroidal topology of population activity in grid cells

doi: 10.1101/2021.02.25.432776

Figure Lengend Snippet: a. Cohomology can help differentiate topological spaces such as the union of three discs (upper left), a circle (upper right), a sphere (lower left) and a torus (lower right) by counting the number of topological holes in different dimensions. The Betti number of the space ( β ) is here given as a list of the number of holes in the first three dimensions. A disc has a 0-D hole (a connected component); a circle additionally has a 1-D hole; a (hollow) sphere is a connected component and has a 2-D hole (a cavity); a torus is a connected component with two 1-D holes (illustrated with red circles) and one 2-D hole (a cavity in the interior of the torus). b. Persistent cohomology tracks the lifetime of topological holes in spaces associated with point clouds. Top: The radius of balls centred at each data point in the point cloud is continuously increased (left to right). The union of the balls forms a space with possible holes. The lifetime of a hole during expansion of the radius is defined as the radial interval from when the hole first appears until it is filled in. Note the short lifetime of the hole marked with a red circle and the long lifetime of the hole indicated with a yellow circle. Second and third row: The lifetime of each hole of dimension zero ( H° ) and one ( H 1 ) in the example in the top row is indicated by the length of a bar (in green) in the barcode diagram. Two 1-D holes are detected: the first bar, corresponding to the red hole in the top row, is short and regarded as noise, while the second, corresponding to the yellow hole, is substantially longer and captures the prominent topology of the point cloud. c., d. Barcodes indicate toroidal topology of grid-cell population activity during open field (OF) foraging ( c , module R2; d , module R3). Left of each panel: colour-coded firing rate maps for two representative grid cells in each module. Colour coding indicated by scale bar. Note difference in grid scale between R2 and R3. Right of each panel: Barcodes as in b . Arrows point to the four most prominent bars. These are substantially longer than a distribution of shuffled data from the same recording. The longest lifetime for each dimension in analyses of 1000 shuffles of the data is indicated by the width of the shaded orange region, with the left edge starting at the same value as each of the original bars. Note that in the original data, there is one long bar in dimension 0, two in dimension 1 and one in dimension 2, indicating Betti numbers equal to those of a torus. Only the 30 longest bars are shown for each dimension. e., f. Toroidal structure in an environment of higher topological complexity than the open field: the “wagon wheel” track (WW), as previously seen in . Data are from the same grid modules as in c and d . Although grid cells had lower periodicity in WW than OF , barcode diagrams are similar to those of open field data, suggesting preserved toroidal structure during distortions of the grid pattern.

Article Snippet: 3-D motion capture (OptiTrack Flex 13 cameras and Motive recording software) was used to track the animal’s head position and orientation, by attaching a set of five retroreflective markers to implant during recordings.

Techniques: Activity Assay

Journal: bioRxiv

Article Title: Toroidal topology of population activity in grid cells

doi: 10.1101/2021.02.25.432776

Figure Lengend Snippet: a. Firing rates of 149 grid cells co-recorded from the same module when a rat was foraging in an open-field arena (rat ‘R’ day 1, module 2; ). Firing rate is colour-coded as a function of the rat’s position in the arena. Colour coding is indicated by scale bar. Each square shows the activity map for one cell. Maximum rates range from 0.2 to 35.0 Hz. Cells are ranked in order of spatial information content (highest spatial information at top left, lowest at bottom right). Time periods when the animal’s speed was below 2.5 cm/s are excluded from all plots in this figure. b. Nonlinear dimensionality reduction reveals torus-like structure in population activity of a single grid module (same 149 cells as in a ). PCA was performed to identify the six principal components with the highest variance explained . These components were further reduced to three dimensions by UMAP, yielding a 3-D nonlinear embedding, which is plotted here (3 different views of the same point cloud). Each dot of each point cloud represents the population activity state at a single time point. Dots are coloured by the value of the first principal component, to aid 3-D visualization. The bold coloured line shows a 5-s example trajectory, demonstrating smooth movement over the toroidal manifold. The corresponding behavioural trajectory of the animal in the open field is shown to the right. In all plots, colour indicates elapsed time (scale bar). c. Toroidal positions of spikes from four grid cells belonging to the module shown in a . Each panel shows the same 3-D point cloud of population activity states as in b , together with black dots which indicate the population state at times when the cell fired. Insets below each point cloud show (left) the 2-D firing locations of the cell in the open field (black dots superimposed on trajectory in grey), (middle) the colour-coded firing rate map of the cell in the open field, and (right) a similarly colour-coded autocorrelogram of the rate map. Colour scales of rate maps range from zero (dark blue) to the maximum value of the rate map for each cell. Colour scales for autocorrelograms range from −1 to +1. Text above rate maps indicates the maximum firing rate of each cell. Text above autocorrelograms indicates the grid score of the autocorrelogram. d. Same as c , plotted for the same four cells in a different behavioural session when the animal ran on an elevated, wheel-shaped track (“wagon-wheel track”).

