The dominance of large-scale phase dynamics in human cortex, from delta to gamma

  1. David M Alexander
  2. Laura Dugué

  • Curated by eLife

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    eLife Assessment

    This study introduces a novel method for estimating spatial spectra from irregularly sampled intracranial EEG data, revealing cortical activity across all spatial frequencies, which supports the global and integrated nature of cortical dynamics. It showcases important technical innovations and rigorous analyses, including tests to rule out potential confounds. However, further direct evaluation of the model, for example by using simulated cortical activity with a known spatial spectrum (e.g., an iEEG volume-conductor model that describes the mapping from cortical current source density to iEEG signals, and that incorporates the reference electrodes and the particular montage used), would even further strengthen the incomplete evidence.

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Abstract

The organization of the phase of electrical activity in the cortex is critical to inter-site communication, but the balance of this communication across large-scale (>8cm), macroscopic (>1cm) and mesoscopic (1cm to 1mm) ranges is an open question. Traveling waves in the cortex are spatial phase gradients, such that phase values change smoothly through the cortical sheet over time. Large-scale cortical traveling waves have been understudied compared to micro- or mesoscopic waves. The spatial frequencies (i.e., the spatial scales) of cortical waves have been characterized in the grey-matter for micro- and mesoscopic scales of cortex and show decreasing spatial power with increasing spatial frequency. This research, however, has been limited by the size of the measurement array, thus excluding large-scale traveling waves. Obversely, poor spatial resolution of extra-cranial measurements prevents incontrovertible large-scale estimates of spatial power via electroencephalogram and magnetoencephalogram. These limitations mean that the relative importance of large-scale traveling waves is unknown, and recent research has suggested waves measured extra-cranially are artefactual. We apply a novel method to estimate the spatial frequency spectrum of phase dynamics in order to quantify the uncertain large-scale range. Stereotactic electroencephalogram (sEEG) is utilized to leverage measurements of local-field potentials within the grey matter, while also taking advantage of the sometimes large extent of spatial coverage. Irregular sampling of the cortical sheet is offset by use of linear algebra techniques to empirically estimate the spatial frequency spectrum. We find the spatial power of the phase is highest at the lowest spatial frequencies (longest wavelengths), consistent with the power spectra ranges for micro- and meso-scale dynamics, but here shown up to the size of the measurement array (up to 8-16cm), i.e., approaching the entire extent of cortex. Low spatial frequencies dominate the cortical phase dynamics. This has important functional implications as it means that the phase measured for a single contact in the grey matter is more strongly a function of large-scale phase organization than local—within the same frequency band at least. This result arises across a wide range of temporal frequencies, from the delta band (1-3Hz) through to the high gamma range (60-100Hz).

Article activity feed

  1. eLife

    eLife Assessment

    This study introduces a novel method for estimating spatial spectra from irregularly sampled intracranial EEG data, revealing cortical activity across all spatial frequencies, which supports the global and integrated nature of cortical dynamics. It showcases important technical innovations and rigorous analyses, including tests to rule out potential confounds. However, further direct evaluation of the model, for example by using simulated cortical activity with a known spatial spectrum (e.g., an iEEG volume-conductor model that describes the mapping from cortical current source density to iEEG signals, and that incorporates the reference electrodes and the particular montage used), would even further strengthen the incomplete evidence.

  2. eLife

    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. The application to data illustrates the solidity of the method and their potential for discovery.

    Comments on revised submission:

    The authors have provided responses to the previous recommendations.

  3. eLife

    Reviewer #3 (Public review):

    Summary:

    The authors propose a method for estimating the spatial power spectrum of cortical activity from irregularly sampled data and apply it to iEEG data from human patients during a delayed free recall task. The 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) 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:

    Although the proposed method is evaluated in several indirect ways, a direct evaluation is lacking. This would entail simulating cortical current source density (CSD) with known spatial spectrum and using a realistic iEEG volume-conductor model to generate iEEG signals.

    Comments on revised version:

    In my original review, I raised the following issue:

    "The proposed method of estimating wavelength from irregularly sampled three-dimensional iEEG data involves several steps (phase-extraction, singular value-decomposition, triangle definition, dimension reduction, etc.) and it is not at all clear that the concatenation of all these steps actually yields accurate estimates. Did the authors use more realistic simulations of cortical activity (i.e. on the convoluted cortical sheet) to verify that the method indeed yields accurate estimates of phase spectra?"

