The effects of spatial leakage correction on the reliability of EEG‐based functional connectivity networks

Abstract

Electroencephalography (EEG) functional connectivity (FC) estimates are confounded by the volume conduction problem. This effect can be greatly reduced by applying FC measures insensitive to instantaneous, zero‐lag dependencies (corrected measures). However, numerous studies showed that FC measures sensitive to volume conduction (uncorrected measures) exhibit higher reliability and higher subject‐level identifiability. We tested how source reconstruction contributed to the reliability difference of EEG FC measures on a large (n = 201) resting‐state data set testing eight FC measures (including corrected and uncorrected measures). We showed that the high reliability of uncorrected FC measures in resting state partly stems from source reconstruction: idiosyncratic noise patterns define a baseline resting‐state functional network that explains a significant portion of the reliability of uncorrected FC measures. This effect remained valid for template head model‐based, as well as individual head model‐based source reconstruction. Based on our findings we made suggestions how to best use spatial leakage corrected and uncorrected FC measures depending on the main goals of the study.

High reliability of functional connectivity (FC) measures not corrected for spatial leakage in resting‐state electroencephalography partly stems from source reconstruction: idiosyncratic noise patterns define a baseline resting‐state functional network that explains a significant portion of the reliability of uncorrected FC measures.

Practitioner Points.

• Networks implied by the source dependencies introduced through the imaging kernel matrices show high similarity to real functional connectivity (FC) matrices for uncorrected measures.

• The identifiability of participants based on FC networks is higher for uncorrected measures compared to corrected measures.

• Networks implied by the source dependencies introduced through the imaging kernel matrices explain a considerable amount of the between‐subject similarity of real FC matrices.

• When the main goal of the investigation is to provide a large‐scale description of the neural activity of the brain in the situation tested, we recommend using spatial leakage corrected measures to estimate electroencephalography (EEG) FC in the source space.

• When the main goal of the investigation is to identify participants based on FC networks, we recommend using uncorrected measures to estimate EEG FC in the source space.

1. INTRODUCTION

Functional connectivity (FC) is a tool that characterizes coordinated interaction between functionally specialized brain areas (e.g., Bastos & Schoffelen, 2016; Gonzalez‐Castillo & Bandettini, 2018; Mueller et al., 2013). While functional magnetic resonance imaging (fMRI) is the most widely used imaging method to study resting‐state functional networks in the human brain (Smitha et al., 2017), magnetoencephalography (MEG) and EEG have also been shown to provide valuable information about brain connectivity (Brookes et al., 2011; Deligianni et al., 2014; Sadaghiani et al., 2022). With their high temporal resolution, M/EEG can capture functional connections on the timescale of the underlying physiological mechanisms providing a more direct view of the functional networks relevant for cognitive functions. Especially the contribution from high‐density EEG could prove essential for the field, as it is less expensive and simpler to measure EEG than fMRI or MEG.

In FC estimation, EEG sensor data are commonly transformed into the space of the underlying neural generators (source space reconstruction) and are grouped further into regions according to brain atlases. The brain regions then serve as the nodes of the functional network while edge weights are estimated from the statistical dependencies across the neural signals attributed to the regions. A general question of the field is how methodological choices in this pipeline affect the FC estimates.

The aim of the current study was to test how the reliability of FC estimation for source reconstructed EEG signals measured in the resting state is affected by source localization based on an anatomic template and the FC measure chosen. Although individual structural MRI and electrode localization is obviously preferred to using templates, often it is not possible to obtain these data. We compared some of the crucial tests between template‐ and individual‐structural‐measurement‐based estimations to assess how reliability of the FC estimation is affected by the lack of individual structural data. As there are several FC measures used in literature, it is important to know how they compare in terms of reliability on the individual (clinical, developmental, etc. studies) and on the group level (e.g., cognitive studies). Finally, we aimed to assess the size of the effect of source reconstruction (based on templates vs. individual structural measures and with parcellation schemes of different resolution) on the similarity of FC networks between different participants. This is a crucial issue for studies focusing on inter‐individual variance as well as those relying on the reliability of the FC networks, because high similarity between individuals artificially inflates the reliability of these networks. To achieve these goals, we tested within‐subject reliability and between‐subject similarity of FC networks based on spatial leakage corrected and uncorrected FC measures and assessed the effects of the similarity between imaging kernel matrices (the linear inverse operator transforming sensor signals into source signals, as employed in source reconstruction; Samuelsson et al., 2021) combined with the chosen FC measure on them.

1.1. EEG functional connectivity

FC estimation from EEG is burdened by methodological challenges (van Diessen et al., 2015). Arguably, the largest problem is that EEG FC estimates are contaminated by spurious connections (artifacts) due to volume conduction. Volume conduction refers to the spatial spread of electrical currents in the brain, causing sensor signals to carry a linear mixture of contributions from multiple neural sources (linear mixing, e.g., Palva & Palva, 2012; Siebenhühner et al., 2016). Consequently, the signal of a specific neural source is detected on multiple electrodes, and, thus, scalp/sensor signals typically display strong interdependencies. A similar issue affects the interpretation of MEG recordings as MEG sensor signals also represent the superposition of the activity from multiple neural sources due to electromagnetic field spread (Winter et al., 2007).

There are multiple ways to combat the effects of volume conduction (Bastos & Schoffelen, 2016; Schoffelen & Gross, 2009). First, an “unmixing” of the signals can be attempted by projecting the signals back into source space, using a variety of established inverse methods (e.g., Michel & Brunet, 2019). However, due to the ill‐posed nature of the inverse problem neighboring sources share some of their activity, leading to spurious local but also long‐range dependencies (Palva et al., 2018; Palva & Palva, 2012). This spatial leakage effect (Wens et al., 2015) can be mitigated to some degree by pooling source signals across larger regions‐of‐interest (ROI) or by clustering similar edges together (Wang et al., 2018). Second, volume conduction artifacts might be canceled out by contrasting FC estimates across appropriate experimental conditions. However, this latter technique is not available for all experimental designs (e.g., in resting state or movie viewing) and may yield misleading results due to signal‐to‐noise ratio changes across conditions (Schoffelen & Gross, 2009). Third, spurious connections due to instantaneous (zero‐lag) dependencies across signals introduced by volume conduction can be further mitigated by reducing the effects of zero‐lag dependencies on FC estimates. This can be achieved by either orthogonalizing the signals before FC estimation (Brookes et al., 2012; Colclough et al., 2015; but see Pascual‐Marqui et al., 2017) or employing FC measures designed to be insensitive to zero‐lag dependencies (e.g., Bruña et al., 2018; Nolte et al., 2004; Stam et al., 2007). However, if there is a nonzero‐lag connection between two brain regions, the neighboring brain regions can also show nonzero‐lag connectivity due to volume conduction. As a result, spurious connections can also appear at nonzero lag.

From the above mitigation approaches, here we focused on the question of the difference between measures that are insensitive to zero‐lag dependencies (corrected FC measures) and measures that are sensitive to them (uncorrected FC measures).

1.2. Reliability of uncorrected FC measures

Despite the above‐described effects of volume conduction, a wide range of EEG FC studies applied uncorrected FC measures. In the context of brain fingerprinting (biomarkers accurately identifying participants; e.g., Finn et al., 2015), several EEG FC studies utilized or recommended uncorrected FC measures for individual identification both for resting‐state data as well as data collected under various task conditions (spectral coherence: Garau et al., 2016; La Rocca et al., 2014; phase synchrony: Fraschini et al., 2018, 2019; Kong et al., 2017; Kumar et al., 2022; Tian et al., 2022; Wang, El‐Fiqi, et al., 2019; Wang et al., 2020; Granger causality estimation: Min et al., 2017). Overall, these studies reported high (in the range of 90–100%) individual recognition or identification accuracy with uncorrected measures, providing evidence for the robustness of EEG FC fingerprints. Furthermore, direct comparisons across corrected and uncorrected FC measures from resting‐state data found a general advantage of uncorrected compared with corrected measures both for identification and test–retest reliability (Duan et al., 2021; Fraschini et al., 2019; Höller et al., 2017; Marquetand et al., 2019; Valizadeh et al., 2019). Because practical fingerprinting is not among the goals of the current study, we only took from these studies the suggestion that zero‐lag connection estimates carry information specific to the individual. These results are also in line with test–retest reliability results from MEG measurements (Colclough et al., 2016; da Silva Castanheira et al., 2021; Garcés et al., 2016; Sareen et al., 2021).

Several factors could simultaneously play a role in the stronger idiosyncrasy and higher reliability of uncorrected FC measures relative to the corrected ones. On the one hand, the difference might be driven by characteristics of signal mixing stemming from anatomical, electrode placement, and/or noise differences specific to individuals or sessions. These factors could influence FC estimates without corresponding to neural functions. Using corrected measures is recommended in order to exclude these factors (e.g., Colclough et al., 2016; O'Neill et al., 2018). On the other hand, the higher reliability of uncorrected than corrected measures may also originate from “true” zero‐lag connections; that is, long‐range in‐phase or antiphase synchrony across regions, an empirically supported brain phenomenon (O'Reilly & Elsabbagh, 2021; Roelfsema et al., 1997). As uncorrected FC measures were used to detect brain network correlates of various clinical disorders (e.g., Chen et al., 2019; Frid et al., 2020; Hassan et al., 2017), emotional states (e.g., Al‐Shargie et al., 2019; Wang, Tong, & Heng, 2019; Wu et al., 2022), cognitive functions (e.g., Gupta et al., 2021; Neubauer & Fink, 2009), and aging (Chow et al., 2022), they undoubtedly capture information about important neural processes. However, it is yet unknown to what degree uncorrected FC measures reflect true zero‐lag connections and/or the effect of various incidental (anatomical or measurement‐related) factors.

1.3. Source reconstruction, source dependencies, and uncorrected FC measures

Physical properties of the head determine how neural electric currents produce electrical potentials at external sensors on the scalp (Ramon et al., 2006). These properties can be modeled and summarized in the leadfield matrix (or gain matrix, Weinstein et al., 2000), a linear operator transforming source currents to potentials measured at external sensors. That is, sensor signals can be modeled as a linear combination of source signals:

$$y\left(t\right)=\mathit{Lx}\left(t\right)$$

where $y\left(t\right)\in R^{\mathit{nx}1}$ is the vector of sensor signals at time $t$, $L\in R^{\mathit{nxm}}$ is the leadfield matrix, and $x\left(t\right)\in R^{\mathit{mx}1}$ is the vector of source activities at time $t$. The family of techniques enabling the transition from sensor space to source space is called source imaging or source reconstruction (Michel et al., 2004). For linear source reconstruction methods, the linear inverse operator transforming sensor signals into source signals is referred to as the imaging kernel matrix or inverse kernel matrix (Samuelsson et al., 2021). With this approach, source signals can be modeled as a linear combination of sensor signals:

$$\widehat{x}\left(t\right)=\mathit{Ky}\left(t\right)$$

where $\widehat{x}\left(t\right)\in R^{\mathit{mx}1}$ is the vector of estimated source activities at time t, and $K\in R^{\mathit{mxn}}$ is the imaging kernel matrix. Popular linear estimation methods (e.g., MNE, dSPM, sLORETA, eLORETA) derive the imaging kernel matrix from the forward model and the noise covariance matrix; thus, the imaging kernel matrix is independent from the measured data of interest if noise is estimated from baseline data. Furthermore, the number of potential sources is much larger than the number of sensors ($m\gg n$), causing the inverse operator to introduce linear dependencies across sources. Therefore, for arbitrary sensor signals $y\left(t\right)$, there is a trivial set of dependencies across estimated source signals $\widehat{x}\left(t\right)$, the structure of which depends only on the imaging kernel matrix $K$, summing anatomical and measurement‐related information (noise, electrode positions, etc.; e.g., Hauk et al., 2011; Schoffelen & Gross, 2009).