Article Snippet: 3-D motion capture (OptiTrack Flex 13 cameras and Motive recording software) was used to track the animal’s head position and orientation, by attaching a set of five retroreflective markers to implant during recordings.

Techniques: Activity Assay

Journal: bioRxiv

Article Title: Toroidal topology of population activity in grid cells

doi: 10.1101/2021.02.25.432776

Figure Lengend Snippet: a. Flow diagram showing method for extracting low-dimensional embeddings of neural activity. The animal forages in an open-field arena ( al ), while spikes from 149 grid cells shown in are recorded ( a2 ; cells are ordered arbitrarily). A 5-second example behavioural trajectory is highlighted in al , with colour indicating elapsed time. The spike trains are binned in time ( N bins) and then smoothed and normalized, yielding a matrix of N- dimensional population activity vectors ( a3 ). Linear dimensionality reduction (PCA) is applied to the matrix shown in a3 , yielding a 6-dimensional linear embedding of the N -dimensional neural activity ( a4 ). The six principal components are then passed through a second, nonlinear, dimensionality reduction step by UMAP, which generates a three-dimensional nonlinear embedding ( a6.i ) which allows the toroidal structure to be visualized. UMAP consists of two steps: first, a fuzzy topological graph representation is constructed (i.e. a “Uniform Manifold Approximation” - UMA) using a distance metric in the high-dimensional space ( a5 ); second, to obtain the lower-dimensional projection (P), the coordinates of corresponding points in fewer dimensions are optimized to have a similar fuzzy topological representation. In the persistence analysis, we applied persistent cohomology to the fuzzy topological representation of the high-dimensional point cloud ( a6.ii ) and subsequently used cohomological decoding to obtain a two-dimensional projection of the original N -dimensional point cloud (left, showing a 5-second snippet). This may subsequently be visualized by using a three-dimensional toroidal projection of the two-dimensional angular values (right), as shown in . The point cloud is coloured according to each angular coordinate, whose direction is indicated by a red arrow. b. Variance explained by the first 30 principal components (PCs) after applying PCA to the N -dimensional neural activity shown in a3 . Note the particularly high values for the first six components. c. Same as b but showing cumulative explained variance. Note the pronounced ‘knee’ in the curve at the sixth PC. d. The first six PCs contain a population grid-like representation. Left: Each panel shows the mean value of one PC as a function of the animal’s position in the OF (top) or WW (bottom). PC value is colour-coded as indicated by the scale bar. PCs are ordered and numbered in red from top-left to bottom-right for each environment (OF, WW). Note the presence of grid-like structure in the first six PCs, which has a similar scale and orientation as individual cells in the grid module. These six grid-like PCs correspond to the set with the highest explained variance in b , each seemingly explaining a similar amount. Right (for OF and WW): Mapping the PCs onto the toroidal sheet given by the cohomological decoding (as in a6.iii ), we see that each of these components completes one cycle in the fundamental tile. PC values are indicated by the scale bar. These components seem to reflect the proposition from CAN models that the isometric activity bump is translation-invariant with respect to the hexagonal symmetry of the neural sheet, which would suggest that the six first PCA components should correspond to the six fundamental Fourier modes of a hexagonal lattice (see Supplementary Methods for theoretical explanation of the six-dimensionality). e. The percentage of subsamples of R2 (resampled randomly 1000 times per number of cells) for which toroidal structure was detected in the parameterization given by the two most persistent 1-D bars in the barcode (as in ). Note that approximately 60 cells were needed for the probability of detecting toroidal structure exceed 50%.

Article Snippet: 3-D motion capture (OptiTrack Flex 13 cameras and Motive recording software) was used to track the animal’s head position and orientation, by attaching a set of five retroreflective markers to implant during recordings.

