doi: 10.1038/s41593-021-00873-x.
Epub 2021 Jun 24.
Cerebellar granule cell axons support high-dimensional representations
Frederic Lanore #
1
2
, N Alex Cayco-Gajic #
1
3
, Harsha Gurnani
1
, Diccon Coyle
1
, R Angus Silver
4
Affiliations
- PMID: 34168340
- PMCID: PMC7611462
- DOI: 10.1038/s41593-021-00873-x
Item in Clipboard
Cerebellar granule cell axons support high-dimensional representations
Frederic Lanore et al.
Nat Neurosci.
2021 Aug.
Display options
Format
Display options
Format
Abstract
In classical theories of cerebellar cortex, high-dimensional sensorimotor representations are used to separate neuronal activity patterns, improving associative learning and motor performance. Recent experimental studies suggest that cerebellar granule cell (GrC) population activity is low-dimensional. To examine sensorimotor representations from the point of view of downstream Purkinje cell 'decoders', we used three-dimensional acousto-optic lens two-photon microscopy to record from hundreds of GrC axons. Here we show that GrC axon population activity is high dimensional and distributed with little fine-scale spatial structure during spontaneous behaviors. Moreover, distinct behavioral states are represented along orthogonal dimensions in neuronal activity space. These results suggest that the cerebellar cortex supports high-dimensional representations and segregates behavioral state-dependent computations into orthogonal subspaces, as reported in the neocortex. Our findings match the predictions of cerebellar pattern separation theories and suggest that the cerebellum and neocortex use population codes with common features, despite their vastly different circuit structures.
© 2021. The Author(s), under exclusive licence to Springer Nature America, Inc.
Conflict of interest statement
Figures
(a) Schematic representing a dorsal view of the cerebellum. The black circle represents the 5 mm cranial window above Crus I. Colored blobs show the approximate location of the virus injection and GCaMP6f expression for the animals in this study. (b) Top view of a cranial window above Crus I. The green channel (left) shows expression of GCaMP6f in lobule Crus I. Green fluorescence is widespread due to the spatial extent of parallel fiber projections. The red channel (right), shows a clump of retrobeads at the injection site (arrow). (c) Confocal tile scanning of a coronal section of Crus I where granule cells (GrCs) were transfected with GCaMP6f. Note the absence of labelled cell bodies in the molecular and Purkinje cell layers. (d) Confocal image with a smaller field of view to show GCaMP6f expression in GrC somata and axons. Labels: Cr. 1: Crus1 lobule; Cr. 2: Crus2; lob. VI: cerebellar lobule VI in the vermis, Simp.: simplex lobule, PM: paramedian lobule, ML: molecular layer, PC: Purkinje cell, GrCL: GrC layer.
(a) Strings of bright varicosities from active axons were traced by hand to obtain orientations of parallel fiber segments. Inset shows the histogram of the angle of individual parallel fiber segments from the average parallel fiber orientation (n = 13, N = 5). White arrow indicates average parallel fiber orientation for this experiment, and purple the acceptance angle for parallel fiber identification (two standard deviations of the distribution in the inset). (b) Examples of candidate varicosity groupings that pass (left, green box, each side 13.7 μm) and fail (right, red box) the first grouping criterion. Varicosities indicated by yellow contours. Title indicates angle between candidate parallel fiber given by linear fit (dotted white line) and the average parallel fiber direction for that experiment (white arrow). (c) Example histogram of correlation coefficient for pairs of varicosities in different patches, used for the second grouping criterion. Dotted line indicates the threshold correlation (95th percentile) for this experiment. (d-f) Example of correlated varicosities that pass the third grouping criterion. (d) Example activity of the two varicosities (r = 0.74). (e) Activity of varicosity 1 plotted against activity of varicosity 2 (grey). Blue line indicates fit from linear regression. Black circle indicates baseline activity distribution (95% confidence interval). Red line indicates vector v onto which activity is projected to calculate the linear deviation ratio for the third criterion. (f) Histogram of activity from (e) projected onto v (grey histogram), and analytically calculated distribution of the baseline distribution projected onto v (orange curve). The ratio of the variances of these distributions is used for the third criterion (linear deviation ratio = 1.03). (g-i) Same as d-f for a pair of varicosities that fail the third grouping criterion (r = 0.69, linear deviation ratio = 3.12). Red arrows in (g) indicate transients that are missed in one varicosity. Red arrow in (i) shows the large tail of the distribution.