The dependencies introduced by source reconstruction can be quantified by the resolution matrix $R\in R^{\mathit{mxm}}$, which is the product of the imaging kernel and leadfield matrices (de Peralta et al., 1997; Hauk et al., 2011; Hauk & Stenroos, 2014):

$$R=\mathit{KL}$$

The diagonal elements of $R$ indicate the sensitivity of each estimated source to itself, and off‐diagonal elements quantify the degree to which estimated sources are affected by the signal from all other sources. If source reconstruction was ideal, the resolution matrix would equal the identity matrix. The rows of $R$ are known as cross‐talk functions (CTFs) and define how the signal of a reconstructed source is necessarily influenced by all other sources as a result of the source reconstruction method. The columns of $R$ are referred to as point‐spread functions (PSFs) and define how the activity of a source is spread across other sources by the source reconstruction procedure. CTFs and PSFs enable the calculation of several spatial resolution metrics that can be used to evaluate and compare different source reconstruction methods (Hauk et al., 2022).

To the degree that any FC measure is sensitive to linear dependencies across source signals, its reliability across epochs, sessions and participants should reflect the effects of the CTFs, and, in turn, the effects of anatomy and measurement‐specific details. Particularly, the reliability of uncorrected FC estimates should be larger than corrected FC estimates across epochs, provided that the imaging kernel matrix is kept constant and uncorrected measures are more sensitive to the linear dependencies introduced by the imaging kernel. Comparing across sessions and participants, the reliability difference between uncorrected and corrected measures should partly reflect the similarity of imaging kernel matrices involved in the comparisons.

1.4. Individual versus template head modeling

Accurate calculation of the leadfield matrix requires a realistic biophysical model of the head which is commonly derived from participants' structural MRI scans (Gramfort et al., 2011). However, scanning each participant is not always feasible. Instead, an MRI template might be used for the derivation of each participants' head model. Template head models are commonly used in EEG‐based FC studies. Based on a non‐exhaustive literature search we estimated that more than two‐thirds of the studies published since 2019 investigating resting‐state EEG‐based FC in the source space used template head models (details of this estimation can be found in the Appendix). Relative to the use of individual MRI scans, a template head model entails that the imaging kernel matrices (and, thus, the CTFs) become more uniform across participants, increasing the apparent similarity of their FC connectivity matrices. In the current study, we explored the reliability of FC estimates derived without individual structural scans. With the help of a smaller data set (from Ignatiadis et al., 2022), we also compared the results to FC estimates from a pipeline including individual head models.

1.5. Testing the reliability of corrected and uncorrected FC measures

Existing studies on resting‐state EEG FC in source space focused on reliability across different source reconstruction procedures (Duan et al., 2021; Mahjoory et al., 2017), and across different imaging modalities (Marquetand et al., 2019; Nentwich et al., 2020). Here, we carried out reliability analyses in the source space across five frequency bands and different epoch lengths and estimated both within‐subject reliability and between‐subject similarity of FC networks. Specifically, the following hypotheses were tested:

Hypothesis H1

The reliability of FC within subjects is larger for uncorrected measures compared to corrected measures.

Hypothesis H2

The group‐level reliability of FC is larger for uncorrected measures compared to corrected measures, therefore, the minimum group size required for reliable group‐level FC estimation is lower for uncorrected measures compared to corrected measures.

Hypothesis H3

FC networks based on a short segment of a participant's EEG are more distinctive to the given individual for uncorrected measures compared to corrected measures.

Regarding the effects of the source reconstruction method, we tested whether the higher reliability of uncorrected over corrected FC measures could be explained by their sensitivity to source dependencies introduced by the imaging kernel matrix. Specifically, the following two hypotheses were tested:

Hypothesis H4

Networks implied by the source dependencies introduced through the imaging kernel matrices show high similarity to real FC matrices for uncorrected measures.

Hypothesis H5

Networks implied by the source dependencies introduced through the imaging kernel matrices explain a considerable amount of the between‐subject similarity of real FC matrices.

As the current data include only one session per participant, we defined reliability here as FC measure consistency across epochs, participants, and groups. Consequently, effects specific to the recording session remained confounding factors.

2. METHODS

2.1. Participants

Resting‐state EEG recordings were collected from 201 young adults (age 18–26 years, M = 22.9, SD = 1.1, 125 females, 23 left‐handed). The recordings were collected as part of different experiments employing common methods; the data were not reported previously. Detailed descriptions of the individual experiments are reported in published papers (Kovács et al., 2023; Szalárdy et al., 2019, 2020, 2021, 2022; Tóth et al., 2019, 2020). All participants were healthy with no history of neural or psychiatric disorder. Informed consent was obtained from all participants after the aims and methods of the relevant study were explained to them. The studies were conducted in accordance with the Helsinki Declaration and all applicable national laws; they were approved by the institutional review board, the United Ethical Review Committee for Research in Psychology.

To compare individual MRI‐based source reconstruction to template MRI‐based source reconstruction, a second, smaller data set was also analyzed (second data set henceforth). The second data set contains resting‐state EEG of 23 young adults, with individual structural MRI and electrode positions for each participant. A detailed description is provided in Ignatiadis et al. (2022), and the data set is available via the OSF platform under https://doi.org/10.17605/OSF.IO/YM26X. One participant was excluded from our analysis due to the low signal‐to‐noise ratio of the EEG signal. Results are reported for the remaining 22 participants.

2.2. Procedures and EEG recording

Resting‐state EEG was recorded with eyes open in one session in an acoustically attenuated, electrically shielded, dimly lit room at the Research Centre for Natural Sciences, Budapest, Hungary. The recording length was 5 min. A 19 in monitor was placed directly in front of the participants at a distance of 195 cm. Participants were instructed to keep eye blinks and all other motor activity to a minimum during the session by focusing on a fixation cross presented continuously at the center of the monitor.

EEG was recorded with a BrainAmp DC 64‐channel EEG system with actiCAP active electrodes. Electrodes were placed according to the International 10/20 system. For EOG monitoring, one electrode was placed lateral to the outer canthus of the right eye and another below the left eye. Electrode impedances were kept below 15 kΩ during the recording. The FCz lead served as the on‐line reference electrode. The sampling rate was 1 kHz, and a 100 Hz online low‐pass filter was applied.

In the second data set (Ignatiadis et al., 2022), resting‐state EEG was recorded with eyes open in one session, for 4 min. EEG was recorded with an actiCHamp 128‐channel EEG system with actiCAP active electrodes. The sampling rate was 1 kHz. In our study, we selected 64 of the available 128 channels in a way that the positions of the selected 64 channels were closest to the electrode positions we used in our own measurements.

All of the analysis steps described in the following chapters were identical for the 201‐subject‐data set and for the 22‐subject‐data set except head modeling.

2.3. EEG preprocessing

Preprocessing was done using MATLAB (2017a; MathWorks) and the EEGLAB toolbox (v2021.1; Delorme & Makeig, 2004). First, the EEG signals were re‐referenced to the common average and band‐pass filtered between 0.5 and 80 Hz (Kaiser windowed, Kaiser β = 5.65, filter length of 4530), with notch filtering at around 50 Hz (47.0–53.0 Hz Kaiser bandstop filter). Maximum two bad EEG channels per participant were interpolated using the spline interpolation algorithm implemented in EEGLAB. The Infomax algorithm of independent component analysis (ICA) implemented in EEGLAB was employed for artifact removal. ICA components constituting blink artifacts were removed via visual inspection of their topographical distribution and frequency contents of the components. Preprocessing resulted in 5‐min‐long continuous artifact‐free EEG for each participant (4 min in the second data set).

2.4. Source reconstruction

EEG source reconstruction was performed with Brainstorm (Tadel et al., 2011; version 10‐Mar‐2022) and followed the protocol of previous studies (e.g., Song et al., 2015; Tóth et al., 2020; see below). Forward boundary element head model was used as provided by the openMEEG algorithm (Fuchs et al., 1998, 2002; Gramfort et al., 2011). For our data set containing 201 recordings, the MNI brain template (Collins et al., 1998) was used for head modeling because no anatomical scans for the participants were available. Default electrode locations were aligned with the default anatomical mesh. In the second data set, individual structural MRI and electrode positions were available for each participant and these were used for head modeling.

The time‐varying source signals (current density) were modeled for all cortical voxels by the sLORETA minimum norm estimate inverse solution (Pascual‐Marqui, 2002) as implemented in Brainstorm. This algorithm models the inverse solution as elementary dipoles perpendicular to the cortical surface and distributed over a nodal mesh representing the cortical volume (15,000 vertices). The resulting activation maps represent cortical current source densities underlying the scalp potentials. Since the concept of pre‐stimulus or baseline state is not applicable to resting‐state EEG, noise was estimated as the variance measured at each sensor, and, thus, diagonal noise covariance matrices were used. This is arguably the most commonly used way to estimate noise covariance in resting‐state EEG, although other methods are also in use (see, e.g., Cai et al., 2021; Engemann & Gramfort, 2015). In Brainstorm, the signal covariance matrix is whitened by the noise covariance matrix, such that the whitened eigenspectrum has elements in terms of SNR (power). We used the default SNR value of Brainstorm resulting in an average SNR (power) of 9. By averaging dipole strengths across voxels, the mean source waveforms (unitless because of the standardization of sLORETA) were obtained for 62 cortical areas (ROIs) based on the anatomical segmentation of the Desikan–Killany–Tourville atlas (DKT atlas, Klein & Tourville, 2012). Imaging kernel matrices were stored for each participant, and a spatially reduced version was calculated using the parcellations scheme. That is, for each ROI, the rows of the imaging kernel matrix corresponding to sources in the ROI were averaged (considering the signs of current flows on opposite sides of the sulci) into an ROI‐specific row. If sources are modeled as dipoles perpendicular to the cortical surface, sources on opposite walls of a sulcus are very close to each other with opposite orientations. If an ROI includes sources with opposite orientations, averaging may cancel out the activity. Therefore, Brainstorm flips the sign of sources with opposite directions before averaging them within an ROI, according to the dominant orientation of the ROI. The same sign flipping was applied to the rows of the imaging kernel matrices. To test the effect of parcellation, a subset of the whole analysis was repeated using the Destrieux atlas (Destrieux et al., 2010) consisting of 148 ROIs.

It is important to note that sLORETA can be considered as a standardized version of the classical minimum norm estimation, where the standardization does not affect the leakage across sources (e.g., Hauk & Stenroos, 2014). Thus, in terms of the reliability of FC measures, results obtained with sLORETA are generalizable to minimum norm estimation procedures and their similar standardizations (e.g., dynamic statistical parametric mapping).

sLORETA is a widely used source reconstruction method in EEG (Hedrich et al., 2017; Samuelsson et al., 2021). For the 64‐channel electrode array used here, mean sLORETA localization error was estimated as 1.40 mm (SD = 3.68) (Song et al., 2015). Localization was applied identically for all participants. Therefore, while overall localization precision might be lower using template anatomy than MRI scans, this noise manifests uniformly across the source reconstruction estimates, increasing the noise of the estimation, but providing no confound for the tests of this study.

