^{1}

^{2}

^{3}

^{2}

^{2}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

Edited by: Pedro Antonio Valdes-Sosa, Clinical Hospital of Chengdu Brain Science Institute, China

Reviewed by: Guido Nolte, Fraunhofer Society (FHG), Germany; Bernadette Van Wijk, University of Amsterdam, Netherlands

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Perceptual, motor and cognitive processes are based on rich interactions between remote regions in the human brain. Such interactions can be carried out through phase synchronization of oscillatory signals. Neuronal synchronization has been primarily studied within the same frequency range, e.g., within alpha or beta frequency bands. Yet, recent research shows that neuronal populations can also demonstrate phase synchronization between different frequency ranges. An extraction of such cross-frequency interactions in EEG/MEG recordings remains, however, methodologically challenging. Here we present a new method for the robust extraction of cross-frequency phase-to-phase synchronized components. Generalized Cross-Frequency Decomposition (GCFD) reconstructs the time courses of synchronized neuronal components, their spatial filters and patterns. Our method extends the previous state of the art, Cross-Frequency Decomposition (CFD), to the whole range of frequencies: it works for any _{1} and _{2} whenever _{1}:_{2} is a rational number. GCFD gives a compact description of non-linearly interacting neuronal sources on the basis of their cross-frequency phase coupling. We successfully validated the new method in simulations and tested it with real EEG recordings including resting state data and steady state visually evoked potentials (SSVEP).

Synchronization between neuronal populations is considered to be a key mechanism underlying interactions between distinct groups of neurons. According to the communication-through-coherence (CTC) hypothesis, efficient communication between two groups of neurons is only possible, when the oscillations are phase-locked (coherent) (Fries,

Among the various synchronization phenomena, interactions within the same frequency band (with ratio 1:1, i.e., gamma-gamma, alpha-alpha or beta-beta) have been mostly studied and well characterized both in humans (Varela et al.,

In addition to within frequency synchronization, coupling between two frequency bands has also been observed in humans (Sauseng et al.,

Several types of cross-frequency coupling have been revealed: amplitude-to-amplitude, phase-to-phase, amplitude-to-phase cross-frequency coupling (Canolty et al.,

A particularly important form of neuronal interactions is phase-to-phase synchronization since it represents stable spike-time relationships between distant neuronal oscillations and, therefore, it directly coordinates phase coupling of fast and slow oscillations (Siebenhühner et al.,

Albeit potentially important for cognitive function and brain computation, phase-to-phase coupling is not easy to characterize in noninvasive recordings, since the prevalent methods suffer from a number of difficulties, such as non-sinusoidal nature of oscillations, non-stationarity of the signals and large amount of noise in EEG data (Nikulin and Brismar,

In order to avoid the limitations of the previous methods, in the present study we propose a new approach for the extraction of components demonstrating cross-frequency phase coupling. We refer to this new method as Generalized Cross-Frequency Decomposition (GCFD). The GCFD is a generalization of CFD (Nikulin et al.,

Let

where

where

To study ordinary 1:1 phase synchrony of two narrow-band signals _{1}(_{2}(

in a long enough time window _{1}(_{2}(_{2}(_{1}(

Now let us generalize this to the case when the central frequencies _{1} and _{2} of _{1}(_{2}(

see Rosenblum et al. (_{1} and φ_{2} but now we consider generalized cyclic phase difference

Now we say that _{1} and _{2} are in _{p, q}(

We use the following _{1} and _{2} in the time window

In case of empirically obtained recordings, raw E/MEG data are inspected and cleaned of blinking and other muscle artifacts. This is done by applying Independent Component Analysis (FastICA algorithm, see Hyvärinen et al.,

Cleaned E/MEG data is then low-pass filtered into a wide band starting from 0.5 Hz to about 150% of the highest frequency to be used in the analysis. If we are interested in cross-frequency interaction between a 20 Hz rhythm and a 30 Hz rhythm, the high cut-off frequency of the filter would be 45 Hz. This is done to further clean the data of any high-frequency noise.

The general workflow of the proposed method is presented in Figure

choose one band that represents a “

identify one or few candidate components for the reference band;

for the other frequency, using non-linear optimization find the unique components which are in the strongest synchrony with the reference rhythm candidates;

output the pair(s) which exhibit the best synchronization.

