The authors have declared that no competing interests exist.
Conceived and designed the experiments: UCL GAM. Performed the experiments: GJN. Analyzed the data: UCL HSL MM. Contributed reagents/materials/analysis tools: GJN. Wrote the paper: UCL HSL MM GAM.
Spectral content in a physiological dataset of finite size has the potential to produce spurious measures of coherence. This is especially true for electroencephalography (EEG) during general anesthesia because of the significant alteration of the power spectrum. In this study we quantitatively evaluated the genuine and spurious phase synchronization strength (PSS) of EEG during consciousness, general anesthesia, and recovery. A computational approach based on the randomized data method was used for evaluating genuine and spurious PSS. The validity of the method was tested with a simulated dataset. We applied this method to the EEG of normal subjects undergoing general anesthesia and investigated the finite size effects of EEG references, data length and spectral content on phase synchronization. The most influential factor for genuine PSS was the type of EEG reference; the most influential factor for spurious PSS was the spectral content. Genuine and spurious PSS showed characteristic temporal patterns for each frequency band across consciousness and anesthesia. Simultaneous measurement of both genuine and spurious PSS during general anesthesia is necessary in order to avoid incorrect interpretations regarding states of consciousness.
The administration of general anesthetics results in dramatic changes in behavioral state, which is accompanied by changes of functional connectivity and information integration capacity in the brain
Random or spurious correlations within multivariate data sets have been studied by random matrix theory (RMT), in which the random part of a correlation can be predicted based on the analytic results obtained from random matrix ensembles. Genuine correlation can then be estimated based on significant deviation from the analytic RMT predictions. Information about the correlation structure of the multivariate data set is imprinted into the structures of a correlation matrix
In the current study, indices of genuine, spurious and total phase synchronization strength (PSS) were quantified in the EEG of conscious and anesthetized human volunteers. First, simulated data based on the Ntori model were used to test the validity of genuine PSS measures. This model simulated multichannel oscillations and dynamic state transitions together with the modulation of phase coherence and the dominant spectral content. Second, in order to quantify spurious elements, we used randomized data with the same spectral content as the original EEG but with zero coherence. Third, the following were investigated as different sources of spurious phase synchronization: four types of EEG reference, six frequency bands and different data lengths. Finally, we investigated the temporal evolution pattern of the genuine, spurious and total PSS during general anesthesia.
To test the validity of the genuine PSS, 20 EEG time series simulated by the Ntori model were used (
(a) The phases of 50 sine waves for a simulated EEG time series are presented. The noncoherent (0–20 s and 120–140 s) and coherent (40–100 s) phases are given to the simulation, with two linear state transitions (20–40 s and 100–120 s). The color bar on the right side indicates a radian within 0 to
In the first and the last 20 s epochs for each EEG time series simulation, the partial phases
In contrast, the spurious phase coherence components were significantly reduced as the mean frequency was increased.
(a) The averaged spectrogram of 21 EEG channels for a subject, and (b) the variance of phase synchronizations between 21 EEG channels for a subject during general anesthesia. This demonstrates the significant changes of the power spectrum and the variance of phase synchronization during general anesthesia. The loss of consciousness and the recovery of consciousness happen at 5 and 10 minutes.
The spurious PSS was estimated from randomized human EEG data in order to evaluate the influence of spectral shifts after general anesthesia that were predicted by the NTori model. Since the randomized data has the same power spectrum with that of the broadband EEG but with zerocoherence by randomization, we can selectively evaluate the effect of spectral shift on the spurious PSS index. The temporal patterns of the spurious, genuine and total PSS were examined by the moving window method to investigate the effect of dynamic spectral change during state transitions (around loss of consciousness [LOC] and return of consciousness [ROC], at 5 and 10 minutes on the time axis, respectively). The means and standard deviations over 20 EEG data sets are presented in
The mean of genuine, spurious and total PSS over all randomized data sets generated from all subjects’ original EEG data are presented over time (the error bar denotes the standard deviation). The spurious PSS (blue square) and the spurious correlation strength by Pearson correlation coefficient (black square, denoted as “CCS”) were compared. (Vertical dotted lines: loss of consciousness and return of consciousness points, sequentially). The linear Pearson correlation and the phase synchronization produce a large spurious component after anesthesia.
Two measures, phase synchronization (nonlinear) and Pearson correlation coefficient (linear), were compared to assess their vulnerability to the finite size effect for this data set. For the genuine, spurious and total linear correlation strengths, the mean phase coherence in the definitions of genuine, spurious and total PSS was replaced with Pearson correlation coefficient
The finite size effects induced by data length, EEG reference and frequency band were investigated using the EEG in the baseline conscious state (
Each symbol denotes the mean genuine PSS (solid lines) or the mean spurious PSS (dotted lines) for the combination of data length, type of EEG reference and frequency band. The EEG reference mainly affects genuine PSS. Each color denotes a type of EEG reference (blue: unipolar reference (A2); green: longitudinal bipolar; red: common averaged reference; and black: transverse bipolar). The unipolar reference has the largest genuine PSS over all frequency bands. By contrast, the frequency band mainly influences spurious PSS. Lower frequency bands have larger spurious PSS over all types of EEG references. The error bars denote the standard deviations over the EEG data sets for 27 data lengths (from 5 to 60 seconds, with 2 second intervals). Six frequency bands were studied: delta band (0.5–4 Hz), theta band (4–8 Hz), alpha band (8–13 Hz), beta band (13–25 Hz), gamma band (25–55 Hz) and whole band (0.5–55 Hz).
In contrast to data length, the EEG reference had a significant effect on the genuine and spurious PSS (
The frequency band demonstrated the most significant effect on spurious PSS. The lower frequency bands (delta, theta and alpha) had a relatively larger spurious PSS, compared to the higher frequency bands (beta, gamma). In general, the unipolar reference produces relatively large coherence artificially.
In this study, there was a biased electrode distribution (primarily frontal and parietal regions), which could produce biased PSS with distant clustered electrodes. As such, it is difficult to distinguish the effect of volume conduction and the effect of biased electrode distribution on the PSS indices in this data. In the analysis of anesthesia EEG, we assumed that the volume conduction was preserved across states and therefore focused on the increase or decrease of PSS indices relative to those of the baseline state.
The three PSS indices were applied to EEG data recorded during baseline conscious, unconscious (loss of consciousness, LOC) and recovery states (return of consciousness, ROC). Each state consists of a 5 minutelong EEG epoch.
Delta  Theta  Alpha  Beta  Gamma  Whole  



