2.5. Filtering, epoching

Source‐reconstructed signals were band‐pass filtered into five frequency bands with zero‐phase FIR filters: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz), and gamma (30–80 Hz). To remove transient events, the first and last 5 s of the filtered signals were trimmed, yielding 290 s long recordings (and 230 s in the second data set). Finally, the recordings were segmented into nonoverlapping epochs of length 1, 2.5, 5, and 9 s. Subsequent analysis steps were carried out separately for each epoch length. Note that as the duration of the recordings was fixed (290 and 230 s), epoch length was inversely proportional to the number of epochs (290, 116, 58 and 32, respectively, for the 201‐subject‐data set and 230, 92, 46, and 25, respectively, for the 22‐subject‐data set).

2.6. FC measures

In this study, FC was estimated for each epoch of each participant with five leakage corrected measures: imaginary part of phase locking value (iPLV), phase lag index (PLI), weighted PLI (wPLI), amplitude envelope correlation corrected for spatial leakage (AECc), imaginary part of coherency (iCOH), and three uncorrected measures: phase locking value (PLV), amplitude envelope correlation (AEC), and coherence coefficient (COH). A detailed description of these FC measures is provided in the Appendix.

2.7. Methods for the first set of hypotheses

2.7.1. Reliability estimation (for H1and H2)

Reliability was defined as the consistency by which an FC measure yielded the same (similar) resting‐state FC matrix given the sets of data involved. Within‐subject reliability refers to consistency estimation carried out using different recording segments from the same participant, and group‐level reliability refers to consistency estimation carried out on data from different participants, or groups of participants.

To test within‐subject reliability, each participants' epochs were divided randomly into two equal sets, then the FC matrices of epochs in both sets were averaged, and the average matrices were compared using Pearson's correlation coefficient (henceforth, correlation refers to Pearson's correlation coefficient). This procedure was repeated 1000 times for each participant and the correlation values were averaged. Here, Pearson's correlation coefficient was taken as the similarity between the two average FC matrices corresponding to the two set of epochs. A large value of the average correlation from the 1000 iterations meant that FC networks were consistent over epochs. As the current data included only one session per participant, the classical concept of test–retest reliability was not applicable. Instead, within‐subject reliability was defined as the consistency of an FC measure across sets of epochs.

As we used three uncorrected and five corrected FC measures, the statistical analysis of within‐subject reliability values was performed in two steps. First, the uncorrected‐corrected pairs of measures (PLV‐iPLV, AEC‐AECc, COH‐iCOH) were analyzed with a two‐way dependent measures ANOVA with factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected). Second, corrected measures (iPLV, PLI, wPLI, AECc, iCOH) were analyzed with a one‐way dependent measures ANOVA with factor Corrected Measure. Within‐subject reliability values were pooled across frequency bands and epoch lengths for both analyses. An alpha level of .05 was used in all statistical tests and effect sizes were estimated with η ². ANOVA interactions were further specified with pairwise t tests (Bonferroni‐corrected). All statistically significant results were reported.

As a special case of within‐subject reliability, the similarity of FC matrices of a given participant with a given FC measure and given epoch length across different frequency bands was investigated. The correlation coefficient of FC matrices was calculated between all possible frequency band pairings for each participant (averaged across epochs), FC measure and epoch length. Finally, correlation values were averaged across participants and epoch lengths. The resulting values characterized the consistency of FC measures across frequency bands.

Group‐level reliability was analyzed by first averaging FC matrices across all epochs, separately for each participant. Then, for a group size of 1, participants' average FC matrices were correlated across all possible participant‐pairings and the median correlation value was selected as the measure of group‐level reliability (between‐subject similarity). For group sizes from 2 to 100 (and group size 5 and 10 for the second, 22‐subject‐data set), separately for each group size, two groups of participants were randomly selected without replacement, their FC matrices were averaged within the groups, and the correlation coefficient was calculated between the two group average FC matrices. This step was repeated 1000 times and the median correlation values were selected as the measure of group‐level reliability for each group size. Here, the correlation coefficient was taken as the similarity between the two average FC matrices corresponding to the two groups. A large value of the median correlation from the 1000 iterations meant that FC networks were consistent over groupings of participants for the given FC measure and group size. The group size at which the between‐group correlation value exceeded r = .9 was also calculated for each FC measure. Group‐level reliability characterized FC measures in terms of their ability to produce consistent estimates of the population's connectivity patterns.

2.7.2. Distinguishing participants based on a short EEG segment (H3)

Individuals can be identified from short EEG recordings (e.g., Paranjape et al., 2001). Here, we tested how well FC networks based on short EEG segments characterized individuals by comparing each epoch to each participant's average FC matrix. Specifically, for each epoch of each participant, the FC matrix corresponding to the selected epoch (epoch FC matrix) was correlated with each participant's average FC matrix, where the average FC matrix refers to the average across all epoch FC matrices excluding the selected epoch for the given participant. The analysis was carried out separately for each epoch, participant, FC measure, frequency band, and epoch length.

Self‐similarity and similarity to others were calculated from the comparisons of epoch FC matrices to average FC matrices. Self‐similarity was computed for each participant as the average similarity (correlation coefficient) of epoch FC matrices to the participant's average FC matrices. Similarity to others was computed for each participant from the similarity values (correlation coefficients) between the participant's epoch FC matrices and other participants' average FC matrices, averaged across all epochs and participants. Finally, the difference between self‐similarity and similarity to others, termed differential identifiability (Amico & Goñi, 2018) was also calculated for each participant.

The statistical analysis of differential identifiability values was performed the same way as for within‐subject reliability. First, the uncorrected‐corrected pairs of measures (PLV‐iPLV, AEC‐AECc, COH‐iCOH) were analyzed with a two‐way dependent measures ANOVA with factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected). Second, corrected measures (iPLV, PLI, wPLI, AECc, iCOH) were analyzed with a one‐way dependent measures ANOVA with factor Corrected Measure. Differential identifiability values were pooled across frequency bands and epoch lengths for both analyses.

2.8. Methods for the second set of hypotheses

2.8.1. Similarity between real FC networks and networks introduced by the imaging kernel matrices (H4)

As outlined in Section 1, there is a trivial set of dependencies across estimated source signals, the structure of which is only dependent on the imaging kernel matrix. To the degree that any FC measure is sensitive to linear dependencies across source signals, its reliability across epochs, sessions, and participants should reflect the effects of the imaging kernel, and, in turn, the effects of anatomy and measurement‐specific details.

To compare the network structures implied by the imaging kernel matrices to real FC networks, imaging kernel matrices were downsampled to the 62 ROIs used for FC estimation. The downsampled imaging kernel matrices were multiplied by artificial random sensor signals (Gaussian white noise with the same number of channels and duration as the real EEG signals). The resulting artificial source activities were band‐pass filtered into the same frequency bands, and epoched with the same epoch lengths as real source activities, and FC was estimated using the same FC metrics as for real data (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). The FC matrices obtained from this procedure were termed imaging kernel networks, as their only participant‐related information comes from the imaging kernel matrix. Correlation coefficients were then calculated for each participant between the participant's average imaging kernel network and average (real) FC matrix, separately for each FC measure, EEG band, and epoch length. The correlation coefficients quantified the similarity of the network structures implied by imaging kernel matrices to the real FC networks calculated from source reconstructed data based on the given kernels. The advantage of imaging kernel networks compared to the kernels themselves is that imaging kernel networks also include the nonlinear effects of FC measures, while the imaging kernel is the result of source reconstruction without FC estimation. The estimation pipeline for both FC networks and imaging kernel networks is summarized in Figure 1.

FIGURE 1.

Estimation pipeline of real FC networks and imaging kernel networks. The imaging kernel was estimated based on the head model and the noise covariance matrix of the EEG recording. Real source activity estimates were calculated by multiplying the imaging kernel by EEG signals. Artificial source activities were calculated by multiplying the imaging kernel by Gaussian white noise. FC was estimated on the level of ROIs using the same cortical atlas for the parcellation of real and artificial source activities. The strength of connectivity defines a network (a real FC network or an imaging kernel network) which can be represented as a matrix. The effect of spatial leakage correction was analyzed based on FC networks and imaging kernel networks.

Statistical analysis of the similarity between imaging kernel networks and real FC matrices was performed the same way as for differential identifiability and within‐subject reliability. First, the uncorrected‐corrected pairs of measures (PLV‐iPLV, AEC‐AECc, COH‐iCOH) were analyzed with a two‐way dependent measures ANOVA with factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected). Second, corrected measures (iPLV, PLI, wPLI, AECc, iCOH) were analyzed with a one‐way dependent measures ANOVA with factor Corrected Measure. Correlation coefficients were pooled across frequency bands and epoch lengths for both analyses.

2.8.2. Between‐subject similarities of real FC matrices explained by the imaging kernel matrices (H5)

First, the similarities across participants' average FC matrices were estimated by correlating them across all possible participant‐pairings, resulting in a 201 × 201 symmetric between‐subject FC similarity matrix, separately for each frequency band, epoch length, and FC measure. The same procedure was replicated with participants' imaging kernel networks, resulting in a 201 × 201 between‐subject kernel network similarity matrix, separately for each frequency band, epoch length and FC measure. Then, the similarity between each between‐subject kernel network similarity matrix and the corresponding between‐subject FC similarity matrix was estimated by Pearson's correlation coefficient. The correlation coefficients quantified the degree of between‐subject similarity of real FC matrices explained by imaging kernel networks. The explained similarity was also quantified by regressing principal components of the between‐subject FC similarity matrix and the between‐subject kernel network similarity matrix. However, this analysis yielded equivalent results to the analysis based on correlations; therefore, these results are not reported.

As an alternative to the above‐described method, the effect of source reconstruction on the identifiability of participants was also estimated, as this is closely related to between‐subject similarity. For this analysis, self‐similarity and the similarity to others were recalculated after subtracting the corresponding imaging kernel networks from epoch FC networks. Since it was expected that the imaging kernel networks for uncorrected measures introduced a nearly identical pattern of dependencies across participants boosting between‐subject similarity, the subtraction of imaging kernel networks from real FC networks was expected to increase differential identifiability. The increased identifiability was quantified by one‐sided paired t tests with differential identifiability values averaged across frequency bands and epoch lengths.

2.8.3. Between‐subject similarities of real FC matrices explained by the parcel resolution matrices (H5)

As outlined in Section 1, the rows (CTFs) of the resolution matrix $R$ define how the signal of a reconstructed source is necessarily influenced by all other sources as a result of the source reconstruction method. To describe the crosstalk among parcels (and not among sources), the Parcel Resolution Matrix $\left(\mathit{\text{PRmat}}\right)$ can be used. We computed $\mathit{\text{PRmats}}$ as described by Farahibozorg et al. (2018): First, the absolute values of the elements of $R$ were computed. Then, submatrices of $R$ were computed by rearranging rows of $R$ such that only rows corresponding to a selected parcel p is contained by the sub matrix $M^p$. Next, singular‐value decomposition was computed along the rows of the $M^p$ matrices for all p parcels. Each parcel was represented by the first eigenvector ${\mathit{CTF}}^p$ along rows. Finally, $\mathit{\text{PRmat}}$ was computed where each element ${\mathit{\text{PRmat}}}_{\mathit{ij}}$ describes the leakage from parcel $i$ to parcel $j$, normalized by the the amount of leakage it receives from all parcels.