Outline of the algorithm.

Below we address each of these steps in detail, as well as some auxiliary steps.

In the following analysis, the frequency bands _{1} and _{2} play two different roles. First, we pick a few strongest rhythmic components from the reference band using Spatio-Spectral Decomposition, see Subsection 2.3.2. These components become our reference signals. Second, for each of the reference signals, we find the component in the fit band which is the most synchronous to the rhythmic activity in the reference signal. This is done with the new Cross-Frequency Phase Fitting (XPF) method which is the core of this paper.

When there is no particular reason to prefer _{1} over _{2} or vice versa as a reference band, we recommend choosing the band with smaller frequency. This way polynomial expressions in the nonlinear optimization procedure have lower degrees and thus the method converges faster and is more accurate. In the following, we always assume that _{1} is the reference band and _{2} is the fit band.

We emphasize here that XPF is asymmetric with respect to the order of the frequency bands _{1} and _{2}. That is, the same analysis but with the bands swapped places is not, in general, guaranteed to yield the same results. However numerical experiments on simulated data indicate that, while the results might be different, both approaches are very close to the ground truth. Spatial patterns for the components in the fit band are, on average, found more accurately then spatial patterns for the components in the reference band, we elaborate on this in the sections below.

For the reference frequency band _{1}, we perform a decomposition procedure which allows us to extract relevant oscillatory components and to reduce the dimensionality of the data. Among such procedures, Spatio-Spectral Decomposition (SSD) method (Nikulin et al.,

Essentially, SSD maximizes the Signal-to-Noise Ratio which is defined as the ratio of the power at the narrow frequency band of interest to the power of the noise at the surrounding flanking frequency ranges. See Nikulin et al. (

For the following analysis, from the reference band we take only the components with the largest eigenvalues, and discard the rest of the reference band signal space. Each particular dataset may have different numbers of these significant components. We recommend first using the default number of 5 components, and adjust it later if needed.

Cross-Frequency Phase Fitting (XPF) is the core procedure of GCFD. Its inputs are a single narrow-band reference signal _{1} and a collection of multiple narrow-band signals _{1}(_{I}(_{2}, where _{1}:_{2} =

We assume that _{i} is a linear combination of an unknown “target” signal _{N} noise components _{i}:

where _{N} + 1)-by-_{i} ∈ ℝ such that _{i} would be the first column of ^{−1}.

For every channel _{i}(

Now for

An equivalent way to write this down is

Note the difference between the notations (·)^{[q]} and (·)^{q}. We keep the latter for the standard complex power

By construction, whenever some other signal

As indicated in Nikulin et al. (

Based on this evidence, we will be maximizing the correlation between ℂ-valued signals _{1}, _{2}. We can find the coefficients _{i} as the solution to the optimization problem

Note that, while both _{i} in the _{i} and iteratively descend to the local minimum. Practically, multiple modern high-level computation suites offer compact solvers for Constrained Nonlinear Least Squares Problems. In our implementation we used

Finally, for each reference component _{1}, we have the coefficients of the spatial filter _{i} and the component _{i}_{i}(_{2} which is in cross-frequency phase synchrony with

The last step is to convert the spatial filters for the newly found components into the corresponding spatial patterns (see Haufe et al.,

where _{i}) is the matrix of sensor space signals in _{2} frequency band. This calculation is equivalent (Haufe et al.,

The final output of the algorithm is the most synchronous pair (

Because the computational complexity of the core nonlinear optimization problem rapidly increases as the number of channels ^{3}, depending on a particular solver), it is sometimes beneficial to reduce the dimension of the problem. For this we project the original sensor space into a linear subspace of fewer dimensions. We apply Spatio-Spectral Decomposition (see Subsection 2.3.2) to the narrow-band signals at frequency _{2} and neglect all but 15 most significant components. For example, for ^{3} = 64 times. For a larger number of sensors, this step becomes even more critical.

