The dominance of global phase dynamics in human cortex, from delta to gamma

Reviewer #1 (Public review):

Summary:

The paper uses rigorous methods to determine phase dynamics from human cortical stereotactic EEGs. It finds that the power of the phase is higher at the lowest spatial phase.

Strengths:

Rigorous and advanced analysis methods.

Weaknesses:

The novelty and significance of the results are difficult to appreciate from the current version of the paper.

(1) It is very difficult to understand which experiments were analysed, and from where they were taken, reading the abstract. This is a problem both for clarity with regard to the reader and for attribution of merit to the people who collected the data.

(2) The finding that the power is higher at the lowest spatial phase seems in tune with a lot of previous studies. The novelty here is unclear and it should be elaborated better. I could not understand reading the paper the advantage I would have if I used such a technique on my data. I think that this should be clear to every reader.

(3) It seems problematic to trust in a strong conclusion that they show low spatial frequency dynamics of up to 15-20 cm given the sparsity of the arrays. The authors seem to agree with this concern in the last paragraph of page 12. They also say that it would be informative to repeat the analyses presented here after the selection of more participants from all available datasets. It begs the question of why this was not done. It should be done if possible.

(4) Some of the analyses seem not to exploit in full the power of the dataset. Usually, a figure starts with an example participant but then the analysis of the entire dataset is not as exhaustive. For example, in Figure 6 we have a first row with the single participants and then an average over participants. One would expect quantifications of results from each participant (i.e. from the top rows of GFg 6) extracting some relevant features of results from each participant and then showing the distribution of these features across participants. This would complement the subject average analysis.

(5) The function of brain phase dynamics at different frequencies and scales has been examined in previous papers at frequencies and scales relevant to what the authors treat. The authors may want to be more extensive with citing relevant studies and elaborating on the implications for them. Some examples below:
Womelsdorf T, et alScience. 2007
Besserve M et al. PloS Biology 2015
Nauhaus I et al Nat Neurosci 2009

Reviewer #2 (Public review):

Summary:

In this paper, the authors analyze the organization of phases across different spatial scales. The authors analyze intracranial, stereo-electroencephalogram (sEEG) recordings from human clinical patients. The authors estimate the phase at each sEEG electrode at discrete temporal frequencies. They then use higher-order SVD (HOSVD) to estimate the spatial frequency spectrum of the organization of phase in a data-driven manner. Based on this analysis, the authors conclude that most of the variance explained is due to spatially extended organizations of phase, suggesting that the best description of brain activity in space and time is in fact a globally organized process. The authors' analysis is also able to rule out several important potential confounds for the analysis of spatiotemporal dynamics in EEG.

Strengths:

There are many strengths in the manuscript, including the authors' use of SVD to address the limitation of irregular sampling and their analyses ruling out potential confounds for these signals in the EEG.

Weaknesses:

Some important weaknesses are not properly acknowledged, and some conclusions are over-interpreted given the evidence presented.

The central weakness is that the analyses estimate phase from all signal time points using wavelets with a narrow frequency band (see Methods - "Numerical methods"). This step makes the assumption that phase at a particular frequency band is meaningful at all times; however, this is not necessarily the case. Take, for example, the analysis in Figure 3, which focuses on a temporal frequency of 9.2 Hz. If we compare the corresponding wavelet to the raw sEEG signal across multiple points in time, this will look like an amplitude-modulated 9.2 Hz sinusoid to which the raw sEEG signal will not correspond at all. While the authors may argue that analyzing the spatial organization of phase across many temporal frequencies will provide insight into the system, there is no guarantee that the spatial organization of phase at many individual temporal frequencies converges to the correct description of the full sEEG signal. This is a critical point for the analysis because while this analysis of the spatial organization of phase could provide some interesting results, this analysis also requires a very strong assumption about oscillations, specifically that the phase at a particular frequency (e.g. 9.2 Hz in Figure 3, or 8.0 Hz in Figure 5) is meaningful at all points in time. If this is not true, then the foundation of the analysis may not be precisely clear. This has an impact on the results presented here, specifically where the authors assert that "phase measured at a single contact in the grey matter is more strongly a function of global phase organization than local". Finally, the phase examples given in Supplementary Figure 5 are not strongly convincing to support this point.