The portion of between‐subject similarities explained by $\mathit{\text{PRmats}}$ was computed similarly to the imaging kernel matrices. Participants' $\mathit{\text{PRmats}}$ were correlated across all possible participant‐pairings, resulting in a 201 × 201 symmetric between‐subject parcel resolution similarity matrix. Then, the similarity between each between‐subject FC similarity matrix and the between‐subject parcel resolution similarity matrix was estimated again by correlation, separately for each FC measure, epoch length, and frequency band. These correlation coefficients quantified the between‐subject similarity of real FC matrices explained by $\mathit{\text{PRmats}}$. Comparing the similarity explained by parcel resolution matrices to the similarity explained by imaging kernel networks helps to estimate the effect of the FC measure, as $\mathit{\text{PRmats}}$ quantify dependencies introduced by the source reconstruction procedure, while imaging kernel networks characterize the combination of the source reconstruction procedure and the FC measure. Note that there is only one between‐subject parcel resolution similarity matrix, because we performed source reconstruction before filtering into frequency bands and epoching, so the parcel resolution matrix of a given participant is the same for all FC measures, epoch lengths, and frequency bands.

2.9. Replication of analyses with structural MRI and with a different parcellation

To assess the effects using individual structural MRI data and electrode position, all analyses regarding within‐subject and group‐level reliability (H1‐2), differential identifiability of participants (H3), the similarity between real FC networks and imaging kernel networks (H4), and between‐subject similarity of real FC networks explained by imaging kernel networks (H5) were performed on the second data set as well. Due to the differences in the hardware and the procedures, the two data sets were not compared directly.

Within‐subject and group‐level reliability (H1‐H2) were also analyzed using the Destrieux atlas (Destrieux et al., 2010) for a subset of FC measures (PLV, iPLV, AEC, AECc). Both the DKT and Destrieux atlases are based on anatomy (gyral and sulcal boundaries) but they offer different resolutions: 62 (as used here) versus 148 regions across the two hemispheres. By repeating a subset of our analyses using the Destrieux atlas, we tested if FC reliability was affected by the number of regions in a parcellation, as that might interact with the low spatial resolution afforded by EEG. This set of analyses was limited both in terms of the hypotheses tested and the FC measures included due to computational constraints and its results are reported in the Appendix.

3. RESULTS

3.1. Results for the first set of hypotheses

3.1.1. Within‐subject reliability (H1)

Here, we tested whether the reliability of FC within subjects was larger for uncorrected measures compared to corrected measures (H1). We found that both epoch lengths and frequency bands influenced within‐subject reliability, with typically higher values in higher frequency bands (Figure 2a).

FIGURE 2.

(a) Group‐averaged (n = 201) within‐subject reliability as a function of epoch length, separately for each frequency band and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Standard error is marked over the measurement points. Note that the scale of uncorrected measures (PLV, AEC, COH) is different from the scale of corrected measures. (b) Left: Group‐mean value and standard deviation of within‐subject reliability values for uncorrected (PLV, AEC, COH) and the corresponding corrected measures (iPLV, AECc, iCOH) pooled across frequency bands and epoch lengths. Right: Group‐mean value and standard deviation of within‐subject reliability values for each corrected measure (iPLV, PLI, wPLI, AECc, iCOH) pooled across frequency bands and epoch lengths.

Within‐subject reliability values were compared across different measures by an ANOVA with within‐subject factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected). The main effect of Leakage Correction was significant (F(2, 200) = 43728.0, p < 1e‐6, η ² = .55), with larger reliability for uncorrected (M = 0.97, SD = 0.02) than corrected measures (M = 0.58, SD = 0.25). The main effect of Measure Type was also significant (F(1, 200) = 3083.0, p < 1e‐6, η ² = .08), and reliability values were largest for coherence (M = 0.87, SD = 0.14), intermediate for phase locking (M = 0.76, SD = 0.27) and smallest for amplitude correlation measures (M = 0.70, SD = 0.32). For the effect of Measure Type, pairwise comparisons revealed that the levels of Measure Type were all significantly different from each other (phase locking vs. amplitude correlation: t(200) = 18.00, p < 1e‐6, d = 1.27; phase locking vs. Coherence: t(200) = 57.86, p < 1e‐6, d = 4.08; amplitude correlation vs. Coherence: t(200) = 49.02, p < 1e‐6, d = 3.76). The interaction was also significant (F(2, 200) = 2413.0, p < 1e‐6, η ² = .06). Follow‐up one‐way ANOVAs with the factor Measure Type were performed on within‐subject reliability values separately for corrected and uncorrected measures. The effect of Measure Type was larger for corrected (F(2, 200) = 2895.0, p < 1e‐6, η ² = .30) than for uncorrected measures (F(2, 200) = 1958.0, p < 1e‐6, η ² = .24). Overall, H1 was supported by the data. Indeed, the within‐subject correlation was vastly higher for corrected compared to uncorrected measures (Figure 2b, left).

Further, we tested how reliability differed across FC measures corrected for spatial leakage. The one‐way dependent measures ANOVA on reliability values with factor Corrected Measure (iPLV, PLI, wPLI, AECc, iCOH) yielded a statistically significant effect (F(4, 200) = 1315.0; p < 1e‐6; η ² = .18). Within‐subject reliability values were largest for iCOH (M = 0.77, SD = 0.13), then wPLI (M = 0.56, SD = 0.26), then iPLV (M = 0.54, SD = 0.21), then PLI (M = 0.50, SD = 0.28), and finally AECc (M = 0.43, SD = 0.27) (Figure 2b, right). Pairwise differences (Table A1) between corrected measures were statistically significant for all pairings (p < 1e‐6 for all pairings except the pairing iPLV‐wPLI, where p = .01).

To assess the effects using individual structural MRI data and electrode positions, the same analyses were performed on the second data set. The two‐way dependent measures ANOVA on reliability values with factors Measure Type and Leakage Correction yielded significant main effects (Leakage Correction: F(1, 21) = 6137.1, p < 1e‐6, η ² = .62; Measure Type: F(2, 21) = 318.1, p < 1e‐6, η ² = .06) and significant interaction (F(2, 21) = 209.1, p < 1e‐6, η ² = .04). For Leakage Correction, within‐subject reliability values were larger for uncorrected (M = 0.96, SD = 0.04) than for corrected measures (M = 0.51, SD = 0.24). As for Measure Type, within‐subject reliability values were largest for coherence (M = 0.83, SD = 0.18), intermediate for phase locking (M = 0.71, SD = 0.30) and smallest for amplitude correlation (M = 0.66, SD = 0.32), replicating the order from the primary data set. Pairwise differences across Measure Type levels were all significant (phase locking vs. amplitude correlation: t(21) = 4.73, p < 1e‐4, d = 1.01; phase locking vs. coherence: t(21) = 20.31, p < 1e‐6, d = 4.33; amplitude correlation vs. coherence: t(21) = 18.60, p < 1e‐6, d = 3.97). To interpret the interaction, follow‐up one‐way ANOVAs with the factor Measure Type were performed on within‐subject reliability values separately for corrected and uncorrected measures. As for the primary data set, the effect of Measure Type was larger for corrected (F(2, 21) = 275.6, p < 1e‐6, η ² = .28) than for uncorrected measures (F(2, 21) = 138.3, p < 1e‐6, η ² = .17).

The one‐way dependent measures ANOVA on reliability values with factor Corrected Measure yielded a statistically significant effect (F(4, 21) = 131.0; p < 1e‐6; η ² = .18). Within‐subject reliability values were largest for iCOH (M = 0.69, SD = 0.15), then wPLI (M = 0.48, SD = 0.25), then iPLV (M = 0.45, SD = 0.22), then PLI (M = 0.42, SD = 0.27), and finally AECc (M = 0.39, SD = 0.15), again replicating the order from the primary data set. Pairwise differences (Table A2) between corrected measures were statistically significant for most pairings except for iPLV‐wPLI and PLI‐AECc. Overall, spatial leakage correction, epoch lengths, and frequency bands affected reliability similarly irrespective of whether individual structural information or a template was used (compare Figure 2a and Figure A1).

As a special case of within‐subject reliability, the similarity of FC matrices between different frequency bands was also estimated. The similarity between frequency bands was significantly higher for uncorrected measures (M = 0.77, SD = 0.046) than for corrected measures (M = 0.26, SD = 0.047; t(400) = 110.40, p < 1e‐6, d = 11.01) (Figure A3). Frequency bands closer to each other (e.g., Delta–Theta) showed higher similarity compared to frequency bands that were far from each other (e.g., Delta–Gamma) (linear regression with frequency band distance as predictor: β = −.12, p < 1e‐6, R ² = .63).

Overall, the within‐subject reliability of corrected FC measures was significantly lower than that of uncorrected FC measures, irrespective of whether individual or a common template‐based head model was used as well as of the parcellation scheme, offering robust support for H1.

3.1.2. Group‐level reliability (H2)

Here, we tested whether the group‐level reliability of FC was larger for uncorrected measures compared to corrected measures, by assessing the minimum group size required for reliable group‐level FC estimation. In general, group size consistently increased group‐level reliability (see Figure 3 for an example: alpha band with an epoch length of 5 s). We further found that both the epoch length and the frequency band influenced the required minimum group size to exceed a group‐level reliability of r = .9 (Figure 4). The trend of influence was unique for each FC measure. The comparison between required minimum group sizes for corrected and uncorrected measures (t test, pooled across all epoch lengths and frequency bands) was statistically significant (t(158) = 14.41; p < 1e‐6; d = 2.35) and revealed that the minimum group size required to reach r = .9 reliability was much lower for uncorrected (M = 4.87, SD = 0.70) than for corrected measures (M = 52.88, SD = 25.76).

FIGURE 3.

Group‐level reliability of connectivity measures with the median of group‐level correlation values in the alpha band with 5 s epoch length, separately for the different FC measures (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). The group size where group‐level reliability exceeds 0.9 is indicated by a red vertical line.

FIGURE 4.

Minimum group sizes required for a group‐level reliability value of 0.9 as a function of the epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Note that the scale of uncorrected measures (PLV, AEC, COH) is different from the scale of corrected measures. Furthermore, note that the maximum possible group size was 100, therefore the required minimum group size values equal to 100 mean that the group‐level reliability value of 0.9 was not reached for a given FC measure‐frequency band‐epoch length combination.

In order to test the effects of using individual structural and electrode position data, group‐level reliability was also analyzed on the second data set. Due to the sample size (n = 22) only three group sizes (1, 5, 10) were tested. Using individual head models instead of the template decreased the between‐subject similarity of FC networks, but the effects of spatial leakage were similar to the template head model‐based case (compare Figure 3 with Figure A4).

Overall, the results supported H2: The minimum group size required for reaching a group‐level reliability value of 0.9 was lower for uncorrected than for corrected FC measures at each frequency band and epoch length. This observation was not affected by whether individual anatomy and electrode locations were used or by the parcellation scheme.

