Traveling waves, measurements and origin.

A-C. Scales of measurement and degree of blurring in EEG, ECoG and stereotactic EEG. A. EEG measures the electrical activity of the cortex at the surface of the scalp (contacts shown in green). Due to volume conduction in the intervening tissues (blue), the cortical signal is highly blurred and preferentially samples gyri. The EEG array covers the entirety of the scalp in a quasi-regular grid, enabling a large extent of cortical coverage (green line, not to scale). B. Electrocorticogram (ECoG) measures the electrical activity of the cortex at the cortical surface. The volume conduction effects are less here, but the measurement array preferentially samples activity at the gyri. The distance between contacts is regular, forming a two-dimensional measurement array, but the size of the array is limited to 10cm or less. C. Stereotactic EEG measures electrical activity within the grey matter of the cortex, and therefore has a lower degree of blurring. The clinical placement of the linear arrays of contacts result in a highly irregular measurement array, though often with wide coverage of the cortex, though in general less than 30cm. D-F. Hypotheses for the origin of macroscopic TWs measured extra-cranially (adapted from 14). D. Extra-cranial TWs reflect real, coordinated, spatio-temporal activity across the entire cortex. The TWs are measurable at the scalp because the dominant spatial frequency of the activity has a long wavelength and is therefore detectable at the scalp. A sinusoidal wave corresponding to this hypothesis is shown in blue, with successive time samples in lighter shades. The fast Fourier transform of the model wave is shown in magenta. The expected spatial frequency spectrum, when measured in the grey-matter, has a peak at a low spatial frequency. E. A local wave (shown here in a sulcus) slowly traverses a small region of grey matter via intracortical connections. The wave appears at the scalp as a fast-moving global wave due to volume conduction effects. The wave’s (blue) expected spatial frequency spectrum (magenta), when measured in the grey-matter, has a peak at a high spatial frequency. F. Localized oscillatory sources that are out of phase appear as macroscopic TWs at the scalp due to volume conduction effects. The oscillators’ (blue) expected spatial frequency spectrum (magenta), when measured in the grey-matter, has a peak at high spatial frequencies corresponding to the size of each oscillatory source, with a weaker elevation in low spatial frequency power due to the interaction between sources.

Overview of the procedure to estimate the spatial frequency spectrum of phase. A-C. Example of using singular value decomposition to empirically estimate Fourier components of a signal.

A. Image of waves. The waves have a peak spatial frequency at 15 cycles per 1000 pixels vertically, and this dimension is treated as if it were time by scanning down the image (nominally) at 1 ms per pixel. The waves are aligned diagonally, moving towards the bottom right of the image. The waves complete approximately one cycle every 512 pixels in the horizontal direction. This is the width for the size of the scanning array, scanning the image vertically three times (white boundaries). The complex-valued phase at 15 Hz along the ‘temporal’ axis is estimated, and the decomposition of the phase dynamics along this dimension is computed. B. First four components of the singular value decomposition of the vector of phases from image in A. Real part of the component is shown in blue, imaginary part in orange. The phase angle corresponding to these real and imaginary values is shown by the colored horizontal line. The first plot shows the dominant spatio-”temporal” frequency, captured as a single cycle wave. Each successive component has an increment in number of cycles from the previous. These components are inherently spatio-temporal, since a spatial vector of smoothly changing phase implies a traveling wave. C. Spatial frequency spectrum of the image in A. Because components from the singular value decomposition are approximate sinusoids, the Fourier spectrum can simply be estimated by reading off the singular values (shown here as % variance explained). The dominant spatial frequency is around wavenumber equals one cycle per 512 pixels. D-F. Cortical measurements and organization of numerical analysis. D. Cortical reconstruction of one participant, showing placement of stereotactic EEG contacts. Grey matter contacts are shown as black dots. White matter contacts are shown with colored circles (blue, or purple for contacts with lowest amplitude signal). E. Placement of grey matter contacts within inflated left hemisphere of cortex. Phase values at some temporal frequency and time-sample are indicated on the rainbow scale. F. Grey matter contacts are grouped into triplets forming approximately equilateral triangles. Edge distances are calculated according to the geodesic distance in the uninflated cortex. G-I. Model construction using higher-order SVD. G. The phase is estimated at some temporal frequency, at each vertex’s contact. When collated over all time samples, each triangle from F has a uniform distribution of phase, shown in the three circular histograms. H. The phase data represented in G comprises a 2-D slice (vertices by samples) in the 3-D tensor of phase. Higher-order SVD is used to reduce the samples and triangles dimensions by removing those components that explain relatively little variance. I. A single set of vertices in the model represents a triplet of phase relations in an abstracted triangle. The phase relations explain a known amount of variance in the original 3-D tensor of phase. J-L. Estimation of spatial frequency for each model component. J. The phase relations for each model triangle define a rate of change of phase, as well as the direction of phase change. K. The reduced model is projected back onto the full space of triangles, enabling a re-association with the distance information for each triangle. L. For each triangle and each model component, the spatial frequency of phase is calculated as cycles/m, along with the weighting of the model component on the triangle. These two estimates, over all triangles and model components, enable construction of the spatial frequency spectrum of phase (as in C).