    And the authors' response was:

    "We now included detailed surrogate testing, in which varying combinations of sEEG phase data and veridical surrogate wavelengths are added together. See our reply from the public reviewer comments. We assess that real neurophysiological data (here, sEEG plus surrogate and MEG manipulated in various ways) is a more accurate way to address these issues. In our experience, large scale TWs appear spontaneously in realistic cortical simulations, and we now cite the relevant papers in the manuscript (line 53)."

    The point that I wanted to make is not that traveling waves appear in computational models of cortical activity, as the authors seem to think. My point was that the only direct way to evaluate the proposed method for estimating spatial spectra is to use simulated cortical activity with known spatial spectrum. In particular, with "realistic simulations" I refer to the iEEG volume-conductor model that describes the mapping from cortical current source density (CSD) to iEEG signals, and that incorporates the reference electrodes and the particular montage used.

    Although in the revised manuscript the authors have provided indirect evidence for the soundness of the proposed estimation method, the lack of a direct evaluation using realistic simulations with ground truth as described above makes that remain sceptical about the soundness of the method.

  4. eLife

    Author response:

    The following is the authors’ response to the original reviews.

    eLife Assessment

    This study introduces a novel method for estimating spatial spectra from irregularly sampled intracranial EEG data, revealing cortical activity across all spatial frequencies, which supports the global and integrated nature of cortical dynamics. The study showcases important technical innovations and rigorous analyses, including tests to rule out potential confounds; however, the lack of comprehensive theoretical justification and assumptions about phase consistency across time points renders the strength of evidence incomplete. The dominance of low spatial frequencies in cortical phase dynamics continues to be of importance, and further elaboration on the interpretation and justification of the results would strengthen the link between evidence and conclusions.

    Public Reviews:

    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.

    We now explicitly state the experiments that were used, lines 715-716.

    (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.

    It is not generally accepted in neuroscience that power is higher at lowest spatial frequencies, and recent research concludes that traveling waves at this scale may be the result of artefactual measurement (Orczyk et al., 2022; Hindriks et al., 2014; Zhigalov & Jensen,2023). The question we answer is therefore timely and a source of controversy to researchers analysing TWs in cortex. While, in our view, the previous literature points in the direction of our conclusions (notably the work of Freeman et. al. 2003; 2000; Barrie et al. 1996), it is not conclusive at the scale we are interested in, specifically >8cm, and certainly not convincing to the proponents of ‘artefactual measurement’.

    We have added to a sentence to make this explicit in the abstract, lines 20-22. Please also note previous text at the end of the introduction, lines 140-148 and in the first paragraph of the discussion, lines 563-569.

    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.

    We have made the core part of the code available on github (line 1154), which should simplify adoption of the technique. We have urged, in the Discussion (lines 653-663), why habitual measurement of SF spectra is desirable, since the same task measured with EEG, sEEG or ECoG does not encompass the same spatial scales, and researchers may be comparing signals with different functional properties. Until reliable methods for estimating SF are available, not dependent on the layout of the recording array, data cannot be analysed to resolve this question. Publication of our results and methods will help this process along.

    (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.

    The new surrogate testing supports our conclusions. The sEEG arrays would not normally be a first choice to estimate SF spectra, for reasons of their sparsity, which may be why such estimates have not been done before. Yet, this is the research challenge that we sought to solve, and a problem for which there was no ready method to hand. Nevertheless, it is a problem that urgently needed to be solved given the current debate on the origin of large-scale TWs. We have now included detailed surrogate testing of real data plus varying strength model waves (Figure 6A and Supplementary Figure 4). We believe this should convince the reader that we are measuring the spatial frequency spectrum with sufficient accuracy to answer the central research question.

    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.

    We have now doubled the number of participants in the main analyses. Since each participant comprises a test of the central hypothesis, now the hypothesis test now has 23 replications (Supplementary Figures 2 and 3). There were four failures to reach significance due to under-powered tests, i.e., not enough contacts. This is sufficient test of the hypothesis and, in our opinion, not the primary obstacle to scientific acceptance of our results. The main obstacle is providing convincing tests that the method is accurate, and this is what we have focussed on. Publication of python code and the detailed methods described here enable any interested researcher to extend our method to other datasets.