3.1.3. Distinguishing participants based on a short EEG segment (H3)

Here, we tested whether FC networks estimated by uncorrected measures were more distinctive for individuals than those calculated by corrected measures. We found that uncorrected measures showed higher values both for self‐similarity as well as for the similarity to others (see Figure A6 for an example: alpha band with an epoch length of 5 s). Differential identifiability values were generally larger for uncorrected than for corrected measures, but not for all frequency band–epoch length combinations (Figure A7).

The two‐way dependent measures ANOVA on differential identifiability with factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected) yielded significant main effects (Leakage Correction: F(1, 200) = 4268.0, p < 1e‐6, η ² = .12; Measure Type: F(2, 200) = 1783.0 p < 1e‐6, η ² = .10) and a significant interaction (F(2, 200) = 1019.0, p < 1e‐4, η ² = .06). For Leakage Correction, differential identifiability was larger for uncorrected (M = 0.22, SD = 0.08) than for corrected measures (M = 0.14, SD = 0.13). As for Measure Type, differential identifiability was largest for coherence (M = 0.23, SD = 0.11), intermediate for phase locking (M = 0.17, SD = 0.12), and smallest for amplitude correlation (M = 0.14, SD = 0.10) (Figure 5, left). Pairwise comparisons revealed that the levels of Measure Type were all significantly different from each other (phase locking vs. amplitude correlation: t(200) = 15.67, p < 1e‐6, d = 1.11; phase locking vs. coherence: t(200) = 50.84, p < 1e‐6, d = 3.59; amplitude correlation vs. coherence: t(200) = 44.04, p < 1e‐6, d = 3.11). To interpret the interaction, follow‐up one‐way ANOVAs with the factor Measure Type were performed on differential identifiability separately for corrected and uncorrected measures. The effect of Measure Type was larger for corrected (F(2, 200) = 1967.0, p < 1e‐6, η ² = .22) than for uncorrected measures (F(2, 200) = 440.8, p < 1e‐6, η ² = .05).

FIGURE 5.

Left: Group‐mean value and standard deviation of differential identifiability values for uncorrected (PLV, AEC, COH) and the corresponding corrected measures (iPLV, AECc, iCOH) pooled across frequency bands and epoch lengths. Right: Group‐mean value and standard deviation of differential identifiability values for each corrected measure (iPLV, PLI, wPLI, AECc, iCOH) pooled across frequency bands and epoch lengths.

The one‐way dependent measures ANOVA on differential identifiability with factor Corrected Measure (iPLV, PLI, wPLI, AECc, iCOH) yielded a statistically significant effect (F(4, 200) = 867.4; p < 1e‐6; η ² = .13). Differential identifiability was largest for iCOH (M = 0.23, SD = 0.13), then wPLI (M = 0.15, SD = 0.14), then PLI (M = 0.14, SD = 0.14), then iPLV (M = 0.11, SD = 0.11), and finally AECc (M = 0.09, SD = 0.09) (Figure 5, right). Pairwise differences (Table A3) between corrected measures were statistically significant for all pairings (p < 1e‐6 for all pairings).

In order to test the effects of using individual structural and electrode position data, the same analyses were performed on the second data set. The two‐way dependent measures ANOVA on differential identifiability with factors Measure Type and Leakage Correction yielded significant main effects (Leakage Correction: F(1, 21) = 2111.4, p < 1e‐6, η ² = .36; Measure Type: F(2, 21) = 326.1, p < 1e‐6, η ² = .11) and significant interaction (F(2, 21) = 197.4, p < 1e‐6, η ² = .07). For Leakage Correction, differential identifiability was larger for uncorrected (M = 0.31, SD = 0.10) than for corrected measures (M = 0.14, SD = 0.13). As for Measure Type, differential identifiability was largest for coherence (M = 0.28, SD = 0.11), intermediate for phase locking (M = 0.23, SD = 0.17) and smallest for amplitude correlation (M = 0.17, SD = 0.12), replicating the order from the primary data set. Pairwise comparisons revealed that the levels of Measure Type were significantly different from each other (phase locking vs. amplitude correlation: t(21) = 11.89, p < 1e‐6, d = 2.53; phase locking vs. coherence: t(21) = 11.85, p < 1e‐6, d = 2.53; amplitude correlation vs. coherence: t(21) = 24.07, p < 1e‐6, d = 5.13). To interpret the interaction, follow‐up one‐way ANOVAs with the factor Measure Type were performed on differential identifiability separately for corrected and uncorrected measures. The effect of Measure Type was larger for corrected (F(2, 21) = 310.1, p < 1e‐6, η ² = .30) than for uncorrected measures (F(2, 21) = 197.1, p < 1e‐6, η ² = .23), similarly to the primary data set.

The one‐way dependent measures ANOVA on reliability values with factor Corrected Measure yielded a statistically significant effect (F(4, 21) = 146.4; p < 1e‐6; η ² = .19). Differential identifiability was largest for iCOH (M = 0.24, SD = 0.13), then wPLI (M = 0.13, SD = 0.13), then PLI (M = 0.12, SD = 0.13), then iPLV (M = 0.11, SD = 0.11), and finally AECc (M = 0.08, SD = 0.08), again replicating the order from the primary data set. Pairwise differences (Table A4) between corrected measures were statistically significant for most pairings except for iPLV‐PLI, iPLV‐AECc, and PLI‐AECc.

Overall, H3 was supported by the results: Participants' differential identifiability values were significantly lower for corrected FC measures relative to uncorrected measures when the values were pooled across all epoch lengths and frequency bands. This was true both for individual and for a common template‐based head model. However, there are frequency band‐epoch length combinations, where corrected measures yielded similar, or even higher differential identifiability values than uncorrected measures.

3.2. Results for the second set of hypotheses

3.2.1. Similarity between real FC networks and imaging kernel networks (H4)

Here, we tested whether the networks implied by the source dependencies introduced through the imaging kernel matrices showed high similarity to real FC matrices for uncorrected measures. We found that subject‐level similarity values between real FC and imaging kernel networks were higher for uncorrected than for corrected measures. For uncorrected measures, the similarity values were typically higher in higher frequency bands. Epoch length had only a weak effect on similarity values for most FC measures, but it had a strong effect on the iPLV measure (Figure A8).

The two‐way dependent measures ANOVA on the similarities between average FC matrices and imaging kernel networks with factors Measure Type (phase locking, amplitude correlation, coherence) and Leakage Correction (corrected vs. uncorrected) yielded significant main effects (Leakage Correction: F(1, 200) = 251224.2, p < 1e‐6, η ² = .89; Measure Type: F(2, 200) = 201.9, p < 1e‐6, η ² = .0014) and significant interaction (F(2, 200) = 2192.4, p < 1e‐6, η ² = .02). For Leakage Correction, similarity values were larger for uncorrected (M = 0.57, SD = 0.10) than for corrected measures (M = 0.04, SD = 0.09). As for Measure Type, FC network–kernel network similarity values were largest for phase locking (M = 0.32, SD = 0.23); intermediate for coherence (M = 0.29, SD = 0.30); and the lowest for amplitude correlation (M = 0.29, SD = 0.30) (Figure 6, left). Pairwise comparisons revealed that the levels of Measure Type were significantly different from each other (phase locking vs. amplitude correlation: t(200) = 28.00, p < 1e‐6, d = 1.97; phase locking vs. coherence: t(200) = 16.6, p < 1e‐6, d = 1.17; amplitude correlation vs. coherence: t(200) = 5.88, p < 1e‐6, d = 0.42). To interpret the interaction, follow‐up one‐way ANOVAs with the factor Measure Type were performed on similarity values separately for corrected and uncorrected measures. The effect of Measure Type was larger for corrected (F(2, 200) = 2411.0, p < 1e‐6, η ² = .28) than for uncorrected measures (F(2, 200) = 539.4, p < 1e‐6, η ² = .06). Importantly, similarity values were above 0.4 for uncorrected measures for each epoch length and frequency band and close to zero for corrected measures, except for iPLV.

FIGURE 6.

Left: Group‐mean value and standard deviation of the similarity between subject‐average FC matrices and subject‐average imaging kernel networks for uncorrected (PLV, AEC, COH) and the corresponding corrected measures (iPLV, AECc, iCOH) pooled across frequency bands and epoch lengths. Right: Group‐mean value and standard deviation of the similarity between subject‐average FC matrices and subject‐average imaging kernel networks for each corrected measure (iPLV, PLI, wPLI, AECc, iCOH) pooled across frequency bands and epoch lengths.

The one‐way dependent measures ANOVA on the similarity values with the factor Corrected Measure (iPLV, PLI, wPLI, AECc, iCOH) yielded a statistically significant effect (F(4, 200) = 2041.0; p < 1e‐6; η ² = .29). Similarity was different from zero only in the case of iPLV (M = 0.11, SD = 0.10, t(200) = 90.16, p < 1e‐6), while similarity values were not significantly different from zero for the rest of the corrected measures (all ps > .08; iCOH: M = 0.003, SD = 0.06; wPLI: M = 0.001, SD = 0.04; PLI: M = −0.0003, SD = 0.04; AECc: M = −0.0004, SD = 0.04; Figure 6, right). Pairwise comparisons (Table A5) revealed that only iPLV differed significantly from all other measures (ps < 1e‐6).

In order to test the effects of using individual structural and electrode position data, the same analyses were performed on the second data set. The two‐way dependent measures ANOVA on the similarities between average FC matrices and imaging kernel networks with factors Measure Type and Leakage Correction yielded significant main effects (Leakage Correction: F(1, 21) = 32832.8, p < 1e‐6, η ² = .91; Measure Type: F(2, 21) = 80.3, p < 1e‐6, η ² = .0044) and significant interaction (F(2, 21) = 245.0, p < 1e‐6, η ² = .01). For Leakage Correction, similarity values were larger for uncorrected (M = 0.63, SD = 0.10) than for corrected measures (M = 0.03, SD = 0.09). As for the measure type, FC network–kernel network similarity values were largest for phase locking (M = 0.36, SD = 0.27), intermediate for coherence (M = 0.32, SD = 0.34), and the lowest for amplitude correlation (M = 0.32, SD = 0.32), replicating the order from the primary data set. Pairwise comparisons revealed that the phase locking vs. amplitude correlation and the phase locking vs. coherence comparisons yielded significant results (t(21) = 20.82, p < 1e‐6, d = 4.44; and t(21) = 9.94, p < 1e‐6, d = 2.12, respectively), while the difference between amplitude correlation and coherence was not significant (t(21) = 0.09; p > 0.5; d = 0.02). To interpret the interaction, follow‐up one‐way ANOVAs with the factor Measure Type were performed on similarity values separately for corrected and uncorrected measures. The ANOVAs showed that the effect of Measure Type is larger for corrected (F(2, 21) = 378.9, p < 1e‐6, η ² = .36) than for uncorrected measures (F(2, 21) = 22.6, p < 1e‐6, η ² = .03), similarly to the primary data set.