B1/B2  
B1/ 















B1/A2 







B1/R1 







B1/R2 






B2/ 














B2/A2 







B2/R1 





B2/R2 




































A2/R1  
A2/R2 
p<0.0001,
p<0.01,
p<0.05.
The mean genuine, spurious and total PSS are denoted by different colors (red: genuine PSS; green: spurious PSS; and blue: total PSS). The error bar denotes the standard deviations of genuine, spurious and total PSS values over all EEG datasets. The vertical dotted lines indicate the loss of consciousness and recovery of consciousness, sequentially. Six frequency bands were studied: delta band (0.5–4 Hz), theta band (4–8 Hz), alpha band (8–13 Hz), beta band (13–25 Hz), gamma band (25–55 Hz) and whole band (0.5–55 Hz).
Repeated measures oneway ANOVA with Tukey’s multiple comparison test was applied to the six substates for each frequency band in the 20 EEG datasets (
Regarding the temporal evolution pattern, the genuine PSS was decreased after LOC in most frequency bands except the delta band. The decreased genuine PSS level of the alpha band was maintained until the end of recording, while the genuine PSS for the theta, gamma and whole frequency bands recovered in a short time. The alpha and gamma bands showed relatively large decreases of genuine PSS.
Regarding the spurious PSS, the delta, beta and whole frequency bands showed an increase after the induction of anesthesia. Conversely, the alpha band showed a large decrease after LOC. The total PSS of the delta and the whole frequency bands increased after LOC, while it significantly decreased for the other bands. The total PSS of the whole frequency band changed in the opposite direction of the genuine PSS after LOC.
The major findings of this study are (1) our measure of genuine PSS reflects true phase synchronization, as demonstrated by the NTori model, (2) spurious PSS significantly increases during general anesthesia, (3) the most influential factor for genuine PSS is the EEG reference while the most influential factor for spurious PSS is the low frequency spectra, and (4) there were individual temporal patterns for genuine and spurious PSS in each frequency band during general anesthesia.
The simulated model data clearly demonstrates the potential problem with conventional mean phase coherence by revealing spurious phase coherence for a time series with periods of zero true coherence. The problem is exacerbated when simulated frequencies are decreased and resolves as frequency increases. Significant spurious PSS began to appear from the mean frequency below about 6 Hz, which corresponds to the theta and delta rhythms of EEG. Consequently, our data suggest that, for an EEG that has a dominant frequency below the theta rhythm, spurious component measures should be taken into account.
The analysis of randomized EEG data acquired during anesthesia demonstrated the effect of spectral content on the three PSS indices (
We investigated which factor among the data length, type of EEG reference and frequency bands amplifies the finite size effect. Genuine PSS depends on the type of EEG reference and the frequency band of EEG. The unipolar EEG reference (A2) had the largest genuine PSS compared to those of the other EEG references. This may be due to the common contribution of a reference’s fluctuation to all of the EEG channels. For genuine PSS, there was no tendency of linear increases or decreases over different frequency bands. For spurious PSS, the frequency band was the most influential factor in comparison with the changes induced by the EEG reference and window size. Lower frequency bands produced more spurious phase synchronization in EEG (in
The genuine, spurious and total PSS evolved with typical temporal patterns corresponding to the states of consciousness and different frequency bands during general anesthesia (
The spurious PSS also showed diverse temporal evolution patterns depending on the state of consciousness and the frequency band. After LOC, the spurious PSS was significantly increased in the delta, beta and whole frequency bands. Conversely, the spurious PSS of the alpha band EEG significantly decreased in unconscious state. The spurious PSS had a distinct temporal evolution pattern corresponding to each frequency band: increased and returned (delta and whole), unchanged (theta and gamma), decreased and then unchanged (alpha), gradually increased (beta).