Then we apply the same nonlinear optimization procedure (see Subsection 2.3.3) to the same reference signal and the first 15 SSD components as fit signals. This is possible because the optimization actually employs no information about the real nature of sensors' positions and works equally well for such “virtual sensors.” Finally, each spatial pattern

where

Note that this speed optimization might come at a cost of reduced accuracy of the algorithm, because the search for the best spatial filter is then performed in some 15-dimensional subspace rather than the whole filter space. Thus for some cases one may consider to disable this option. However, this option is extremely useful for a fast rough search for synchronized components across many frequency bands' combinations.

In some experimental scenarios, we may want to study synchronization of brain oscillations to a certain signal which is already known. For example, Bayraktaroglu et al. (

We conducted numerical simulations to assess the capability of detached XPF to tackle these situations with the known reference signals. These tests showed that in this mode the overall pattern reconstruction quality for synchronized sources is significantly better than of a more sophisticated full GCFD algorithm (see below). Moreover, the detached XPF is capable of precise source reconstruction at greater values of

In the above, we searched for components in cross-frequency phase synchrony:

A weaker condition called

To search for cross-frequency phase-locked components in a E/MEG signal, we modify the formula (3) for the frequency warp of reference components. We choose an integer

for

Our preliminary analysis has shown that the distribution of phase differences between components is quite broad thus indicating that there is no need to exactly align the phases of both signals to have 0 or π difference. A similar number of

For algorithm performance tests we picked

We aimed at simulating phase couplings between theta, alpha, beta and low gamma oscillations. Some of them, like alpha:beta (1:2), were previously observed in E/MEG recordings. For each ratio

For each simulation we first generated 5 independent pairs of cross-frequency synchronized oscillatory signals with different frequency ratio

In addition we used 100 mutually independent noise sources with 1/

Each of 10 synchronized oscillatory signals and 100 noise signals coresponded to a respective current dipole randomly chosen from the nodes of triangularly tesselated cortical mantle. Dipole orientation was also randomized. We used a realistic three compartment volume conductor forward model (Nolte and Dassios,

In addition, we normalized the signal-to-noise ratios (SNR) of all our signals, which we define as the ratio of the mean variance of signals across channels for each projected signal dipole and the mean variance across channels for the whole projected noise cumulative. We tested the performance of our algorithm for SNR values of 0.1, 0.5, 1.0, 2.0, see below.

Then for each simulation we ran the source reconstruction GCFD algorithm explained in Subsections 2.2–2.4. The resulting spatial patterns for the recovered sources in _{1} frequency band are then compared to the true patterns which are known

In addition to these simulations we have also produced simulations for 2:3 case (SNR=0.5) mixing five components at 20 Hz frequency range but the corresponding 30 Hz components were produced by the frequency warping of another five 10 Hz components which were independent of the first 10 Hz components. This way we produced five components at 20 and 30 Hz frequency bands which were independent within and between these frequency bands. Then we performed 100 of such simulations and calculated the mean pattern divergence (see next section).

We measured the difference ϵ between the true pattern

see Nikulin et al. (

Note that the recovered patterns come in no specific order related to the original patterns. A sorting procedure is required to find the actual recovery error for each signal. Namely, we calculated all the pairwise pattern divergences between all the recovered patterns and all the original patterns. Then we used a greedy algorithm to match the recovered patterns with the original patterns: we first find the pair of a recovered pattern and an original pattern with the smallest divergence, and then we remove both of them from the pattern sets. Then we repeat the procedure with the remaining patterns to find the second best match

Each simulation yields a vector of 5 pattern divergence numbers. Multiplied by 100 simulations, in the end we have 500 numbers for each frequency ratio

For the empirical data, we lack the information about the ground truth patterns, and thus we cannot directly measure the divergences between the ground truth patterns and the estimated patterns. In this case we rather use pattern divergence as a measure of similarity between the two patterns relating to cross-frequency coupled components.

The GCFD algorithm has been tested on EEG data obtained at the Centre for Cognition and Decision Making at Higher School of Economics (HSE, Moscow). All the experimental procedures were approved by the local Ethics Committee. The participants signed an informed consent form. 32 healthy subjects (12 men, right-handed, mean age 23 years) participated in the EEG experiment. The EEG data were recorded with 60 active electrodes of BrainVision actiCHamp (Brain Products GmbH) according to the extended version of the 10–20 system. The data were sampled at 500 Hz. Active channels were referenced against the mean of two mastoid electrodes. The electrooculogram was recorded with electrodes placed at the outer canthi and below the right eye. The EEG recordings were offline filtered in the frequency range 0.5–40 Hz. Spectral analysis by means of FFT (fast Fourier transform) was performed with Hammings̀ window of 3 seconds. Participants were seated comfortably before a dark screen for 10 minutes while fixating their eyes on the cross in front of them.