Another weakness is in the discussion on spatial scale. In the analyses, the authors separate contributions at (approximately) > 15 cm as macroscopic and < 15 cm as mesoscopic. The problem with the "macroscopic" here is that 15 cm is essentially on the scale of the whole brain, without accounting for the fact that organization in sub-systems may occur. For example, if a specific set of cortical regions, spanning over a 10 cm range, were to exhibit a consistent organization of phase at a particular temporal frequency (required by the analysis technique, as noted above), it is not clear why that would not be considered a "macroscopic" organization of phase, since it comprises multiple areas of the brain acting in coordination. Further, while this point could be considered as mostly semantic in nature, there is also an important technical consideration here: would spatial phase organizations occurring in varying subsets of electrodes and with somewhat variable temporal frequency reliably be detected? If this is not the case, then could it be possible that the lowest spatial frequencies are detected more often simply because it would be difficult to detect variable organizations in subsets of electrodes?

Another weakness is disregarding the potential spike waveform artifact in the sEEG signal in the context of these analyses. Specifically, Zanos et al. (J Neurophysiol, 2011) showed that spike waveform artifacts can contaminate electrode recordings down to approximately 60 Hz. This point is important to consider in the context of the manuscript's results on spatial organization at temporal frequencies up to 100 Hz. Because the spike waveform artifact might affect signal phase at frequencies above 60 Hz, caution may be important in interpreting this point as evidence that there is significant phase organization across the cortex at these temporal frequencies.

A last point is that, even though the present results provide some insight into the organization of phase across the human brain, the analyses do not directly link this to spiking activity. The predictive power that these spatial organizations of phase could provide for spiking activity - even if the analyses were not affected by the distortion due to the narrow-frequency assumption - remains unknown. This is important because relating back to spiking activity is the key factor in assessing whether these specific analyses of phase can provide insight into neural circuit dynamics. This type of analysis may be possible to do with the sEEG recordings, as well, by analyzing high-gamma power (Ray and Maunsell, PLoS Biology, 2011), which can provide an index of multi-unit spiking activity around the electrodes.

Reviewer #3 (Public review):

Summary:

The authors propose a method for estimation of the spatial spectra of cortical activity from irregularly sampled data and apply it to publicly available intracranial EEG data from human patients during a delayed free recall task. The authors' main findings are that the spatial spectra of cortical activity peak at low spatial frequencies and decrease with increasing spatial frequency. This is observed over a broad range of temporal frequencies (2-100 Hz).

Strengths:

A strength of the study is the type of data that is used. As pointed out by the authors, spatial spectra of cortical activity are difficult to estimate from non-invasive measurements (EEG and MEG) due to signal mixing and from commonly used intracranial measurements (i.e. electrocorticography or Utah arrays) due to their limited spatial extent. In contrast, iEEG measurements are easier to interpret than EEG/MEG measurements and typically have larger spatial coverage than Utah arrays. However, iEEG is irregularly sampled within the three-dimensional brain volume and this poses a methodological problem that the proposed method aims to address.

Weaknesses:

The used method for estimating spatial spectra from irregularly sampled data is weak in several respects.

First, the proposed method is ad hoc, whereas there exist well-developed (Fourier-based) methods for this. The authors don't clarify why no standard methods are used, nor do they carry out a comparative evaluation.

Second, the proposed method lacks a theoretical foundation and hinges on a qualitative resemblance between Fourier analysis and singular value decomposition.

Third, the proposed method is not thoroughly tested using simulated data. Hence it remains unclear how accurate the estimated power spectra actually are.

In addition, there are a number of technical issues and limitations that need to be addressed or clarified (see recommendations to the authors).

My assessment is that the conclusions are not completely supported by the analyses. What would convince me, is if the method is tested on simulated cortical activity in a more realistic set-up. I do believe, however, that if the authors can convincingly show that the estimated spatial spectra are accurate, the study will have an impact on the field. Regarding the methodology, I don't think that it will become a standard method in the field due to its ad hoc nature and well-developed alternatives.

Author response:

We thank the editor and reviewers for their feedback. We believe we can address the substantive criticisms in full, first, by providing a more explicit theoretical basis for the method. Then, we believe criticism based on assumptions about phase consistency across time points are not well founded and can be answered. Finally, in response to some reviewer comments, we will improve the surrogate testing of the method.

