Article Figures & Data
Figures
- Figure 1.
Overview of recording locations, task design, data analysis, and sample data. A, 32-channel custom-designed electrode array (HIP: hippocampus; PAR: parietal cortex; PFC: prefrontal cortex). The line drawing underneath illustrates the approximate locations of the electrodes on a sagittal slice. B, Task flow and timing (HC1-4: home cage sessions 1-4; TR: training; TE: testing). The red diamonds and green square indicate objects placed in the arena. The picture underneath is from a camera placed overhead. C1, A data matrix with combined LFP and multiunits (smoothed with a 30-ms FWHM Gaussian) from three different regions. C2, Data covariance matrices for the data snippet shown in C1, either narrowband-filtered (S) or broadband (R). A generalized eigendecomposition of these two matrices (panel C3) provides a set of eigenvectors (w) and corresponding eigenvalues (λ), from which three pieces of information are extracted: The component spatial map (the eigenvector multiplied by the covariance matrix), the component time series (the eigenvector multiplied by the data matrix), and the separability of narrowband vs. broadband activity (the eigenvalue for one frequency; the eigenvalues over frequencies creates an eigenspectrum). Illustrated here is one eigenvalue solution for one frequency; in practice, the number of solutions (w/λ pairs) corresponds to the number of data channels, and this entire procedure is repeated across a range of frequencies. D, Multiple components can be isolated from each frequency, with distinct temporal dynamics. Example component power time series are illustrated from 20 seconds of a recording; each row corresponds to a distinct component. Frequency groups are based on empirical frequency boundaries (described later) and components are sorted within each frequency band based on total component energy.
- Figure 2.
Generalized Eigendecomposition enables spectrally resolved source separation across 3 areas for a single recording. A, Spatial maps over all three regions per frequency (each column corresponds to one frequency). The thick horizontal dashed lines show inter-regional boundaries, and thin horizontal dashed lines show within-region boundaries between LFP (top) and multiunit (bottom) channels. Within-region rows are ordered according to the channel index in the dataset, not according to anatomical location. The colors indicate the strength of the contribution of that channel to the brain-wide component (data were per-frequency normalized prior to GED, so the color values are comparable across frequencies), vertical dashed lines show the empirically defined frequency boundaries (detailed later): red lines indicate the lower bounds of the frequency band and blue lines indicate the upper bounds. B, Eigenspectra from the largest three components per frequency, which highlights that there can be multiple separable components at the same frequency. The map in panel A is only for the top eigenspectrum (blue line). C, Example topographical maps of the anatomical distribution of the filter projections for the indicated frequency ranges. Each black dot is the location of an electrode. In all columns, medial is to the left and anterior is to the top.
- Figure 3.
Distinct frequency bands separate clearly in the LFP data with specific spectrotemporal profiles. A, R2 correlation matrix across all pairs of frequency-specific eigenvectors, with pink boxes drawn around empirically derived clusters (based on the dbscan algorithm), from one recording. The cluster boundaries separate spatially distinct topographies across different frequency ranges. B, Topographical maps of the spatial filter from the frequency bands in panel A. White/black numbers indicate corresponding bands/maps. C, Aggregated results of the number of empirical frequency bands per experiment session (H1-4 indicate home sessions; Tr indicates training session; Te indicates test session). Error bars show standard deviations across the six animals. D, Center frequencies for each group as defined by k-means clustering analysis over animals. Error bars show standard deviations across 100 repeats of the k-means clustering algorithm with different random initializations, and the numbers above each data point shows the average center frequency from that band.
- Figure 4.
Component topographies are reproducible within animals in different sessions, yet differ across animals. A, R2 spatial correlations per frequency. The analysis was run on the components with the largest eigenvalue per frequency (“top comp.”), and by selecting the largest correlation amongst the top two components (“max”). B, Each individual correlation, separated according to the experiment sessions from which the spatial map pairs were drawn (“T-T” indicates train-test pairs, “H-H” indicates home-home pairs). Black bars indicate the mean R2. The color of each dot is the average of the eigenvalues of the component pair (which indicates the separability of the narrowband from broadband signals), and the r-value on top of each column is the correlation between the spatial map R2 and the average eigenvalue.
- Figure 5.