    (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.

    The results are now clearly split into sections, where we first deal with all the single participant analyses, then the surrogate testing to confirm the basic results, then the participant aggregate results (Figure 7 and Supplementary Figure 7). The participant aggregate results reiterate the basic findings for the single participants. The key finding is straightforward (SF power decreases with SF) and required only one statistical analysis per subject.

    (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

    We have added two paragraphs to the discussion, in response to the reviewer suggestion (lines 606-623). These paragraphs place our high TF findings in the context of previous research.

    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 overinterpreted 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.

    “using wavelets with a narrow frequency band … 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”

    Our method uses very short time-window Morlet wavelets to avoid the assumptions of oscillations, i.e., long-lasting sinusoids in the signal, in the sense of sinusoidal waveforms, or limit cycles extending in time. Cortical TWs can only last one or two cycles (Alexander et al., 2006), requiring methods that are compact in the time domain to avoid underreporting the desired phenomena. Additionally, the short time-window Morlet wavelets have low frequency resolution, so they are robust with respect to shifts in frequency between sites. We now discuss this issue explicitly in the Methods (lines 658-674). This means the phase estimation methods used in the manuscript precisely do not have the problem of assuming narrow-band oscillations in the signal. The methods are also robust to the exact shape of the waveforms; the signal needs be only approximately sinusoidal; to rise and fall. This means the Fourier variant we use does not introduce ringing artefact that can be introduced using longer timeseries methods, such as FFT.

    “This step makes the assumption that phase at a particular frequency band is meaningful at all times”

    This important consideration is entrenched in our choice of methods. By way of explanatory background, we point out that this step is not the final step. Aggregation methods can be used to distinguish between signal and noise. In the simple case, event-locked time-series of phase can be averaged. This would allow consistent (non-noise) phase relations to be preserved, while the inconsistent (including noise) phase relations would be washed out. This is part of the logic behind all such aggregation procedures, e.g., phase-locking, coherence. SVD has the advantage of capturing consistent relations in this sense, but without loss of information as occurs in averaging (up to the choice of number of singular vectors in the final model). Specifically, maps of the spatial covariances in phase are captured in the order of the variance explained. Noise (in the sense conveyed by the reviewer) in the phase measurements will not contribute to highest rank singular vectors. SVD is commonly used to remove noise, and that is one of its purposes here. This point can be seen by considering the very smooth singular vectors derived from MEG (Figure 3F) in this new version of the manuscript. These maps of phase gradients pull out only the non-noisy relations, even as their weighted sums reproduce any individual sample to any desired accuracy.

    To summarize, the next step (of incorporating the phase measure into the SVD) neatly bypasses the issue of non-meaningful phase quantification. This is one of the reasons why we do not undertake the spatial frequency estimates on the raw matrices of estimated phase.

    We now include a new sub-paragraph on this topic in the methods, lines 831-838.

    In addition, we have reworded the first description of the methods with a new paragraph at the end of the introduction, which better balances the description of the steps involved. The two sentences (lines 162-166 highlight the issue of concern to the reviewer.

    “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.”

    The correct description of the full sEEG signal is beyond the scope of the present research. Our main goal, as stated, is to show that the hypothesis that ‘extra-cranial measurements of TWs is the result of projection from localized activity’ is not supported by the evidence of spatial patterns of activity in the cortex. Since this activity can be accessed as single frequency band (especially if localized sources create the large-scale patterns), analysis of SF on a TF-by-TF basis is sufficient.

    “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".

    We agree with the reviewer, even though we expect that the strongest influences on local phase are due to other cortical signals in the same band. The implicit assumption of the focus on bands of the same temporal frequency is now made explicit in the abstract (lines 31-34).

    A sentence addressing this issue had been added to the first paragraph of the discussion (lines 579-582).

    Inclusion of cross-frequency interactions would likely require a highly regular measurement array over the scales of interest here, i.e., the noise levels inherent in the spatial organization of sEEG contacts would not support such analyses.

    “Finally, the phase examples given in Supplementary Figure 5 are not strongly convincing to support this point.”