For the consecutive offline analysis the EEG data were downsampled to 200 Hz, the data length was 10 minutes. We reduced the dimension of the signal using the 5 strongest SSD components in both frequency ranges of interest ^{*}_{1} ± 1 Hz and ^{*}_{2} ± 1 Hz. The settings for SSD were as follows: cut-off frequency range for the band-pass filter was ^{*}_{1} ±1 Hz and ^{*}_{2} ± 1 Hz; cut-off frequency range for the lowest and highest frequencies defining flanking intervals was ^{*}_{1} ± 3 Hz and ^{*}_{2} ± 3 Hz; cut-off frequency range for the band-stop filter was ^{*}_{1} ± 2 Hz and ^{*}_{2} ± 2 Hz. We looked for the strongest synchronous components for

To demonstrate performance of the GCFD we used EEG data obtained at the Centre for Cognition and Decision Making at Higher School of Economics (HSE, Moscow) with Steady State Visually Evoked Potentials (SSVEP), which were recorded for BCI experiments (Işcan and Nikulin,

For the consecutive GCFD analysis we concatenated all 3-second trial recordings into a single 75-second multi-channel signal for each subject and each flickering frequency. Since the raw data was filtered with high-pass at 0.53 Hz before the concatenation, there were no offsets between the epochs and thus the following filtering did not produce artifacts as was also confirmed by the visual inspection.

We reduced the dimension of the signal using the 15 strongest SSD components in both frequency ranges of interest ^{*}_{1} ± 1Hz and ^{*}_{2} ± 1Hz. For SSD we used following settings: cut-off frequency range for the band-pass filter was ^{*}_{1} ± 1Hz and ^{*}_{2} ± 1Hz; cut-off frequency range for the lowest and highest frequencies defining flanking intervals was ^{*}_{1} ± 3Hz and ^{*}_{2} ± 3Hz; cut-off frequency range for the band-stop filter was ^{*}_{1} ± 2Hz and ^{*}_{2} ± 2Hz. For computational convenience in our analysis we approximated the real flickering frequencies 5.45, 8.57, 12, 15 Hz with the integer frequencies 6, 9, 12, 15 Hz respectively.

We used the nonparametric permutation test to evaluate statistical significance of the results (Maris and Oostenveld,

We divided recordings and combined segments in random order from the data relating to finding a spatial filter

Next, we ran our algorithm on the permuted recording and obtained new paired signal for the reference signal. Then we created permutation distribution by repeating this procedure 1000 times and computing for each pair a corresponding phase locking value (1). The null hypothesis under this permutation test was that all permuted pairs and original pair belonged to the same distribution. Finally we computed the P-value for original pair of signals and if it was smaller than 0.05 we concluded that the result was statistically significant.

This is a frequently used approach for non-parametric permutation testing (Hesterberg et al.,

First we tested how accurately the detached XPF procedure (recall Subsection 2.3.3) recovers source spatial patterns in the scenario when true sources are provided as reference signals and thus we only have to find cross-frequency coupled components in the fit band. Note that this is also a valid simulation of an experiment when the entraining signal is known from other sources such as a cardiogram, a myogram, oscillatory signal from the transcranial alternating current stimulation, a visual or an auditory input etc. Recall Subsection 2.5 for details. For each pair

and each SNR = 1.0, 0.5, 0.1 we performed 100 simulations similar to the ones described in Subsection 2.7. The results are presented on Figure

Pattern reconstruction accuracy of detached XPF.

In this test we essentially eliminated all the errors which relate to the performance of SSD algorithm at the step of the initial extraction of the reference components. As we will show later, insufficiently clean extraction of SSD components can lead to reconstruction errors for the GCFD algorithm, compare Figures

In general the results of this test demonstrate that the core optimization procedure performs well for all tested frequency ratios which are often met in E/MEG signal synchronization studies.