This study has a number of limitations. The genuine PSS may not control for all possible factors that can generate spurious elements. In particular, the spurious PSS mainly focuses on the random phase synchronization components generated by spectral content. It does not, however, consider the volume conduction effect at all. We therefore use the term “genuine” with respect to PSS based on previously published terminology and acknowledge that it may not reflect true PSS with perfect accuracy. Furthermore, in order to normalize the total, spurious and genuine coherence components, the three PSS indices were divided with different denominators. Thus, the PSS indices represent relative rather than absolute genuine or spurious coherence for a given EEG dataset. The total PSS is therefore not exactly the sum of genuine and spurious PSS. Another limitation is that the Ntori model cannot represent the hierarchical functional structure of the brain; thus, a more sophisticated model is needed for realistic anesthesia EEG simulation
In conclusion, our proposed measures of genuine and spurious PSS can clarify the interpretations of phase synchronization structure for a given EEG dataset. Genuine and spurious PSS are associated with complex temporal evolution patterns depending on the state of consciousness. Because of significant spurious phase synchronization, simultaneous monitoring of genuine and spurious PSS is necessary. This approach may also be beneficial in elucidating true functional connectivity based on coherence measures for nonstationary physiological data in which very low frequency spectra dominate.
The Institutional Review Board (IRB) of Asan Medical Center approved this study in human volunteers. After IRB approval and written informed consent, ten normal human subjects were studied on two separate occasions with 21channel EEG. Three states were investigated: 1) baseline consciousness, 2) general anesthesia, defined as loss of response to a command after 2 mg/kg propofol, and 3) recovery, defined as return of responsiveness. The EEG data were originally gathered and analyzed using different methods for a study of the frontoparietal system; full details of anesthetic protocol can be found in
The EEGs of 21 channels (Fp1, Fp2, F3, F4, F5, F6, F7, F8, Fz, C3, C4, Cz, T7, T8, P3, P4, P5, P6, P7, P8, Pz referenced by A2, 10–20 system) were recorded on the bed with closed eyes, with a sampling frequency of 256 Hz and 16bits analogtodigital precision by WEEG32® (LXE3232RF, Laxtha Inc., Daejeon, Korea). Baseline EEG was recorded for 5 min before an intravenous bolus of propofol. EEG was recorded continuously during and after the intravenous bolus of propofol, and up to 10 min after ROC. For the band pass filtering we used the fourth order Butterworth filter to avoid a possible shifting of the signal phases (in Matlab Signal Processing Toolbox).
The genuine, spurious and total PSS were defined based on the decomposition of spatial synchronization structures of multichannel EEG data and the comparison with its randomized data set. The decomposition of a phase synchronization matrix
The phases
If all channel data are “completely independent,” all nondiagonal elements of
For any finite
If all
These properties were used to evaluate how much a given EEG data set deviates from completely independent or identical data (the property (1) and (3)). With the property (2), we can estimate the spurious PSS.
Randomized data sets that have the same spectral contents and amplitude distribution, but without the phase coherence between signals, were used for estimating the spurious PSS
Genuine PSS quantifies the phase synchronization that deviates from that of the randomized data set
Spurious PSS quantifies the spurious phase synchronization produced by finite size and certain spectral contents of the data. Spurious PSS is defined as the fraction of the difference between independent data and randomized EEG to the difference between independent data and completely correlated data,
Total PSS measures the amount of total phase synchronization strength (spurious and genuine PSS) contained in a given set of data. Total PSS is defined as the fraction of the difference between independent data and original EEG to the difference between independent data and completely correlated data,
Term  Definition 