    We have removed the phase examples that were previously in Supplementary Figure 5 (and Figure 5 in the previous version of the main text), since further surrogate testing and modelling (Supplementary Figure 11) shows the LSVs from irregular arrays will inevitably capture mixtures of low and high SF signals. The final section of the Methods explains this effect in some detail. Instead, the new version of the manuscript relies on new surrogate testing to validate our methods.

    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?

    The motivation for our study was to show that large-scale TWs measured outside the cortex cannot be the result of more localized activity being ‘projected up’. In this case, the temporal frequency of the artefactual waves would be the same as the localized sources, so the criticism does not apply.

    “while this point could be considered as mostly semantic in nature”

    We have changed the terminology in the paper to better coincide with standard usage. Macroscopic now refers to >1cm, while we refer to >8cm as large-scale.

    “15 cm is essentially on the scale of the whole brain, without accounting for the fact that organization in sub-systems may occur.”

    We can assume that subtle frequency variation (e.g., within an alpha phase binding) is greatest at the largest scales of cortex, or at least not less varying than measurements within regions. This means that not considering frequency-drift effects will not inflate low spatial frequency power over high spatial frequency power. Even so, the power spectrum we estimated is approximately 1/SF, so that unmeasured cross-frequency effects in binding (causal influences on local phase) would have to overcome the strength of this relation for this criticism to apply, which seems unlikely.

    “would spatial phase organizations occurring in varying subsets of electrodes and with somewhat variable temporal frequency reliably be detected?”

    See our previous comments about the low temporal frequency resolution of two cycle Morlet wavelets. The answer is yes, up to the range approximated by half-power bandwidth, which is large in the case of this method (see lines 760-764).

    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.

    We have now added a sentence on this issue to the discussion (lines 600-602).

    However, our reading of the Zanos et al. paper is that the low temporal frequency (60-100Hz) contribution of spikes and spike patterns is negligible compared to genuine post-synaptic membrane fluctuations (see their Figure 3). These considerations come more strongly into play when correlations between LFP and spikes are calculated or spike triggered averaging is undertaken, since then a signal is being partly correlated with itself, or, partly averaged over the supposedly distinct signal with which it was detected.

    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.

    “even if the analyses were not affected by the distortion due to the narrow-frequency assumption”

    See our earlier comment about narrow TFs; this is not the case in the present work.

    The spiking activity analysis would be an interesting avenue for future research. It appears the 1000Hz sampling frequency in the present data is not sufficient for method described in Ray & Maunsell (2011). On a related topic, we have shown that large-scale traveling waves in the MEG and 8cm waves in ECoG can both be used to predict future localized phase at a single sensor/contact, two cycles into the future (Alexander et al., 2019). This approach could be used to predict spiking activity, by combining it with the reviewer’s suggestion. However, the current manuscript is motivated by the argument that measured large-scale extra-cranial TWs are merely projections of localized cortical activity. Since spikes do not arise in this argument, we feel it is outside the scope of the present research. We have added this suggestion to the discussion as a potential line of future research (lines 686-688).

    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 threedimensional 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.

    We disagree that the method is ad hoc, though the specific combination of SVD and multiscale differencing is novel in its application to sEEG. The SVD method has been used to isolate both ~30cm TWs in MEG and EEG (Alexander et al., 2013; 2016), as well as 8cm waves in ECoG (Alexander et al., 2013; 2019). In our opening examples in the results now reiterate these previous related findings, by way of example analysis of MEG data (Figure 3). This will better inform the reader on the extent of continuity of the method from previous research.

    Standard FFT has been used after interpolating between EEG electrodes to produce a uniform array (Alamia et al., 2023). There exist well-developed Fourier methods for nonuniform grids, such as simple interpolation, the butterfly algorithm, wavefield extrapolation and multi-scale vector field techniques. However, the problems for which these methods are designed require non-sparse sampling or less irregular arrays. The sEEG contacts (reduced in number to grey matter contacts) are well outside the spatial irregularity range of any Fourierrelated methods that we are aware of, particularly at the broad range of spatial scales of interest here (2cm up to 24cm). This would make direct comparison of these specialized Fourier method to our novel methods, in the sEEG, something of a straw-man comparison.

    We now include a summary paragraph in the introduction, which is a brief review of Fourier methods designed to deal with non-uniform sampling (lines 159-162).