We will enhance the theoretical justification for the application of higher-order singular value decomposition (SVD) to the problem of irregular sampling of the cortical area. The initial version of the manuscript was written to allow informal access to these ideas (if possible), but the reviewers find a more rigorous account appropriate. We will add an introduction to modern developments in the use of functional SVD in geophysics, meteorology & oceanography (e.g., empirical orthogonal functions) and quantitative fluid dynamics (e.g., dynamic mode decomposition) and computational chemistry. Recently SVD has been used in neuroscience studies (e.g., cortical eigenmodes). To our knowledge, our work is the first time higher-order SVD has been applied to a neuroscience problem. We use it here to solve an otherwise (apparently) intractable problem, i.e., how to estimate the spatial frequency (SF) spectrum on a sparse and highly irregular array with broadband signals.

We will clarify the methodological strategy in more formal terms in the next version of the paper. But essentially SVD allows a change of basis that greatly simplifies quantitative analysis. Here it allows escape from estimating the SF across millions of data-points (triplets of contacts, at each sample), each of which contains multiple overlapping signals plus noise (noise here defined in the context of SF estimation) and are inter-correlated across a variety of known and unknown observational dimensions. Rather than simply average over samples, which would wash out much of the real signal, SVD allows the signals to be decomposed in a lossless manner (up to the choice of number of eigenvectors at which the SVD is truncated). The higher-order SVD we have implemented reduces the size of problem to allow quantification of SF over hundreds of components, each of which is guaranteed certain desirable properties, i.e., they explain known (and largest) amounts of variance of the original data and are orthonormal. This last property allows us to proceed as if the observations are independent. SF estimates are made within this new coordinate system.

We will also more concretely formalise the relation between Fourier analysis and previous observations of eigenvectors of phase that are smooth gradients.

We will very briefly review Fourier methods designed to deal with non-uniform sampling. The problems these methods are designed for fall into the non-uniform part of the spectrum from uniform–non-uniform–irregular–highly-irregular–noise. They are highly suited to, for example, interpolating between EEG electrodes to produce a uniform array for application of the fast Fourier transform (Alamia et al., 2023). However, survey across a range of applied maths fields suggests that no method exists for the degree of irregular sampling found in the sEEG arrays at issue here. In particular, the sparseness of the contact coverage presents an insurmountable hurdle to standard methods. While there exists methods for sparse samples (e.g., Margrave & Fergusen, 1999; Ying 2009), these require well-defined oscillatory behavior, e.g., for seismographic analysis. Given the problems of highly irregular sampling, sparseness of sampling and broadband, nonstationary signals, we have attempted a solution via the novel methods introduced in the current manuscript. We were able to leverage previous observations regarding the relation between eigenvectors of cortical phase and Fourier analysis, as we outline in the manuscript.

We will extend the current 1-dimensional surrogate data to better demonstrate that the method does indeed correctly detect the ordinal relations in power on different parts of the SF spectrum. We will include the effects of a global reference signal. Simulations of cortical activity are an expensive way to achieve this goal. While the first author has published in this area, such simulations are partly a function of the assumptions put into them (i.e., spatial damping, boundary conditions, parameterization of connection fields). We will therefore use surrogate signals derived from real cortical activity to complete this task.

Some more specific issues raised:
(1) Application of the method to general neuroscience problems:
The purpose of the manuscript was to estimate the SF spectrum of phase in the cortex, in the range where it was previously not possible. The purpose was not specifically to introduce a new method of analysis that might be immediately applicable to a wide range of available data-sets. Indeed, the specifics of the method are designed to overcome an otherwise intractable disadvantage of sEEG (irregular spatial sampling) in order to take advantage of its good coverage (compared to ECoG) and low volume conduction compared to extra-cranial methods. On the other hand, the developing field of functional SVD would be of interest to neuroscientists, as a set of methods to solve difficult problems, and therefore of general interest. We will make these points explicit in the next version of the manuscript. In order to make the method more accessible, we will also publish code for the key routines (construction of triplets of contacts, Morlet wavelets, calculation of higher-order SVD, calculation of SF).

(2) Novelty:
We agree with the third reviewer: if our results can convince, then the study will have an impact on the field. While there is work that has been done on phase interactions at a variety of scales, such as from the labs of Fries, Singer, Engels, Nauhaus, Logothetis and others, it does not quantify the relative power of the different spatial scales. Additionally, the research of Freeman et al. has quantified only portions of the SF spectrum of the cortex, or used EEG to estimate low SFs. We would appreciate any pointers to the specific literature the current research contributes to, namely, the SF spectrum of activity in the cortex.