The one‐way dependent measures ANOVA on the similarity values with factor Corrected Measure (iPLV, PLI, wPLI, AECc, iCOH) yielded a statistically significant effect (F(4, 21) = 305.6; p < 1e‐6; η ² = .36). As with the primary data set, similarity values were different from zero only in the case of iPLV (M = 0.11, SD = 0.11, t(21) = 30.08, p < 1e‐6), while similarity values were not significantly different from zero for the rest of the corrected measures (all ps > .05; iPLV: M = 0.11, SD = 0.11; wPLI: M = 0.002, SD = 0.04; AECc: M = 0.0015, SD = 0.04; PLI: M = 0.0002, SD = 0.04; iCOH: M = −0.02, SD = 0.06). Pairwise comparisons (Table A6) revealed that only iPLV differed significantly from all other measures (ps < 1e‐6).

Overall, H4 was supported by the results: The similarity between real FC networks and imaging kernel networks was higher for uncorrected than corrected FC measures, irrespective of whether individual or a common template‐based head model was used. Similarity values for corrected measure are largely indistinguishable from zero; however, iPLV values were influenced by imaging kernel networks.

3.2.2. Between‐subject similarity of real FC networks explained by imaging kernel networks (H5)

Here, we assessed how much of the between‐subject similarity of real FC matrices was explained by networks implied by the source dependencies introduced through the imaging kernel matrices. We found that the explained similarity values pooled across epoch lengths and frequency bands were significantly higher for uncorrected (M = 0.15, SD = 0.08) than for corrected measures (M = 0.0003, SD = 0.001; t(158) = 18.19, p < 1e‐6, d = 2.57). The apparent effect of frequency bands (see Figure 7) on explained similarity was tested post hoc with one‐way ANOVAs on the explained similarity values pooled across epoch lengths and measures, separately for corrected and uncorrected measures. For uncorrected measures, frequency had a significant effect on explained similarity (F(4, 55) = 99.55, p < 1e‐6, η ² = .88) and pairwise comparisons (t tests with Bonferroni correction) revealed that explained similarity was larger in the beta and gamma bands compared to all other frequency bands (ps < 1e‐5). For corrected measures, the effect of frequency was not significant (F(4, 95) = 1.93, p = .11, η ² = .07).

FIGURE 7.

Correlation of between‐subject kernel network similarity values with between‐subject FC similarity as a function of epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Correlation was transformed into R ².

The effects of using individual structural MRI and electrode positions for each participant on how much of the between‐subject FC similarity is explained by between‐subject kernel network similarity was assessed on the second data set. While we refrained from direct statistical comparisons across the two data sets, one might expect a larger effect with individual head models than with a common template (as individual anatomy better captures individual differences). We again found that the explained similarity values were higher for uncorrected measures (M = 0.18, SD = 0.08) than for corrected measures (M = 0.007, SD = 0.023; t(158) = 20.12, p < 1e‐6, d = 2.91), and the average explained similarity slightly increased by using individual head models (compare Figure 7 and Figure A9). As for the primary data set, the post hoc one‐way ANOVAs on explained similarity values with factor Frequency yielded a significant main effect for uncorrected measures (F(4, 55) = 15.95, p < 1e‐6, η ² = .53), but not for corrected measures (F(4, 95) = 0.54, p = .71, η ² = .02). Pairwise comparisons across frequency bands for uncorrected measures revealed a different pattern relative to the primary data set. Explained similarity was significantly lower in the alpha band compared to all other bands (ps < .015).

We also assessed how much of the between‐subject similarity of real FC networks was explained by the between‐subject similarity of parcel resolution matrices (Farahibozorg et al., 2018), as parcel resolution matrices estimate spatial leakage in different way from imaging kernel networks. Again, explained similarity was significantly higher for uncorrected measures (M = 0.09, SD = 0.058) than for corrected measures (M = 0.015, SD = 0.015; t(158) = 12.30, p < 1e‐6, d = 1.77). The two ways of calculating explained similarity, either with the imaging kernel or the parcel resolution matrices, were compared in a two‐way ANOVA with factors Matrix Type (Imaging kernel vs. Parcel resolution matrix) and Measure Type (corrected vs. uncorrected). There was a significant interaction (F(1, 316) = 53.90, p < 1e‐6, η ² = .15) and significant main effects (Matrix Type: F(1, 316) = 7.08, p = .008, η ² = .02; Measure Type: F(1, 316) = 481.76, p < 1e‐6, η ² = .60). Pairwise comparisons revealed that explained similarity values were larger for the imaging kernel matrix method than for the parcel resolution matrix method only for uncorrected (p < 1e‐6) but not corrected measures (p = .1) (compare Figure 7 and Figure 8).

FIGURE 8.

Correlation of between‐subject parcel resolution similarity values with between‐subject FC similarity as a function of epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Correlation was transformed into R ².

Finally, we assessed the effect of source reconstruction on the differential identifiability of participants, by calculating the difference between self‐similarity and the similarity to others after subtracting the imaging kernel networks from the real FC networks. We found that when subtracting the imaging kernel networks from the real FC networks, differential identifiability increased for all FC measures at each frequency band and epoch length, except for the iPLV and the AECc in the gamma band with 1 s epoch length (see Figure A10 for an example: alpha band with an epoch length of 5 s). The increased differential identifiability was quantified by one‐sided paired t tests (Table A7) with differential identifiability values averaged across frequency bands and epoch lengths. The comparisons revealed that subtracting the imaging kernel networks from FC networks increased differential identifiability significantly (p < 1e‐6 for all measures). The effect was stronger for uncorrected measures (d = 10.37, d = 9.64, d = 10.06 for PLV, AEC, and COH, respectively) than for corrected measures (d = 7.94, d = 4.68, d = 4.87, d = 7.06, d = 5.06 for iPLV, PLI, wPLI, AECc, iCOH, respectively).

In sum, the following results supported H5: The between‐subject similarity of imaging kernel networks explained a considerable amount of the between‐subject similarity of real FC networks. This observation remained valid when individual anatomy and electrode locations were used. The amount of between‐subject similarity explained by imaging kernel networks was also demonstrated by subtracting the imaging kernel networks from FC networks. The subtraction significantly increased the differential identifiability of participants by decreasing between‐subject similarity. Thus, H5 was fulfilled.

4. DISCUSSION

The current study aimed to examine the effects of the FC measures, correction for spatial leakage, and source reconstruction on the reliability of FC networks. These networks were extracted from the EEG signals recorded in the resting state. Reliability was estimated on the individual‐ and on the group level. The results supported our hypotheses that individual‐ and group‐level reliability as well as the differential identifiability of participants (their self‐similarity minus similarity to others) is lower for FC measures corrected for spatial leakage than for uncorrected measures. Further, imaging kernel matrices used in source reconstruction significantly contribute to the between‐subject similarity of FC networks. None of these conclusions were altered when source reconstruction was based on individually measured structural MRI scans and electrode positions or when turning to a higher resolution parcellation scheme. In the following, the current results are discussed in the context of previous similar studies.

We found a large influence of spatial leakage correction on individual‐ and group‐level reliability of FC networks. FC measures corrected for spatial leakage (iPLV, PLI, wPLI, AECc, and iCOH) showed lower individual‐ and group‐level reliability than uncorrected measures (PLV, AEC, and COH, see H1 and H2). Similar observations were previously reported for MEG (Colclough et al., 2016; Garcés et al., 2016) and EEG data (Duan et al., 2021; Yu, 2020). These studies concluded that spatial leakage could contribute to the high individual‐ and group‐level reliability of FC networks. Spatial leakage introduces spurious zero‐lag dependencies into the FC networks which are independent of the recorded data and, thus, boost individual‐ and group‐level reliability as well. In our study, we have estimated this effect by driving the source reconstruction and FC estimation pipeline with noise, yielding the imaging kernel networks. These networks correlated strongly with the real FC networks for uncorrected measures, while, as expected, they remained mostly independent for measures corrected against spatial leakage (with the exception of iPLV). This effect was comparable across the two data sets tested here, both with template head models (primary data set) and with individual anatomy and electrode locations (second data set).

While individual differences between imaging kernel networks included anatomical differences only in case of the individual head models, differences in the noise statistics appeared in both cases. The imaging kernel networks were thus slightly different for each participant even in the primary data set, with template head models, and the contribution of their similarity to between‐subject differences could be estimated. The similarity of individual FC matrices was weakly determined by the source reconstruction and FC estimation pipeline for uncorrected measures, both with and without individual anatomy. Overall, our results suggest that FC networks calculated based on measures not corrected for spatial leakage reflect more the characteristics of the participant and the signal analysis procedure and less the ongoing neural activity during the recording. Between‐subject FC similarity showed lower correlation with between‐subject parcel resolution similarity values than with between‐subject kernel network similarity values for uncorrected measures, but higher correlation for corrected measures. The difference in correlation values can be explained by the fact that parcel resolution matrices quantify dependencies introduced by the source reconstruction procedure, while imaging kernel networks characterize the combination of the source reconstruction procedure and the FC measure.

In our study, individual‐ and group‐level reliability of FC networks was estimated in five frequency bands with four different epoch lengths. The effects of frequency band and epoch length showed FC measure‐specific results. Higher frequency bands typically showed higher reliability but not in all FC measure‐epoch length combinations. A similar trend was reported by Colclough et al. (2016) and Garcés et al. (2016) for MEG data. Epoch length affects the accuracy of FC estimation. The frequency of oscillations and the epoch length determine the number of oscillatory cycles in one epoch. This is especially problematic in the delta band with 1 s epoch length where one period of the slowest oscillations can exceed the epoch length. However, increasing the epoch length resulted in an increased reliability only for a subset of the FC measure‐frequency band combinations. (It is important to note though that the duration of the recordings was fixed in the current study, therefore longer epochs correspond to lower number of epochs.) These results are also in line with the findings of Miljevic et al. (2022), who concluded that both epoch length and the number of epochs influence the reliability of EEG FC estimates, and the effects of these parameters depend both on the FC metric and on the frequency band. The similarity between FC networks corresponding to different frequency bands was generally higher for uncorrected measures compared to corrected measures. This effect might be due to the spurious connections introduced by source space reconstruction. As the CTFs of the resolution matrices are independent of the properties of neural generators, the spurious connections are the same for all frequency bands. To the degree that the spurious connections consist of zero‐lag connections, uncorrected measures should yield a larger similarity of FC networks across frequency bands than corrected measures.

Group size consistently increased group‐level reliability. Based on the group‐level reliability analysis, the required minimum group size can be determined for a given FC measure‐frequency band‐epoch length combination. A group size of 60 yielded a group‐level reliability above 0.9 for most of these combinations (H2). For MEG data, Colclough et al. (2016) found that a group size of 30 resulted in group‐level reliability above 0.9 for all uncorrected FC measures and for some of the corrected FC measures in each frequency band.

The analysis of differential identifiability revealed a similar trend for the correlation of one selected epoch to the subject‐level average of epochs as was seen for within‐subject reliability: lower values for corrected FC measures than for uncorrected FC measures (H3). For MEG data, Sareen et al. (2021) and for EEG data, Fraschini et al. (2019) similarly reported better identification success rate for non‐leakage corrected than for leakage‐corrected FC measures. Fraschini et al. (2019) did not rule out the hypothesis that muscle artifacts (particularly at high frequencies) may play a role in finding distinctive idiosyncratic characteristics. These authors also hypothesized that distinctive patterns of connectivity may be strongly influenced by spurious connectivity values. In sum, the pattern of spatial leakage likely contains idiosyncratic information, which helps to identify the individual. We further analyzed this effect by recomputing differential identifiability after subtracting the imaging kernel networks from real FC networks. Subtracting the imaging kernel network significantly increased the identifiability of participants by decreasing between‐subject similarity, and the effect was stronger for uncorrected measures than for corrected measures. This result also demonstrates that the between‐subject similarity of FC networks can be partly explained by the between‐subject similarity of imaging kernel networks and also hints at the possibility that real zero‐lag dependencies could contribute to individual differences independent from the patterns introduced only by the estimation pipeline.