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

    We have improved our description of the theoretical relation between Fourier analysis and SVD (additional material at lines 839-861 and 910-922). In fact, there are very strong links between the two methods, and now it should be clearer that our method does not rely on a mere qualitative resemblance.

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

    We now include a new surrogate testing procedure, which takes as inputs the empirical data and a model signal (of known spatial frequency) in various proportions. Thus, we test both the impact of small amount of surrogate signal on the empirical signal, and the impact of ‘noise’ (in the form of a small amount of empirical signal) added to the well-defined surrogate signal.

    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.

    Simulations of cortical activity do not seem the most direct way to achieve this goal. The first author has published in this area (Liley et. al., 1999; Wright et al., 2001), and such simulations, for both bulk and neuronally based simulations, readily display traveling wave activity at low spatial frequencies (indeed, this was the origin of the present scientific journey). The manuscript outlines these results in the introduction, as well as theoretical treatments proposing the same. Several other recent studies have highlighted the appearance of largescale travelling waves using connectome-based models (https://www.biorxiv.org/content/10.1101/2025.07.05.663278v1; https://www.nature.com/articles/s41467-024-47860-x), which we do not include in the manuscript for reasons of brevity. In short, the emergence of TW phenomenon in models is partly a function of the assumptions put into them (i.e., spatial damping, boundary conditions, parameterization of connection fields) and would therefore be inconclusive in our view.

    Instead, we rely on the advantages provided by the way our central research question has been posed: that the spatial frequency distribution of grey matter signal can determine whether extra-cranial TWs are artefactual. The newly introduced surrogate methods reflect this advantage by directly adding ground truth spatial frequency components to individual sample measurements. This is a less expensive option than making cortical simulations to achieve the same goal.

    For the same reasons, we include testing of the methods using real cortical signals with MEG arrays (for which we could test the effects of increasing sparseness of contacts, test the effects of average referencing, and also construct surrogate time-series with alternative spectra).

    Recommendations for the authors:

    Reviewer #2 (Recommendations for the authors):

    Major points

    Methods, Page 18: "... using notch filters to remove the 50Hz line signal and its harmonics ...": The sEEG data appear to have been recorded in North America, where the line frequency is 60 Hz. Is this perhaps a typo, or was a 50 Hz notch filter in fact applied here (which would be a mistake)?

    This has now been fixed in the text to read 60Hz. This is the notch filter that was applied.

    Minor points

    (1) While the authors do state that they are analyzing the "spatial frequency spectrum of phase dynamics" in the abstract, this could be more clearly emphasized. Specifically, the difference between signal power at different spatial frequencies (as analyzed by a standard Fourier analysis) and the organization of phase in space (as done here) could be more clearly distinguished.

    We now address this point explicitly on lines 167-172. We now include at the end of the results additional analyses where the TF power is included. This means that the effects of including signal power at different temporal frequencies can be directly compared to our main analysis of the SF spectrum of the phase dynamics.

    (2) Figure 1A-C: It was not immediately clear what the lengths provided in these panels (e.g."> 40 cm cortex", "< 10 cm", "< 30 cm") were meant to indicate. This could be made clearer.

    Now fixed in the caption.

    (3) Figure 2A: If this is surrogate data to explain the analysis technique, it would be helpful to note explicitly at this point.

    This Figure has been completely reworked, and now the status of the examples (from illustrative toy models to actual MEG data) should be clearer.

    (4) Figure 4A: Why change from "% explained variance" for the example data in Figure 2C to arbitrary units at this point?

    This has now been explicitly stated in the methods (lines 1033-1036).

    (5) Page 15: "This means either the results were biased by a low pass filter, or had a maximum measurable...": If the authors mean that the low-pass filter is due to spatial blurring of neural activity in the EEG signal, it would be helpful to state that more directly at this point.

    Now stated directly, lines 567-568.

    (6) Page 23: "...where |X| is the complex magnitude of X...": The modulus operation is defined on a complex number, yet here is applied to a vector of complex numbers. If the operation is elementwise, it should be defined explicitly.

    ‘Elementwise’ is now stated explicitly (line 1020).