Spatial frequency estimation via higher-order singular vector decomposition (HOSVD).

A. The stereotactic EEG contacts, indicated by x’ ss, are grouped into triplets. Three x’s are positioned in the panel to accurately reflect the three inter-contact cortical (geodesic) distances, where the scale of the panel is 1m2. In the corresponding vertices of the large triangle are shown the phase distributions measured at each of three contacts over all times, at a temporal frequency of 9.2Hz. The relative phase distribution for pairs of contacts is shown in the middle of each edge of the triangle. In this case, the mode of one relative phase distribution (left<->top) is approximately zero, while the other two modes are approximately Π. B. One of the model components of phase (at 9.2Hz) for the triplet of contacts in A. This triplet was strongly weighted on this model component in the HOSVD. The phase values of the left and top vertices in the triangle were similar (red-orange), and the right vertex phase value was offset by approximately Π (green). The phase gradient shown within the triangle is interpolated from these values, indicating visually the direction and wavelength of the phase gradient. C. The grey arrow shows the direction and wavelength of the estimated spatial phase gradient over the three contacts for this model component. Relative cortical distances between contacts are also shown. The estimated wavelength for this component was ∼0.3m, corresponding to a spatial frequency of 3.3 cycles/m. D. The estimated distribution of spatial frequencies for this participant at 9.2Hz, with the spatial frequency value estimated from A-C shown as a black dot. E-G. Three other triplets of contacts from this participant, showing phase (at 9.2Hz) and relative-phase distributions and the model component on which it was weighted most strongly. E. Here the estimated spatial frequency was near zero, indicating near synchrony. F. Here the estimated spatial frequency was ∼4 cycles/m. G. Here the estimated spatial frequency was ∼5 cycles/m. Conventions in E-G are the same as A-C.

Estimated spatial frequency spectra.

A. Spatial frequency spectra across the temporal frequency range for an example participant. The maximum inter-contact distance in the measurement array is shown with a black dot, in units of contacts per metre for comparison with spatial frequency. B. Stacked histogram of the same data shown in A. C. Aggregate spatial frequency spectra for 11 participants. Each participant’ ss maximum inter-contact distance is shown with a black dot. D. Plot of temporal frequency versus peak spatial frequency power, derived from C. E. Plot of temporal frequency versus spatial frequency at peak power, derived from C. F. Cross-participant regression for each participant’s maximum inter-contact distance versus each participant’s spatial frequency at peak power. The regression lines are shown for each separate temporal frequency. Individual’s distance by frequency data points are shown with colored dots.

Example phase samples at 8.0Hz.

A. Spectra derived from model components that had a peak in either macroscopic range or mesoscopic range. B. Sample that had good fit to the macroscopic spectra (left) or the mesoscopic spectra (right). Phase values shown on the colour wheel, contacts positioned on inflated cortical surface.

Cross-frequency correlations of model fits in the macroscopic and mesoscopic spatial frequency bands.

A. Single participant cross-frequency correlations and difference between macroscopic and mesoscopic correlations. At each temporal frequency (i.e. axes in Hz), the participant’s model fit for a spatial frequency band was correlated with the model fits in that band for all temporal frequencies. This is shown for model components drawn from the macroscopic spatial frequency band (first) and mesoscopic (second). The difference between the two cross-frequency correlations is shown in the third panel. The cross-frequency correlation of macroscopic fits (x-axis) with microscopic fits (y-axis) is shown in the fourth panel. Significant correlations are indicated by opaque colors (p<0.001). B. Participant averages of the cross-frequency correlations, in the macroscopic and mesoscopic band, their difference and macroscopic versus mesoscopic.