(a) Left: Example traces of different behavioral variables: whisker set point (WSP), whisking amplitude (WA), wheel motion index (WMI), and locomotion speed (LS). Right: Histograms of correlations of parallel fiber Ca2+ activity (ΔF/F) with WSP, WA, WMI and LS (n = 13, N = 5). Red and blue indicate parallel fibers that are positively or negatively correlated with each behavioral variable respectively (p < 0.05, two-sided shuffle test). Grey indicates parallel fibers that are not significantly correlated with that behavior. Pie charts reveal a similar fraction of positively modulated (PM, 58 – 67%), negatively modulated (NM, 16 – 22%) and non-modulated GrCs (13 – 20%) regardless of behavioral variable. (b) Correlation between all pairs of behavioral variables for each experiment (grey circles). Black bars indicate mean across experiments (p = 2.4 x 10-4 for all pairs of behavioral variables, two-sided Wilcoxon signed rank test, n = 13, N = 5). Error bars indicate s.e.m.
(a) ΔF/F traces of positively modulated (PM) and negatively modulated (NM) parallel fibers in grey (top) together with locomotion speed and bead fluorescence, from a single experiment aligned at locomotion onset. Bottom: Grey indicates individual traces and the black indicates the mean. (b) Example experiment showing temporal dispersion of parallel fiber activation during locomotion onsets. Top and middle panels show average ΔF/F (zscored) of PM and NM parallel fibers calculated over locomotion onsets. Locomotion onsets were randomly split into training (50%) and test (50%) data, and parallel fibers were sorted according to the time lag of their peak correlation (PM) or anticorrelation (NM) with locomotion speed during the training data. Bottom panels show average locomotion speed during training and test onsets. (c) Distribution of pairwise correlations for pairs of positively (black, top) and negatively (black, bottom) modulated parallel fibers during 1s interval surrounding locomotion onsets (n = 12, N = 5). Red and blue curves indicate distributions of correlations during random periods in the active state (for positively modulated and negatively modulated parallel fiber pairs, respectively). Arrowheads represent the means. (d) Relationship between correlations between putative axons at locomotion onsets as a function of inter-fiber distance, for positively modulated pairs (red), negatively modulated pairs (blue), and all pairs (grey; n = 12, N = 5). Shaded regions indicate s.e.m. Thick lines indicate double exponential fit to the data.
(a) Example of three non-modulated parallel fibers (top) compared to positively modulated and negatively modulated parallel fibers (same example shown in Fig. 1e for full experiment). Magenta/cyan indicates AS/QW. (b) Distribution of signal-to-noise ratios (SNRs; Methods) for all non-modulated parallel fibers (top), as well as positively modulated (centre) and negatively modulated parallel fibers (bottom) (n = 13, N = 5).
Histograms of changes in ΔF/F response during the AS relative to QW across all parallel fibers for all 13 experiments across 5 mice. Positively modulated (red) and negatively modulated (blue) parallel fibers, as well as parallel fibers which were not significantly modulated by behavioral state (grey). Pie charts indicate the proportion of each class across experiments.
(a) Schematic illustrating how distances between parallel fibers were calculated. Left: example of two patches with three parallel fibers, each with different numbers of varicosities. Black unidirectional arrow indicates average parallel fiber direction. To calculate the distance between parallel fibers, the position of the centre of its varicosities is projected onto the dimension orthogonal to the average fiber vector (red line). The XY distance (dXY) is the distance in the projected dimension. Right: Same schematic, rotated to show Z-dimension. The XYZ distance (dXYZ) is the distance in the projection plane (red). (b and c) Correlations between varicosities or putative axons as a function of inter-fiber distance, for positively modulated pairs (red), negatively modulated pairs (blue), and all pairs (grey; n = 13, N = 5). Shaded regions indicate s.e.m. Thick lines indicate double exponential fit to the data. (b) Correlations and XY distances (dXY) for ungrouped varicosities (within the same patch). Note similar trend to grouped data, except for stronger peak at small distances (< 2 μm) (c.f. Fig. 1c). (c) Correlations and XYZ distances (dXYZ) for putative axons across all patches.
Parallel fiber population activity visualized by plotting first three principal components. Each panel indicates a different mouse (N = 5 in combination with Fig. 3b). Color indicates projection along the quiet wakefulness (QW; cyan) to active state (AS; magenta) state dimension.