Techniques: Activity Assay, Construct

Journal: bioRxiv

Article Title: Toroidal topology of population activity in grid cells

doi: 10.1101/2021.02.25.432776

Figure Lengend Snippet: a. Individual grid cells have distinct firing fields on an inferred torus. Toroidal coordinates for population activity vectors may be decoded by computing time-varying, circular coordinates from the two significant 1D holes (red circles) identified by barcodes in the persistent cohomology analysis ( ; ). The Cartesian product of these circular coordinates localizes neural activity in the two-dimensional toroidal state space . Left: 3-D embedding of the toroidal state space displaying mean firing rate of a single grid cell as a function of toroidal position, colour-coded as indicated by scale bar. Middle: A 2-D torus may be formed by gluing opposite sides of a rhombus. Right: Arranging multiple rhombi to tesselate a 2-D surface reveals a grid-like pattern in the activity of the grid cell, akin to its spatial firing. b. Tuning of representative grid cells from module R2 to coordinates on the inferred torus. The panel shows every 10 th cell in a distribution where the cells are ranked according to their spatial information content (total of 15 cells are shown; see for the full set of data). Each row of 4 plots corresponds to one cell. From left to right: Rate maps showing firing rates across the spatial environment (open field, OF; wagon-wheel track, WW) and on the inferred torus (toroidal OF, toroidal WW). Firing rates are colour-coded as in a . The toroidal parametrizations were aligned to common axes before creating the rate maps. Note that firing locations on the torus are consistently preserved between OF and WW. c. Toroidal information rate and explained deviance in OF (left half) and WW (right half). Left of each: Scatterplots showing information rate for position on the inferred torus (T) vs position in the physical environment (S). The torus was inferred from WW for OF and from OF for WW; random noise was added to the T coordinates to compensate for tracking error in the S coordinates. Dots show individual cells; modules are colour-coded (inset). Right of each half: Mean information rate (±S.E.M) for position in S and T coordinates for each module. Note that 5/6 modules during OF and 4/6 during WW show higher information for the torus than for position. d. Scatterplots comparing the goodness-of-fit of two GLM models, based either on position in the spatial environment (S) or on the torus (T). We fitted a Poisson GLM model to the spike count using 3-fold cross validation. Goodness-of-fit is expressed as explained deviance, an R 2 -statistic whose values range from 0 to 1, with higher values indicating better fit. The two scatter plots show data from OF (left) and WW (right), respectively. Toroidal coordinates were processed as in c. Symbols as in c . Line diagrams to the right: Mean deviance explained for each module (±S.E.M.). In 6/6 modules from OF and 5/6 from WW, the explained deviance was larger for the toroidal regressor than the spatial regressor. e. Distribution of centres-of-mass for grid-cell firing fields on the torus (all cells of R2; OF, red; WW, green). Blue lines connect red and green peaks of the same cell (occasionally extending over the periodic boundaries of the toroidal sheet). Note the proximity of red-green pairs in the original data (after separate alignment of the two toroidal parametrizations “Separate” in left panel, or using the parametrization in OF for decoding in both OF and WW, “Common”, in middle panel) compared to shuffled data where rate maps are reordered randomly (right panel). f. Left: Cumulative frequency distribution showing distances between field centres on the torus for grid cells in module R2, in OF and WW (as in e; red: separate parametrizations “Separate”; grey: same parametrizations “Common”). Note that there are two grey curves (one parametrization for each environment). Values range between 0 and · 180° (the maximum possible distance between two points on the toroidal sheet). Green curves show distribution from 1,000 shuffled versions of the data. Right: mean distance between field centres (±S.E.M.) for each module (modules are colour-coded; Or, original data; Sh, shuffled data). Note substantially shorter distances for the original spike trains. g. Left: Cumulative frequency distribution showing Pearson correlations between the toroidal rate maps in OF and WW for grid cells in module R2 (colour-coding as in f ). Right: Mean correlations (±S.E.M.) for each module; symbols as in f . Note substantially larger correlations for the original spike trains.

Article Snippet: 3-D motion capture (OptiTrack Flex 13 cameras and Motive recording software) was used to track the animal’s head position and orientation, by attaching a set of five retroreflective markers to implant during recordings.

Techniques: Activity Assay