Illustration of use of higher order SVD to extract specific spatial frequency bands.

Artificial phase dynamics are generated for a one-dimensional cortex. Higher order SVD is used to extract and reconstitute the dominant component using the methods described, across pairs of contacts. Phase of signal is indicated by coloured scale, and is shown as rainbow bands for all cortical sites at four randomly chosen time samples. The phase distribution at two sites, over all samples, is shown in the circular histograms. The relative phase distribution for the two sites is shown in the rectangular histogram. A. Raw phases on the one-dimensional cortex. The spatial frequency spectrum is broad, but the highest power is at a low spatial frequency, as can be seen visually. B. Extracted low spatial frequency component. The two sites shown comprise a pair that weighted strongly on this component. The model component values for the same four time samples from A are indicated in the region of one-dimensional cortex directly above each circular histogram, other values are interpolated between these two values. C. Same as A, but for phase dynamics generated with highest power at a medium spatial frequency. D. Same as B, but the medium spatial frequency component has been extracted from the data in C. The pair of sites that weighted strongly on this component are shown.

Left: An example phase map with a moderate standing wave component, shown by the sharp transition between blue contacts (phase angle -3π/4) in the top row and the orange contacts (phase angle π/4) in the row immediately below. The sharp transition leads to a local underestimate of the spatial frequency. Right: The same phase map, after decomposition into pure traveling wave components. The smoother transition between the phase values over successive rows of contacts allows more accurate estimates of spatial frequency from local estimates of rate of change of phase per metre.

Spatial frequency stacked spectrum (power in arbitrary units) for all participants (n=35), frequency range 2.0 to 42.2Hz. The highest power occurred for the lowest spatial frequencies (p<0.01; Chi-square), limited only by the maximum extent of the stereotactic EEG array. The spatial frequency spectrum had a similar shape across the temporal frequency range.

Linear fits to spatial frequency spectra at temporal frequency 9.2Hz, for all 35 participants.

Estimated spatial frequency spectra for all other participants.

Spatial frequency spectra, as a stacked histogram across the temporal frequency range. The participants have been ordered by the maximum inter-contact distance in the measurement array (black dot, 1/metre), starting with the participant with the smallest array size. Figure conventions are otherwise as for Figure 4B, except two participants had a maximum temporal frequency of 97 Hz, rather than 128 Hz.

Example samples of phase measurements with either good fit to macroscopic components or to mesoscopic components.

Examples of spectra for macroscopic or mesoscopic model components are shown in Figure 4. Samples of phase are selected from evenly spaced temporal frequencies (on a log scale). Phase values are shown on a circular colour scale, with zero phase at red. Phase values are displayed at the contact locations on the inflated cortex, in both left and right hemispheres, in dorsal and posterior views.

Effects of measurement array size on spatial frequency spectrum.

A-B. Spatial frequency histogram for the same participant as Figure 3. These spectra are produced using only the largest (for panel A) 20% of contact triplets (i.e., by inter-contact distance) and smallest (for panel B). Conventions are otherwise the same as Figure 3B, and the same participant’s data is shown. C. Example participant spatial frequency spectra (envelope over all temporal frequencies) for five spectra produced from five equally sized bins of contact triplets (grouped according to linearly decreasing inter-contact distance). D. Mean inter-contact distance versus spatial frequency at peak in power. The panel shows the data-points derived from the five spectra in C as black dots, as well as the regression line. E. Mean inter-contact distance versus peak spatial frequency power. The panel shows the data-points points derived from the five spectra in C as black dots, as well as the regression line (log scale). F. Summary data for mean inter-contact distance versus spatial frequency at peak in power, for 9 participants. Depending on the number of contacts, each participant’s contact triplets were arranged into 5, 10 or 20 bins. Each participant’s data are shown with a single shade of grey dots, along with the associated regression line, which is extended out to the value expected at larger array sizes. G. Summary data for mean inter-contact distance versus peak spatial frequency power, for 9 participants. Conventions otherwise the same as F.