    Reviewer #3 (Recommendations for the authors):

    In the submitted manuscript, the authors propose a method to estimate spatial (phase) spectra from irregularly sampled oscillatory cortical activity. They apply the method to intracranial (iEEG) data and argue that cortical activity is organized into global waves up to the size of the entire cortex. If true, this finding is certainly of interest, and I can imagine that it has profound implications for how we think about the functional organization of cortical activity.

    We have added a section to the discussion outlining the most radical of these implications: what does it mean to do source localization when non-local signals dominate? Lines 670-681.

    The manuscript is well-written, with comprehensive introduction and discussion sections, detailed descriptions of the results, and clear figures. However, the proposed method comprised several ad hoc elements and is not well-founded mathematically, its performance is not adequately assessed, and its limitations are not sufficiently discussed. As such, the study failed to convince (me) of the correctness of the main conclusions.

    We now have a direct surrogate testing of the method. We have also improved the mathematical explanation to show that the link between Fourier analysis and SVD is not ad hoc, but well understood in both literatures. We had addressed explicitly in the text all of the limitations raised by the reviewers.

    Major comments

    (1) The main methodological contribution of the study is summarized in the introduction section:

    "The irregular sampling of cortical spatial coordinates via stereotactic EEG was partly overcome by the resampling of the phase data into triplets corresponding to the vertices of approximately equilateral triangles within the cortical sheet."

    There exist well-established Fourier methods for handling irregularly sampled data so it is unclear why the authors did not resort to these and instead proposed a rather ad hoc method without theoretical justification (see next comment).

    We have re-reviewed the literature on non-uniform Fourier analysis. We now briefly review the Fourier methods for handling irregularly sampled data (lines 155-162) and conclude that none of the existing methods can deal with the degree of irregularity, and especially sparsity, found for the grey-matter sEEG contacts.

    (2) In the Appendix, the authors write:

    "For appropriate signals, i.e., those with power that decreases monotonically with frequency, each of the first few singular vectors, v_k, is an approximate complex sinusoid with wavenumber equal to k."

    I don't think this is true in general and if it is, there must be a formal argument that proves it. Furthermore, is it also true for irregularly sampled data? And in more than one spatial dimension? Moreover, it is also unclear exactly how the spatial Fourier spectrum is estimated from the SVD.

    In response to these reviewer queries, we now spend considerably more time in the conceptual set-up of the manuscript, giving examples of where SVD can be used to estimate the Fourier spectrum. We have now unpacked the word ‘appropriate’ and we are now more exact in our phrasing. This is laid out in lines 843-850 of the manuscript. In addition, the methods now describe the mathematical links between Fourier analysis and SVD (lines 851861 and 910-922).

    The authors write:

    "The spatial frequency spectrum can therefore be estimated using SVD by summing over the singular values assigned to each set of singular vectors with unique (or by binning over a limited range of) spatial frequencies. This procedure is illustrated in Figure 1A-C."

    First, the singular vectors are ordered to decreasing values of the corresponding singular values. Hence, if the singular values are used to estimate spectral power, the estimated spectrum will necessarily decrease with increasing spatial frequency (as can be seen in Figure 2C). Then how can traveling waves be detected by looking for local maxima of the estimated power spectra?

    TWs are not detected by looking for local maxima in the spectra. Our work has focussed on the global wave maps derived from the SVD of phase (i.e., k=1-3), which also explain most of the variance in phase. This is now mentioned in the caption to Figure 3 (lines 291-294).

    Second, how are spatial frequencies assigned to the different singular vectors? The proposed method for estimating spatial power spectra from irregularly sampled data seems rather ad hoc and it is not at all clear if, and under what conditions, it works and how accurate it is.

    The new version of the manuscript uses a combination of the method previously presented (the multi-scale differencing) and the method previously outlined in the supplementary materials (doing complex-valued SVD on the spatial vectors of phase). We hope that along with the additional expository material in the methods the new version is clearer and seems less ad hoc to the reviewer. Certainly, there are deep and well-understood links between Fourier analysis and SVD, and we hope we have brought these into focus now.

    (3) The authors define spatial power spectra in three-dimensional Euclidean space, whereas the actual cortical activity occurs on a two-dimensional sheet (the union of two topological 2spheres). As such, it is not at all clear how the estimated wavelengths in three-dimensional space relate to the actual wavelengths of the cortical activity.