Simulations based on realistic head modeling showed that GCFD algorithm reliably recovers cross-frequency coupled components at different frequencies, relating to each other through rational numbers

Figure

Source patterns (SP) and recovered patterns (RP) for 5 pairs of simulated synchronized sources at 20 and 30 Hz. SNR = 0.1. The color-scale is in arbitrary units.

To measure the overall pattern recovery quality of GCFD algorithm, we performed series of 100 simulations for each of frequency ratios (6)

Pattern reconstruction accuracy for the whole GCFD.

Naturally, as the SNR decreased, for each fixed ratio

Overall we concluded that for all tested frequencies and SNRs ≥0.1 the pattern recovery accuracy is sufficient for the analysis of synchronized sources in real E/MEG recordings.

When simulating uncoupled sources we observed that at SNR = 0.5 pattern divergence was on average 0.33 which was at least 15 times larger than the pattern divergence typical for coupled sources. Such values of pattern divergence indicate that the extracted topographies were very different from the topographies of the original uncoupled sources. This in turn indicates that in simulations where the sources are not coupled, GCFD is not able to recover simulated components.

First we tested how the GCFD works for the resting state EEG recordings described in 2.9.1. We chose 8 subjects with the most pronounced power peaks in the alpha, beta and gamma frequency range and ran GCFD analysis to identify cross-frequency coupled synchronous sources. The base frequencies were taken from the alpha range 8–12 Hz. Figure

Examples of cross-frequency coupled synchronous oscillations detected with the GCFD algorithm for 2:1 and 2:3 search.

All components' pairs (2:1 and 2:3) from all subjects extracted with GCFD for 2:1 (left) and 2:3 (right) search.

For the frequency ratio 2:1 most (92%) computed phase locking values were statistically significant. However, the correlation between PLV and pattern divergence was not particularly strong (^{2} = 0.49) thus indicating that only part of the data is likely to represent a coupling due to non-sinusoidal shape of neuronal oscillations.

For the frequency ratio 2:3 only 20% of the PLVs were statistically significant. We also analyzed the relationship between the strength of phase coupling and relevant pattern divergence. We observed a negative correlation for the case of 2:1 (^{−6}) and no correlation for 2:3 (

We also tested the GCFD on the Steady State Visual Evoked Potentials (SSVEP) signals described in 2.9.2. The goal was to demonstrate that our approach is able to find cross-frequency phase synchronized harmonics of SSVEP signals with frequencies relating to each other through a rational relationship

We presented a new algorithm for the detection and extraction of cross-frequency phase-to-phase synchronized neuronal components. Generalized Cross-Frequency Decomposition is able to reconstruct both the time courses of synchronized neuronal components and corresponding spatial filters and patterns.

We showed that the GCFD was capable of detecting synchrony between frequencies related by a rational relationship p:q, for

The most common approach is to calculate cross-frequency synchronization in sensor space (Schanze and Eckhorn,

Numerical simulations showed that GCFD can recover interacting sources even when they are masked by a very strong noise (SNR = 0.1), see Figure

While looking for cross-frequency synchronization, there is always a possibility to detect CFS not due to genuine neuronal interactions, but also due to non-sinusoidal shape of oscillations (Gaarder and Speck,

In order to control for this side-effect, we calculated the pattern divergence between the spatial patterns of the reference signal and the synchronized rhythmic components. If the patterns were similar we considered them to be harmonics. For 2:1 case we observed many similar patterns with high PLV. This was not the case for 2:3 coupling where we observed a smaller number of similar patterns and the average PLV was lower.

We showed that GCFD can be applied to find phase coupling in cases when we use pre-selected peaks on the basis of the spectra as shown for resting state Figure

As the direction for the future research, it would be interesting to apply GCFD to investigate the role of cross-frequency phase synchrony between different networks, demonstrating strong within-frequency coupling, in a variety of cognitive tasks suggested to engage such integration, e.g., in visual working memory (Siebenhühner et al.,

This study was carried out in accordance with the recommendations of HSE Ethics Committee with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the HSE Ethics Committee.

DV designed study, performed research, wrote the paper. ID performed research, wrote the paper. AM performed research, wrote the paper. BG and VN designed study, wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Dr. Zafer İşcan for discussing SSVEP data.