(a) Average cross-validated unexplained variance for locomotion speed based on the first principal component (PC), the first 10 PCs, and the optimal number of PCs. Each circle indicates a different experiment (n = 11, N = 5; two-sided Wilcoxon signed rank test). (b) Average cross-validated unexplained variance for locomotion speed based on the best parallel fiber (PF) for each recording and for lasso regression on the population activity (n = 11, N = 5; two-sided Wilcoxon signed rank test). (c) Range of optimal number of parallel fibers to minimize the cross-validated unexplained variance. Each marker represents a different experiment. (d) Correlation between the lasso regression coefficients of the optimal decoders for locomotion speed and for whisker set point, plotted against average decoder error (unexplained variance averaged for speed and whisker spoint) (two-sided Spearman correlation: r = - 0.73, p = 0.02; n =11, N = 5). For each decoder, regression coefficients were averaged over 10 random samples of training/test data. Error bars in a and b denote s.e.m.
We tested our procedure for estimating dimensionality in a simple model of random 60-dimensional representations in populations of 300 neurons corrupted with increasing levels of noise. Each black line represents the mean variance explained for a fixed standard deviation of the noise distribution. Shading represents s.e.m. across different random representations. Inset: linear relationship between lower bound of the dimensionality and the maximum variance explained.
(a) Schematic of the experimental configuration for the acousto-optic lens (AOL) 3D imaging showing head-fixed mouse on a wheel, along with high-speed camera to track whisker movement (left). Spatial arrangement of multiple simultaneously acquired imaging planes (‘patches’) within the imaging volume in relation to granule cell (GrC) axons (in green) and Purkinje cell dendritic tree (in grey) in the molecular layer with example of imaged patch showing varicosities expressing GCaMP6f (average fluorescence image; right). (b) Example of varicosity grouping (n=1, N=1 of n=13, N=5). Top: Correlation image of a patch (13.7 μm x 68.4 μm) with identified varicosities outlined in white dots. Greyscale indicates correlation with the fluorescence of neighbouring pixels. The colored outlines show examples of grouped varicosities per axon, with each color corresponding to one axon. Bottom: ΔF/F traces for each varicosity highlighted in color. (c) Matrix showing correlation between ΔF/F traces of varicosities in b. Colored bars on the side show the grouping into putative axons. The strongest correlations were between varicosities on the same putative axon. (d) Distribution of distances between varicosities grouped onto the same putative parallel fiber (n = 13, N = 5). The red arrow shows the mean intervaricosity distance. Black line and arrow indicate the range and mean intervaricosity distances as determined previously in fixed tissue with anatomical methods. The close match suggests our detection of varicosities and method of grouping into axons identifies the majority of boutons per active axon in the imaged patch. (e) Example of activity (ΔF/F) of 700 putative GrC axons (parallel fibers) in a single experiment, grouped into positively modulated (PM, red), negatively modulated (NM, blue), and non-modulated parallel fibers (non-M, grey). Bottom: Whisker set point (WSP; slow-frequency component of whisker angle) and locomotion speed.
(a) Example of behavioral state segmentation and parallel fiber responses. Top: time series of whisker set point (WSP) and locomotion speed labelled as periods of active state (AS, magenta), quiet wakefulness (QW state, cyan) or unclassified timepoints (black). Bottom: ΔF/F traces of parallel fibers that exhibited a significant increase or decrease during the AS, compared to QW (p < 0.05, two-sided shuffle test). (b) Histogram of changes in ΔF/F response during the AS relative to QW across all parallel fibers (n = 13, N = 5). Positively modulated (PM; red) and negatively modulated (NM; blue) parallel fibers, as well as axons which were not significantly modulated by behavioral state (grey). (c) Average pairwise correlation between parallel fiber activity as a function of the distance between axons (n = 13, N = 5), shown for positively modulated (red), negatively modulated (blue), and all parallel fibers (grey). Shading indicates s.e.m. and solid lines indicate double-exponential fits. (d) Within-group nearest-neighbor (NN) distances for positively modulated (red) and negatively modulated (blue) parallel fibers, and shuffle controls (black) (n = 13, N = 5).