    We define spatial power spectra on the folded cortical sheet, rather than Cartesian coordinates. We use geodesic distances in all cases where a distance measurement is required. We have included two new figures (Figure 5 and Supplementary Figure1) showing the mapping of the triangles onto the cortical sheet, which should bring this point home.

    (4) The authors' analysis of the iEEG data is subject to a caveat that is not mentioned in the manuscript: As a reference for the local field potentials, the average white-matter signal was used and this can lead to artifactual power at low spatial frequencies. This is because fluctuations in the reference signal are visible as standing waves in the recording array. This might also explain the observation that

    "A surprising finding was that the shape of the spatial frequency spectrum did not vary much with temporal frequency."

    because fluctuations in the reference signal are expected to have power at all temporal frequencies (1/f spectrum). When superposed with local activity at the recording electrodes, this leads to spurious power at low spatial frequencies. Can the authors exclude this interpretation of the results?

    The new version of the manuscript deals explicitly with this potential confound (lines 454467). First, the artefactual global synchrony due to the reference signal (the DC component in our spatial frequency spectra of phase) is at a distinct frequency from the lowest SF of interest here. The lowest spatial frequency is a function of the maximum spatial range of the recording array and not overlapping in our method with the DC component, despite the loss of SF resolution due to the noise of the spatial irregularity of the recording array. This can be seen from consideration of the SF tuning (Figure 4) for the MEG wave maps shown in Figure 3, and the spectra generated for sparse MEG arrays in Supplementary Figure 5. Additionally, this question led us to a series of surrogate tests which are now included in the manuscript. We used MEG to test for the effects of average reference, since in this modality the reference free case is available. The results show that even after imposing a strong and artefactual global synchrony, the method is highly robust to inflation of the DC component, which either way does not strongly influence the SF estimates in the range of interest (4c/m to 12c/m for the case of MEG).

    (5) Related to the previous comment: Contrary to the authors' claims, local field potentials are susceptible to volume conduction, particularly when average references are used (see e.g. https://www.cell.com/neuron/fulltext/S0896-6273(11)00883-X)

    Methods exist to mitigate these effects (e.g. taking first- or second-order spatial differences of the signals). I think this issue deserves to be discussed.

    We have reviewed this research and do not find it to be a problem. The authors cited by the reviewer were concerned with unacknowledged volume conduction up to 1 cm for LFP. The maximum spatial frequency we report here is 50c/m, or equivalent to 2cm. While the intercontact distance on the sEEG electrodes was 0.5cm, in practice the smallest equilateral triangles (i.e., between two electrodes) to be found in the grey matter was around 2cm linear size. We make no statements about SF in the 1cm range. We do now cite this paper and mention this short-range volume conduction (lines 602-605). The method of taking derivatives has the same problems as source localization methods. They remove both artefactual correlations (volume conduction) and real correlations (the low SF interactions of interest here). We mention this now at lines 667-669. In addition, our method to remove negative SF components from the LSVs ameliorates the effects of average referencing. There are now more details in the Methods about this step (lines 924-947), as well as a new supplementary figure illustrating its effects on signal with a known SF spectrum (MEG, supplementary Figure 6).

    (6) Could the authors add an analysis that excludes the possibility that the observed local maxima in the spectra are a necessary consequence of the analysis method, rather than reflecting true maxima in the spectra? A (possibly) similar effect can be observed in ordinary Fourier spectra that are estimated from zero-mean signals: Because the signals have zero mean, the power spectrum at frequency zero is close to zero and this leads to an artificial local maximum at low frequencies.

    We acknowledge the reviewer’s mathematical point. We do not agree that it could be an issue, though it is important to rule it out definitively. First, removing the DC component will only produce an artefactual low SF peak if the power at low SF is high. This may occur in the reviewer’s example only because temporal frequency has a ~1/f spectrum. If the true spectrum is flat, or increasing in power with f, no such artificial low SF will be produced (see Supplementary Figure 5G). Additionally,

    (1) The DC component is well separated from the low SF components in our method;

    (2) We now include several surrogate methods which show that our method finds the correct spectral distribution and is not just finding a maximum at low SFs due to the suggested effect (subtraction of the DC component). Analysis of separated wave maps in MEG (Figures 3 & 4) shows the expected peaks in SF, increasing in peak SF for each family of maps when wavenumber increases (roughly three k=1 maps, three k=2 etc.). A specific surrogate test for this query was also undertaken by creating a reverse SF spectrum in MEG phase data, in which the spectrum goes linearly with f over the SF range of interest, rather than the usual 1/f. Our method correctly finds the former spectrum (Supplementary Figure 5). Additionally, we tested for the effects of introducing the average reference and the effects of our method to remove the DC component of the phase SF spectrum (Supplementary Figure 6). We can definitively rule out the reviewer’s concern.

