Fig. 1: Finding 3-D toroidal representation of grid cell ensemble with conjunctive head direction tuning.
From: Uncovering 2-D toroidal representations in grid cell ensemble activity during 1-D behavior
a Gardner et al. recorded 483 grid cells4 in rat `R'. Illustration of single grid cell tuning in an OF arena (upper) with directional preference (lower). b Embedding the (randomly sampled) whole grid cell population activity (n = 483 cells, top row) and (preprocessed) identified ensemble activity (n = 75 cells, bottom row) into three dimensions using CEBRA20 with cosine (left) and Euclidean (middle) distance and UMAP16 (right, cosine metric). Colored by simultaneously recorded head direction (HD) angle (as indicated by color wheel). c Correlation distance matrix (one minus Pearson correlation of the time-lagged cross-correlation of the firing rate activity between all recorded grid cells, values indicated by color bar), sorted by cluster indices from hierarchical clustering. Largest ensemble (‘M’, n = 75 cells) indicated. d The spike trains for all grid cells may be represented by a matrix X of size N × T (neurons × time; left). Gray bars indicating time of spike. The ensemble activity \({{{{{{{\bf{X}}}}}}}}{\prime}\) (upper right) is a subset of X (n < N neurons) and is further reduced by downsampling the number of activity states (bottom right, X″, t < T time points). e Barcode from applying PH to ensemble activity after temporal downsampling (as in d), with bars corresponding to the lifetime of 0- and 1-dimensional features (H0 and H1). Three circular features are identified (ϕ1−3, arrows). Shaded region indicates 99th percentile of shuffled distribution. f Comparing decoding of the circular features in (e) obtained by cohomological coordinatization37, with recorded physical variables, reveals encoding of head direction (HD, top) and spatial position (bottom, color indicating cosine value of specified decoding, ‘dec’). g Eight examples (rows) of normalized single-cell firing rate tuning to spatial position (left column, value indicated by color bar), head direction (middle) and the decoded 3-D toroidal state space (right, each axis of the cube corresponds to a circular coordinate as in f and boundaries are continuous across opposite sides of the cube). Source data are provided as a Source Data file.