Anatomical parcellation and the spatial resolution of the measurement configuration critically influence spatial leakage and the corresponding spurious connections in FC analysis. Some distinct anatomical parcels may produce highly similar CTFs due to the limited spatial resolution of the measurement configuration, while large parcels may be split into subregions with distinguishable CTFs. If deeper parcels are also involved in the analysis, they may receive much larger crosstalk from areas close to the sensors than the diagonal element of $\mathit{\text{PRmat}}$ corresponding to the given parcel (Farahibozorg et al., 2018; Hauk et al., 2022). In our study, the analysis was based on the parcellation defined by the DKT atlas consisting of 62 ROIs. However, to test the effect of parcellation, a subset of the whole analysis was repeated using the Destrieux atlas consisting of 148 ROIs. Within‐subject reliability and the minimum group sizes required for a group‐level reliability value exceeding 0.9 did not differ significantly for the tested PLV, iPLV, AEC, and AECc measures in the Destrieux atlas‐based analysis compared to the DKT atlas‐based case. The DKT atlas, as well as the Destrieux atlas are gyral (and sulcal)‐based atlases. It is possible that other parcellation strategies with similar resolution (e.g., the functionally parcelled Schaefer atlas; Schaefer et al., 2018) might yield different results. Although it was not investigated in our study, using an adaptive parcellation (such as the one described by Farahibozorg et al., 2018) might result in lower reliability values due to the reduction of spatial leakage and false positive functional connections.

Our study has four main limitations. First, anatomical scans for participants in the 201‐subject‐data set were not available, for this reason, the same head model was used for each participant. Therefore, we repeated a subset of our analysis using a smaller data set where individual structural MRI and electrode positions were available for each participant. While we acknowledge that the small number of participants (22) in this data set did not enable the perfect reproduction of our analysis, our observations about the effect of spatial leakage correction on the individual‐ and group‐level reliability, on the differential identifiability of participants, as well as on the between‐subject similarity of FC networks explained by imaging kernel networks remained valid when using individual head models. Second, only one resting‐state recording was available for each participant. As SNR can considerably change between recordings of the same individual, testing participant identifiability with more than one recording per participant would enable one to assess the identifiability of participants based on different FC measures in a more realistic way. Third, the analysis of reliability was carried out only for resting‐state EEG data. As certain functional networks in the brain are substantially reorganized during various tasks, carrying out the same analysis for task‐based EEG recordings would provide valuable information about the potential of FC measures corrected for spatial leakage to reliably yield functional networks reflecting neural activity. Fourth, only one source reconstruction method (sLORETA) was used. As localization accuracy and spatial extent of the reconstructed source activities may be different with other source reconstruction methods, carrying out the same analysis with other source reconstruction methods would provide information about the relationship of the different source reconstruction methods and FC measures and their combined effect on individual‐ and group‐level reliability.

Similarly to previous studies, we found that the imaging kernel matrices used in source reconstruction significantly contribute to the higher reliability and better participant identifiability for FC measures not corrected for spatial leakage over corrected measures. Some of the differences likely reflect stable physiological characteristics of participants. Therefore, in light of the current and previous results, when the main goal of the investigation is to provide a large‐scale description of the neural activity of the brain in the situation tested, we recommend using spatial leakage corrected measures to estimate EEG FC in the source space. On the other hand, when the goal is to identify participants, we recommend using uncorrected FC measures.

FUNDING INFORMATION

This work was funded by the Office of Naval Research (N62909‐23‐1‐2025) and the National Research, Development and Innovation Office of Hungary grants K132642 (to IW) and ANN131305 (to BT).

CONFLICT OF INTEREST STATEMENT

The authors declare that they have no conflict of interest.

INDIVIDUAL VERSUS TEMPLATE HEAD MODELING

We performed a non‐exhaustive literature search using Google scholar with the keyword combination EEG “functional connectivity” “resting‐state” source, and also using Web of Science with the keyword combination EEG AND functional connectivity AND resting state AND source, and for each search engine, we selected the first 50 articles published since 2019, sorted by relevance. Altogether, 71 articles were checked (the results of Google scholar and Web of Science overlap), and in these articles, 51 studies used a template head model while 20 studies used individual head models.

FC MEASURES

Several FC measures (including corrected and uncorrected measures) were successfully applied in previous studies to investigate brain networks based on EEG data. Uncorrected measures include the amplitude or power envelope correlation (Bruns et al., 2000), spectral coherence (Guevara & Corsi‐Cabrera, 1996), phase‐locking value (PLV; Lachaux et al., 1999), various estimators of Granger causality when applied in sensor space (see Brunner et al., 2016; Van de Steen et al., 2019), phase synchrony (Tass et al., 1998), mutual information (e.g., Palus, 1997), and their various extensions. Corrected measures, in general, penalize zero‐lag contributions to various synchrony measures. These include imaginary spectral coherence (Nolte et al., 2004), lagged coherence (Pascual‐Marqui, 2007), corrected AEC (Brookes et al., 2012), imaginary and corrected imaginary PLV (Bruña et al., 2018), phase lag index (PLI; Stam et al., 2007) and wPLI (Vinck et al., 2011), phase slope index (Nolte et al., 2008), and time‐lagged mutual information (Wilmer et al., 2012). In this study, the following FC measures were used:

Phase‐locking value

Since the brain is a nonlinear dynamical system, phase‐based connectivity measures provide a promising approach to quantify its interactions (Aydore et al., 2013). The most commonly used phase interaction measure is PLV, the magnitude of the mean phase difference between the two signals, with phase differences expressed as complex unit‐length vectors (Mormann et al., 2000). The discrete‐time PLV is defined as follows:

$$\mathit{PLV}=\left|\frac{1}{T}{\sum }_{t=1}^T{e}^{j\left({\varphi }_a\left(t\right)-{\varphi }_b\left(t\right)\right)}\right|$$

where T is the data length and Φ _a and Φ _b are the instantaneous phases of the signals s _a and s _b . The instantaneous phase of an arbitrary signal (s(t)) can be expressed as

$$\varphi \left(t\right)=\mathit{arctan}\frac{\widehat{s}\left(t\right)}{s\left(t\right)}$$

where $\widehat{s}\left(t\right)$ is the Hilbert transform of the signal s(t).

PLV takes values between 0 and 1, with 0 reflecting no phase synchrony and 1 reflecting perfect phase synchrony (i.e., the relative phase between the two signals is identical in all samples).

Imaginary part of PLV

The imaginary part of PLV (Bruña et al., 2018) is defined as

$$\mathit{\text{iPLV}}=\frac{1}{T}I\left\{{\sum }_{t=1}^T{e}^{j\left({\varphi }_a\left(t\right)-{\varphi }_b\left(t\right)\right)}\right\}$$

where $I\left\{x\right\}$ stands for the imaginary part of x. Imaginary part of PLV (iPLV) ignores zero phase differences because the complex unit‐length vector for zero phase produces a real value.

Phase lag index

Another very commonly used FC measure ignoring zero phase differences is the PLI. PLI averages the sign of phase differences and takes the absolute value of the average (Stam et al., 2007). PLI is defined as

$$\mathit{PLI}=\left|\frac{1}{T}{\sum }_{t=1}^T\mathit{sign}\left[{\varphi }_a\left(t\right)-{\varphi }_b\left(t\right)\right]\right|$$

where sign[x] stands for the sign function of x.

Weighted PLI

The wPLI scales the contributions of angle differences according to their distance from the real axis. The scaling is intended to express the confidence of the observed phase lead or phase lag, as for small angle differences, the sign can be changed by low‐amplitude noise while the sign of large angle differences is more robust against noise (Vinck et al., 2011). wPLI at a given frequency ω is defined as

$$\mathit{\text{wPLI}}\left(\omega \right)=\frac{\left|{\sum }_{t=1}^T\left|I\left\{S_{\mathit{ab}}\left(\omega ,t\right)\right\}\right|\mathit{sign}\left[I\left\{S_{\mathit{ab}}\left(\omega ,t\right)\right\}\right]\right|}{{\sum }_{t=1}^T\left|I\left\{S_{\mathit{ab}}\left(\omega ,t\right)\right\}\right|}$$

where $S_{\mathit{ab}}\left(\omega \right)$ is the cross‐spectrum of the signals s _a and s _b , I{x} and sign[x] have the same meaning as for iPLV and PLI.

Amplitude envelope correlation

Synchronization between the amplitude envelopes of band‐limited oscillations has been initially mostly studied using MEG data (O'Neill et al., 2015). More recently, amplitude envelope‐based connectivity estimation has been successfully applied to investigate resting‐state FC networks based on EEG signals (Coquelet et al., 2020). In the first step of AEC estimation, for each epoch and participant, the envelope of the signal in each ROI is computed as the magnitude of the analytical signal. Then, Pearson's correlation coefficients are calculated between the envelopes.

AEC corrected for spatial leakage based on pairwise orthogonalized signals (AECc)

The effect of spatial leakage can be reduced by signal orthogonalization. If the linear projection of the vector v onto the vector u is removed from v , the corrected vector v _R is orthogonal to u (Brookes et al., 2012).

$$v_R=v-\frac{⟨u,v⟩}{⟨u,u⟩}u$$

where $⟨u,v⟩$ stands for the inner product (dot product) of u and v . Orthogonalization is performed for each pair of signals (considering their samples as a vector), then AEC is calculated between the pairs of orthogonalized signals. The orthogonalization can be performed in two directions (u to v and v to u). In our study, AECs were calculated for both directions, and the average of the two correlation values was used. (Note that the multivariate orthogonalization procedure proposed by Colclough et al. (2015) was not applicable in our study due to the low rank of the forward model).

Coherence coefficient

The covariation between two signals in the frequency domain can be quantified by the coherence which can be considered the frequency‐domain equivalent of the cross‐correlation (Guevara & Corsi‐Cabrera, 1996). The COH at a given frequency ω is defined as

$$\mathit{COH}\left(\omega \right)=\frac{\left|S_{\mathit{ab}}\left(\omega \right)\right|}{\sqrt{S_{\mathit{aa}}\left(\omega \right)S_{\mathit{bb}}\left(\omega \right)}}$$

where $\left|S_{\mathit{ab}}\left(\omega \right)\right|$ is the magnitude of the cross‐spectral density between signals s _a and s _b ; $S_{\mathit{aa}}\left(\omega \right)$ is the autospectrum of the signal s _a ; and $S_{\mathit{bb}}\left(\omega \right)$ is the autospectrum of the signal s _b at frequency ω.