    A related issue (perhaps) is the observation that the location of the maximum (i.e. the peak spatial frequency of cortical activity) depends on array size: If cortical activity indeed has a characteristic wavelength (in the sense of its spectrum having a local maximum) would one not expect it to be independent of array size?

    This is only true when making estimates for relatively clean sinusoidal signals, and not from broad-band signals. Fourier analysis and our related SVD methods are very much dependent on maximum array size used to measure cortical signals. This is why the first frequency band (after the DC component) in Fourier analysis is always at a frequency equivalent to 1/array_size, even if the signal is known to contain lower frequency components. We now include a further illustration of this in Figure 3, a more detailed exposition of this point in the methods, and in Supplementary Figure 11 we provide a more detailed example of the relation between Fourier analysis and SVD when grids with two distinct scales are used.

    In short, it is not possible, mathematically, to measure wavelengths greater than the array size in broad-band data. This is now stated explicitly in the manuscript (lines 143-144). A common approach in Neuroscience research is to first do narrowband filtering, then use a method that can accurately estimate ‘instantaneous’ phase change, such as the Hilbert transform. This is not possible for highly irregular sEEG arrays.

    (7) The proposed method of estimating wavelength from irregularly sampled threedimensional iEEG data involves several steps (phase-extraction, singular value decomposition, triangle definition, dimension reduction, etc.) and it is not at all clear that the concatenation of all these steps actually yields accurate estimates.

    Did the authors use more realistic simulations of cortical activity (i.e. on the convoluted cortical sheet) to verify that the method indeed yields accurate estimates of phase spectra?

    We now included detailed surrogate testing, in which varying combinations of sEEG phase data and veridical surrogate wavelengths are added together.

    See our reply from the public reviewer comments. We assess that real neurophysiological data (here, sEEG plus surrogate and MEG manipulated in various ways) is a more accurate way to address these issues. In our experience, large scale TWs appear spontaneously in realistic cortical simulations, and we now cite the relevant papers in the manuscript (line 53).

    Minor comments

    (1) Perhaps move the first paragraph of the results section to the Introduction (it does not describe any results).

    So moved.

    (2) The authors write:

    "The stereotactic EEG contacts in the grey matter were re-referenced using the average of low-amplitude white matter contacts"

    Does this mean that the average is taken over a subset of white-matter contacts (namely those with low amplitude)? Or do the authors refer to all white-matter contacts as "low-amplitude"? And had contacts at different needles different references? Or where the contacts from all needles pooled?

    A subset of white-matter contacts was used for re-referencing, namely those 50% with lowest amplitude signals. This subset was used to construct a pooled, single, average reference. We have rephrased the sentences referring to this procedure to improve clarity (line 202 and 743745).

  5. Version published to 10.7554/elife.100674.2 on eLife
  6. Version published to 10.7554/elife.100674 on eLife
  7. eLife

    eLife Assessment

    This study introduces a novel method for estimating spatial spectra from irregularly sampled intracranial EEG data, revealing cortical activity across all spatial frequencies, which supports the global and integrated nature of cortical dynamics. The study showcases important technical innovations and rigorous analyses, including tests to rule out potential confounds; however, the lack of comprehensive theoretical justification and assumptions about phase consistency across time points renders the strength of evidence incomplete. The dominance of low spatial frequencies in cortical phase dynamics continues to be of importance, and further elaboration on the interpretation and justification of the results would strengthen the link between evidence and conclusions.

  8. eLife

    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

  9. eLife

    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.

  10. eLife

    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.

  11. eLife

    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

  12. Version published to 10.7554/elife.100674.1 on eLife
  13. Version published to 10.1101/2024.06.04.597334 on bioRxiv