All recorded regions contributed to the components per frequency, with some frequencies showing regional dominance. A, The region bias index for each animal (A1) and averaged over animals for each experiment session (A2). Values close to 0 indicate equal spread of components across all three brain regions, whereas values close to 1 indicate that a single region dominates the component. B, The fraction of total component energy attributable to each region, normalized to the sum over all three regions (thus, the sum per frequency is 1). Each panel is a different animal, averaged over experiment sessions. Patches indicate one standard deviation above and below the mean across sessions, which illustrates the reproducibility of these characteristics over time (six sessions spanning 2 hours). All panels have the same tick marks and axis labels as the lower-left panel. The group-average regional fractions are shown in panel C. Horizontal lines at 1/3 and 2/3 indicate equal contribution of all three regions to the component. D, The modality dominance spectrum quantitatively showed that components were predominantly driven by LFP instead of by multiunits. E, Entropy spectrum shows that LFP channels had higher entropy compared to the multiunits (multiunits’ entropy is the same for all frequencies). F, The multiunits made significant contributions to the components over most frequencies except in the range of 20-90 Hz. Positive values indicate better separability when multiunits are included. The black line is the average over all animals, and the surrounding patch indicates one standard deviation around that average. Red lines show significant changes relative to zero at p<.05, FDR corrected for multiple comparisons over frequencies.
- Figure 6.
Generalized eigendecomposition reveals that narrowband subspaces are multidimensional (quantified as the number of statistically significant components), and components within each frequency are partially synchronized but non- redundant. A-B, Subspace dimensionality across animals (A) and experiment sessions (B). C, The distribution of all component dimensionalities, normalized to percent of the maximum possible dimensionality (the rank of the covariance matrices), revealed that the narrowband components spanned around 10% of the total possible signal dimensionality. D-F, Phase synchronization between the top two components per frequency indicates both coordination and independence across within-frequency networks. Volume-conduction-independent phase synchronization tended to decline with frequency except for a prominent peak in theta/alpha (∼7-13 Hz) and a smaller prominence in beta ((∼15-30 Hz). The patterns were similar over different animals (D) and different sessions (E). F, Average synchronization in the theta/alpha range for the different sessions.
Extended Data
Extended Data Figure 1-1
Left plot shows an example multiunit correlation matrix from one recording session. The right plot shows a histogram of all unique off-diagonal correlation values. These plots illustrate that our spike-sorting approach was not overly contaminated by identifying the same units on multiple channels. Download Figure 1-1, PDF file.
Extended Data Figure 2-1
Kurtosis, a measure of non-Gaussianity of a distribution (see text below), computed on frequency-specific component time series. The red and blue lines in panel A show kurtosis per frequency for the narrowband-filtered time series (blue) and amplitude envelope (red), averaged over all animals and sessions. The horizontal dashed line indicates the expected kurtosis of a pure Gaussian distribution. B, Kurtosis over frequencies for each animal separately. Note the striking decrease in kurtosis in the theta band in all animals. C, Example time series histograms illustrating the platykurtic effect at 8 and 11 Hz for two different animals and sessions.
Distribution shape via kurtosis. Non-Gaussianity is considered an indicator of an information-rich signal. This comes from the central limit theorem, which leads to the assumption that random noise, and random linear mixtures of signals, will produce Gaussian distributions. We therefore quantified the kurtosis (4th statistical moment of a distribution; the kurtosis of a pure Gaussian distribution is 3) as a measure of the non-Gaussianity of the component time series. We computed kurtosis for the narrowband filtered signal and its amplitude envelope at each component.
Component time series kurtosis was computed as the 4th statistical moment of the component time series. We extracted kurtosis from both the real part of the narrowband signal and the amplitude envelope (extracted via the Hilbert transform). The amplitude envelope had overall higher kurtosis (Extended Data Fig. 2-1), which is not surprising considering that amplitude is a strictly non-negative quantity.
Nearly all frequencies had kurtosis higher than 3, indicating leptokurtic distributions characterized by narrow peaks and fatter tails. This is consistent with suggestions that brain activity is characterized by extreme events and long-tailed distributions (Buzsáki and Mizuseki, 2014). Curiously, all six animals exhibited a dip in kurtosis in the theta band (∼9 Hz; Extended Data Fig. 2-1B), indicating a platykurtic distribution with data values clustered towards zero and relatively fewer data points having extreme values (the tails of the distributions; Extended Data Fig. 2-1C). This may be related to the known sawtooth-like shape of hippocampal theta (Scheffer-Teixeira and Tort, 2016).
Note that unlike ICA, GED is based purely on the signal covariance (second moment) and not on any higher-order statistical moments. Thus, non-Gaussian distributions are not trivially imposed by the decomposition method, but instead arose from the data without bias or selection. Download Figure 2-1, DOCX file.
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