Imaginary part of coherency

If the magnitude operator is omitted from the formula of COH, a complex‐valued quantity is obtained called the coherency. If the coherency is projected onto the imaginary axis, the imaginary part of coherency (iCOH) is obtained (Nolte et al., 2004). iCOH is defined as

$$\mathit{\text{iCOH}}\left(\omega \right)=\frac{I\left\{S_{\mathit{ab}}\left(\omega \right)\right\}}{\sqrt{S_{\mathit{aa}}\left(\omega \right)S_{\mathit{bb}}\left(\omega \right)}}$$

where $I\left\{x\right\}$ stands for the imaginary part of x, $S_{\mathit{ab}}\left(\omega \right)$, $S_{\mathit{aa}}\left(\omega \right)$, and $S_{\mathit{bb}}\left(\omega \right)$, have the same meaning as for COH. iCOH ignores zero phase differences.

RESULTS FOR H1‐H2 WITH THE DESTRIEUX ATLAS

The effects of the parcellation scheme on H1 were tested by analyzing within‐subject reliability for a subset of FC measures (PLV, iPLV, AEC, AECc) using the Destrieux atlas (with 148 regions) on the primary data set. The two‐way dependent measures ANOVA on reliability values with factors Measure Type and Leakage Correction yielded significant main effects (Leakage Correction: F(1, 200) = 31675.1 p < 1e‐6, η ² = .64; Measure Type: F(1, 200) = 516.1 p < 1e‐6, η ² = .01) and significant interaction (F(1, 200) = 211.2, p < 1e‐6, η ² = .004). For Leakage Correction, within‐subject reliability values were larger for uncorrected (M = 0.97, SD = 0.03) than for corrected measures (M = 0.50, SD = 0.25). As for Measure Type, within‐subject reliability values were larger for phase locking (M = 0.76, SD = 0.26) than for amplitude correlation (M = 0.70, SD = 0.32). The general trends of the effect of spatial leakage correction, epoch lengths, and frequency bands on within‐subject reliability values were very similar when using the DKT atlas (62 ROIs) or the Destrieux atlas (148 ROIs) (compare Figure 2a and Figure A2).

The effects of the parcellation scheme on H2 were tested by analyzing group‐level reliability for a subset of FC measures (PLV, iPLV, AEC, AECc) using the Destrieux atlas on the primary data set. The required group sizes to reach r = 0.9 reliability for the DKT (M = 25.31, SD = 27.91) and the Destrieux atlas (M = 24.91, SD = 26.75) were not significantly different: t(79) = 1.07, p = 0.29, d = 0.12 (paired t test, PLV, iPLV, AEC, AECc measures, pooled across all epoch lengths and frequency bands). The effect of spatial leakage correction on the minimum group size to reach 0.9 reliability was very similar when using the DKT atlas (62 ROIs) compared to the Destrieux atlas (148 ROIs) (compare Figure 4 with Figure A5).

FIGURE A1.

Group‐averaged (n = 22) within‐subject reliability as a function of epoch length, separately for each frequency band and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH) for a data set containing individual structural MRI and electrode positions for each participant. Standard error is marked over the measurement points. Note that the scale of uncorrected measures (PLV, AEC, COH) is different from the scale of corrected measures.

FIGURE A2.

Group‐averaged (n = 201) within‐subject reliability as a function of epoch length, separately for each frequency band and selected FC measure (PLV, iPLV, AEC, AECc) using the Destrieux atlas consisting of 148 ROIs. Standard error is marked over the measurement points. Note that the scale of uncorrected measures (PLV, AEC) is different from the scale of corrected measures.

FIGURE A3.

Group‐averaged (n = 201) similarity between frequency bands for each FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Subject‐level similarity values were averaged across participants and epoch lengths.

FIGURE A4.

Group‐level reliability of connectivity measures with the median of group‐level correlation values for a data set containing individual structural MRI and electrode positions for each participant in the alpha band with 5 s epoch length, separately for the different FC measures (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH).

FIGURE A5.

Minimum group sizes required for a group‐level reliability value of 0.9 as a function of the epoch length, separately for each frequency band (marked by color) and selected FC measure (PLV, iPLV, AEC, AECc) using the Destrieux atlas consisting of 148 ROIs. Note that the scale of uncorrected measures (PLV, AEC) is different from the scale of corrected measures.

FIGURE A6.

Distribution of the similarity (measured by Pearson's correlation coefficient) of epoch‐networks with the subject‐level average network of the same participant (red) and with the other participants (blue) in the alpha band with 5 s epoch length, separately for the different FC measures (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH).

FIGURE A7.

Group‐average (n = 201) of the differential identifiability as a function of epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH).

FIGURE A8.

Group‐averaged (n = 201) similarity between subject‐average FC matrices and subject‐average imaging kernel networks as a function of epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH). Standard error is marked over the measurement point. Note that the scale of PLV, AEC, COH, and iPLV is different from the scale of PLI, wPLI, AECc, and iCOH.

FIGURE A9.

Correlation of between‐subject kernel network similarity values with between‐subject FC similarity as a function of epoch length, separately for each frequency band (marked by color) and FC measure (PLV, iPLV, PLI, wPLI, AEC, AECc, COH, iCOH) for a data set containing individual structural MRI and electrode positions for each participant. Correlation was transformed into R ².

FIGURE A10.

Distribution of the similarity of epoch‐networks with the subject‐level average network of the same participant (red) and with the subject‐level average networks of other participants (blue), before (light colors) and after (dark colors) subtracting the imaging kernel networks from PLV, iPLV, PLI, wPLI, AEC, AECc, COH, and iCOH FC networks for the alpha band with 5 s epoch length. The differences between the medians (self‐others) are illustrated by bars before (light blue) and after (dark blue) subtracting the imaging kernel networks from real FC networks.

TABLE A1.

Results of paired t tests of within‐subject reliability values between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH). Reliability values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	7.43	200	<1e‐6	0.52
iPLV‐wPLI	−3.34	200	.01	−0.24
iPLV‐AECc	14.80	200	<1e‐6	1.06
iPLV‐iCOH	−57.31	200	<1e‐6	−4.04
PLI‐wPLI	−19.59	200	<1e‐6	−1.38
PLI‐AECc	6.85	200	<1e‐6	0.48
PLI‐iCOH	−39.10	200	<1e‐6	−2.76
wPLI‐AECc	15.47	200	<1e‐6	1.09
wPLI‐iCOH	−37.22	200	<1e‐6	−2.63
AECc‐iCOH	−46.03	200	<1e‐6	−3.25

TABLE A2.

Results of paired t tests of within‐subject reliability values between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH) for the second data set. Reliability values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	3.14	21	.05	0.67
iPLV‐wPLI	−2.67	21	.14	−0.57
iPLV‐AECc	3.14	21	.05	0.67
iPLV‐iCOH	−19.33	21	<1e‐6	−4.12
PLI‐wPLI	−8.18	21	<1e‐6	−1.74
PLI‐AECc	0.89	21	>0.5	0.19
PLI‐iCOH	−14.11	21	<1e‐6	−3.01
wPLI‐AECc	3.72	21	.01	0.79
wPLI‐iCOH	−15.70	21	<1e‐6	−3.35
AECc‐iCOH	−3.47	21	<1e‐6	−3.47

TABLE A3.

Results of paired t tests of differential identifiability values between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH). Differential identifiability values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	−12.78	200	<1e‐6	−0.90
iPLV‐wPLI	−17.10	200	<1e‐6	−1.21
iPLV‐AECc	6.75	200	<1e‐6	0.48
iPLV‐iCOH	−58.50	200	<1e‐6	−4.13
PLI‐wPLI	−10.98	200	<1e‐6	−0.77
PLI‐AECc	10.96	200	<1e‐6	0.77
PLI‐iCOH	−40.78	200	<1e‐6	−2.88
wPLI‐AECc	16.48	200	<1e‐6	1.16
wPLI‐iCOH	−39.10	200	<1e‐6	−2.76
AECc‐iCOH	−38.95	200	<1e‐6	−2.75

TABLE A4.

Results of paired t tests of differential identifiability values between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH) for the second data set. Differential identifiability values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	−2.22	21	.37	−0.47
iPLV‐wPLI	−4.55	21	.001	−0.97
iPLV‐AECc	1.95	21	.65	0.42
iPLV‐iCOH	−18.64	21	<1e‐6	−3.97
PLI‐wPLI	−5.53	21	.0002	−1.18
PLI‐AECc	2.50	21	.21	0.53
PLI‐iCOH	−20.84	21	<1e‐6	−4.44
wPLI‐AECc	4.45	21	.002	0.95
wPLI‐iCOH	−24.75	21	<1e‐6	−5.28
AECc‐iCOH	−16.40	21	<1e‐6	−3.50

TABLE A5.

Results of paired t tests between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH) for the similarity values between FC matrices and corresponding imaging kernel networks. Similarity values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	74.76	200	<1e‐6	5.27
iPLV‐wPLI	73.26	200	<1e‐6	5.17
iPLV‐AECc	71.07	200	<1e‐6	5.01
iPLV‐iCOH	47.56	200	<1e‐6	3.35
PLI‐wPLI	−1.45	200	>.5	−0.10
PLI‐AECc	0.02	200	>.5	0.0013
PLI‐iCOH	−1.26	200	>.5	−0.09
wPLI‐AECc	0.59	200	>.5	0.04
wPLI‐iCOH	−0.83	200	>.5	−0.06
AECc‐iCOH	−1.31	200	>.5	−0.09

TABLE A6.

Results of paired t tests between pairings of FC measures (iPLV, PLI, wPLI, AECc, iCOH) for the similarity values between FC matrices and corresponding imaging kernel networks for the second data set. Similarity values were averaged across the different epoch lengths and frequency bands. Significance values are reported with Bonferroni‐correction against multiple comparisons.

Comparison	t	df	p	d
iPLV‐PLI	23.34	21	<1e‐6	4.98
iPLV‐wPLI	20.99	21	<1e‐6	4.48
iPLV‐AECc	27.44	21	<1e‐6	5.85
iPLV‐iCOH	15.26	21	<1e‐6	3.25
PLI‐wPLI	−0.84	21	>.5	−0.18
PLI‐AECc	−0.88	21	>.5	−0.19
PLI‐iCOH	2.39	21	.27	0.51
wPLI‐AECc	−0.32	21	>.5	−0.07
wPLI‐iCOH	2.60	21	.17	0.56
AECc‐iCOH	3.00	21	.07	0.64

TABLE A7.

Results of paired one‐sided t tests of the differential identifiability values between before and after subtracting the imaging kernel networks (the tested direction was that differential identifiability increased when the imaging kernel network was subtracted), separately for the PLV, iPLV, PLI, wPLI, AEC, AECc, COH, and iCOH measures. Differential identifiability values were averaged across epoch lengths and frequency bands.

FC measure	t	df	p	d
PLV	146.99	200	<1e‐6	10.37
iPLV	112.52	200	<1e‐6	7.94
PLI	66.90	200	<1e‐6	4.68
wPLI	69.03	200	<1e‐6	4.87
AEC	136.64	200	<1e‐6	9.64
AECc	100.02	200	<1e‐6	7.06
COH	142.61	200	<1e‐6	10.06
iCOH	71.69	200	<1e‐6	5.06

Nagy, P. , Tóth, B. , Winkler, I. , & Boncz, Á. (2024). The effects of spatial leakage correction on the reliability of EEG‐based functional connectivity networks. Human Brain Mapping, 45(8), e26747. 10.1002/hbm.26747

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.