(a) Schematic diagram illustrating possible overlapping (left) and separate (right) representations in neural activity space of the active state (AS, magenta) and quiet wakefulness (QW, cyan). (b) First three principle components (PCs) of parallel fiber population activity for a single experiment. Manifolds representing AS and QW and the transitions between them. Magenta to cyan color change indicates a continuous AS-QW scale for the state dimension (Methods). (c) Plot showing the average Euclidean distance between all pairs of neural activity patterns (ΔF/F) within the QW manifold (cyan; 4.8 ± 0.5; mean ± s.e.m.) within the AS manifold (magenta; 5.4 ± 0.5), or between the two manifolds (black; 6.9 ± 0.7). Each circle represents a different experiment (n = 13, N = 5; two tailed Wilcoxon signed rank test). (d) Schematic depicting quantification of the angle between the AS and QW subspaces (i.e., hyperplanes in which the AS and QW manifolds are embedded). (e) Example of null distribution obtained by calculating the angle between two halves of the data, after shuffling time for an individual experiment. Dashed line indicates the observed angle between AS and QW manifolds in the same experiment. (f) Plot showing angle between AS and QW manifolds, compared to the mean angle between random halves of the data after shuffling timepoints. Each circle indicates a different experiment (n = 13, N = 5; two-sided Wilcoxon signed rank test). (g) Angle between AS and QW manifolds (black) as increasing fractions of the most strongly positively and negatively modulated fibers are excluded. Schematic (right) depicts a distribution of the change of ΔF/F with positively modulated (PM, red), negatively modulated (NM, blue), and non-modulated (grey) parallel fibers listed (cf. Fig. 1b). Brown box indicates the parallel fibers analysed when the 70th percentile is excluded (two-sided Wilcoxon signed rank test with Bonferroni correction). The gray curve indicates the shuffle control, and dotted black curve indicates the random control, in which the same number of neurons are analysed, but randomly sampled across the distribution. Grey boxes at bottom show the number of experiments (n) and animals (N) analysed. Shading indicates s.e.m.
(a) Whisker set point (WSP; black) and first principal component (PC1; green) of parallel fiber population activity from a single experiment, together with binary representation of state. (b) WSP plotted against the first principal component (PC1) for the same experiment as (a). Color indicates a continuous active state (AS) to quiet wakefulness (QW) scale for the state dimension (Methods). (c) Correlation values between PC1 and different behavioral variables: binary state, or WSP over all time, during QW or AS. Each circle indicates a different experiment (n = 13, N = 5; two-sided Wilcoxon signed rank test). Error bars denote s.e.m.
(a) Whisker set point (WSP) during the same experiment shown in figure 4a and 4b over a different period. Measured WSP (black), and its reconstruction using linear regression over the best performing parallel fiber (grey), first 10 principal components (PCs) (orange), and first 100 PCs (brown). Reconstruction error for each case is indicated as root mean square error (RMSE). (b) Example of unexplained variance (cross-validated) for WSP (an assay of the error in decoding performance) as a function of the number of PCs used for linear regression (same experiment as in Figure 4a, 4b and 5a). Shading indicates s.e.m. over random draws of held-out data. (c) Plot of the average cross-validated unexplained variance for WSP based on the first PC, the first 10 PCs, and the optimal number of PCs. Each circle indicates a different experiment (n = 13, N = 5; two-sided Wilcoxon signed rank test). (d) Plot of the average cross-validated unexplained variance for WSP during QW for a decoder trained only on QW times, compared to a decoder trained on random times across the experiment (n = 13, N = 5; two-sided Wilcoxon signed rank test). Both decoders were based on their optimal number of PCs, and were tested on the same held-out data during QW. (e) Plot of the average cross-validated unexplained variance for WSP based on the best parallel fiber (PF) for each recording and for lasso regression on parallel fiber population activity (n = 13, N = 5; two-sided Wilcoxon signed rank test). (f) Range of optimal number of parallel fibers to minimize the cross-validated unexplained variance in f. Each marker represents a different experiment. Error bars in c, d and e denote s.e.m.
(a) Relationship between the variance of the population activity explained and number of principal components (PCs) based on cross-validated principal component analysis (PCA). Each black line represents the mean variance explained for a single experiment (all data randomly subsampled to 300 axons). Shading represents s.e.m. across different randomly subsampled populations and colors indicate different animals (n = 10, N = 3). The arrowheads represent the lower bound of the dimensionality for each experiment. Inset: Expanded region from main panel. Black bars indicate average over experiments. (b) Relationship between the lower bound of the dimensionality and the maximum variance explained. Grey and colored arrowheads indicate individual subsamples of held-out data and means for each experiment, respectively. Linear extrapolation predicts that 62 dimensions are necessary to explain all the variance for populations of 300 parallel fibers. (c) The ratio of number of neurons to the extrapolated dimensionality for all subsampled population sizes, ranging from 100 (n = 13, N = 5) to 650 parallel fibers (n = 2, N = 1).
References
Publication types
MeSH terms
Associated data
Grants and funding
LinkOut - more resources
Full Text Sources