(3) Further analyses:
The main results of the research are relatively simple: monotonically falling SF-power with SF; this effect occurs across the range of temporal frequencies. We provide each individual participant’s curves in the supplementary Figures. By visual inspection, it can be seen that the main result of the example participant is uniformly recapitulated. One is rarely in this position in neuroscience research, and we will make this explicit in the text.

The research stands or falls by the adequacy of the method to estimate the SF curves. For this reason most statistical analyses and figures were reserved for ruling out confounds and exploring the limits of the methods. However, for the sake of completeness, we will now include the SF vs. SF-power correlations and significance in the next version, for each participant at each frequency.

Since the main result was uniform across participants, and since we did not expect that there was anything of special significance about the delayed free recall task, we conclude that more participants or more tasks would not add to the result. As we point out in the manuscript, each participant is a test of the main hypothesis. The result is also consistent with previous attempts to quantify the SF spectrum, using a range of different tasks and measurement modalities (Barrie et al., 1996; Ramon & Holmes 2015; Alexander et al., 2019; Alexander et al., 2016; Freeman et al., 2003; Freeman et al. 2000). The search for those rare sEEG participants with larger coverage than the maximum here is a matter of interest to us, but will be left for a future study.

(4) Sampling of phase and its meaningfulness:
The wavelet methods used in the present study have excellent temporal resolution but poor frequency resolution. We additionally oversample the frequency range to produce visually informative plots (usually in the context of time by frequency plots, see Alexander et al., 2006; 2013; 2019). But it is not correct that the methods for estimating phase assume a narrow frequency band. Rather, the poor frequency resolution of short time-series Morlet wavelets means the methods are robust to the exact shape of the waveforms; the signal need be only approximately sinusoidal; to rise and fall. The reason for using methods that have excellent resolution in the time-domain is that previous work (Alexander et al., 2006; Patten et al. 2012) has shown that traveling wave events can last only one or two cycles, i.e., are not oscillatory in the strict sense but are non-stationary events. So while short time-window Morlet wavelets have a disadvantage in terms of frequency resolution, this means they precisely do not have the problem of assuming narrow-band sinusoidal waveforms in the signal. We strongly disagree that our analysis requires very strong assumptions about oscillations (see last point in this section).

Our hypothesis was about the SF spectrum of the phase. When the measurement of phase is noise-like at some location, frequency and time, then this noise will not substantially contribute to the low SF parts of the spectrum compared to high SFs. Our hypothesis also concerned whether it was reasonable to interpret the existing literature on low SF waves in terms of cortically localised waves or small numbers of localised oscillators. This required us to show that low SFs dominate, and therefore that this signal must dominate any extra-cranial measurements of apparent low SF traveling waves. It does not require us to demonstrate that the various parts of the SF spectrum are meaningful in the sense of functionally significant. This has been shown elsewhere (see references to traveling waves in manuscript, to which we will also add a brief survey of research on phase dynamics).

The calculation of phase can be bypassed altogether to achieve the initial effect described in the introduction to the methods (Fourier-like basis functions from SVD). The observed eigenvectors, increasing in spatial frequency with decreasing eigenvalues, can be reproduced by applying Gaussian windows to the raw time-series (D. Alexander, unpublished observation). For example, undertaking an SVD on the raw time-series windowed over 100ms reproduces much the same spatial eigenvectors (except that they come in pairs, recapitulating the real and imaginary parts of the signal). This reproducibility is in comparison to first estimating the phase at 10Hz using Morlet wavelets, then applying the SVD to the unit-length complex phase values.

(5) Other issues to be addressed and improved:
clarity on which experiments were analyzed (starting in the abstract) discussion of frequencies above 60Hz and caution in interpretation due to spike-waveform artefact or as a potential index of multi-unit spiking discussion of whether the ad hoc, quasi-random sampling achieved by sEEG contacts somehow inflates the low SF estimates

References (new)
Patten TM, Rennie CJ, Robinson PA, Gong P (2012) Human Cortical Traveling Waves: Dynamical Properties and Correlations with Responses. PLoS ONE 7(6): e38392. https://doi.org/10.1371/journal.pone.0038392
Margrave GF, Ferguson RJ (1999) Wavefield extrapolation by nonstationary phase shift, GEOPHYSICS 64:4, 1067-1078
Ying Y (2009) Sparse Fourier Transform via Butterfly Algorithm SIAM Journal on Scientific Computing, 31:3, 1678-1694