Estimation of spurious elements produced by a specific spectral content in a finite data length in the phase synchronization for a given two EEG signals. 

Evaluation of how much the phase synchronization of a given EEG signal deviates from the estimated spurious PSS. 

Estimation of the amount of total phase synchronization (spurious and genuine PSS) between two EEG signals. 

These concepts are used in linear algebra, representing the properties of a matrix. In a diagonalized matrix, the eigenvalues are the numbers on the diagonal and the eigenvectors are the basis vector to which these numbers refer. This enables the analysis of a given matrix in a way similar to a diagonal matrix, which simplifies the process. In a diagonalized coherence matrix, we can more easily determine the principle coherence elements with large eigenvalues. 

A replica of a given EEG signal that retains the original spectral contents but with randomized phases. 

A given time series is replicated many times over in order to generate an enormous number of copies. The replica is allowed to differ microscopically in the time series, while retaining the same general properties (such as spectral content). Such a collection of replicated time series is called an ensemble, and the average is called an ensemble average. 
In the
The validity of the genuine and spurious PSS was tested with multivariate model data, in which the mean frequency and the phase coherence were modulated. We hypothesized that lower mean frequency produces higher spurious PSS and that genuine PSS correlates well with the given phase coherence. With the simulated EEG time series, we investigated the effects of mean frequency, phase coherence and their combination on the spurious, genuine and total PSS. Twenty time series were generated from Ntori
The amplitudes
A time series,

0–20  20–40  40–60  60–80  80–100  100–120  120–140 

U  U→N  N  N  N  N→U  U 

10  10  10  10→1  1  1  1 

0/↔  ⇑/↔  ↑/↔  ↑/⇑  ↑/↑  ⇓/↑  0/↑ 
The first period (0–20 s): the phases
The second period (20–40 s): the distribution of phases was transformed from the uniform distribution to the normal distribution with mean of
The third period (40–60 s): the same normal distribution of phases was given. Therefore, the phase coherence is maintained at the same level. No changes were expected in genuine and spurious PSS.
The fourth period (60–80 s): Using the normal distribution of phases, the mean frequency
The fifth period (80–100 s): the same normal distribution of phases and the mean frequency
The sixth period (100–120 s): the distribution of phases is transformed from the normal distribution to the uniform distribution, while
The seventh period (120–140 s): the uniform distribution of phases is maintained, while
The PSS indices were applied to anesthesia EEG data, which consisted of three different states: baseline conscious state, unconscious state (loss of consciousness, LOC) and recovery state (return of consciousness, ROC). Each state consisted of a 5 minutelong EEG epoch. The moving window method was used to investigate the temporal evolution patterns of the genuine, spurious and total PSS during general anesthesia. The window size of 10 seconds without overlap was used to achieve the appropriate time resolution for the fast state transitions during general anesthesia, which take place in a short term after LOC and before ROC. To obtain an ensemble of phase synchronization matrices in the
The effect of EEG reference on the genuine, spurious and total PSS indices was studied with four different types of EEG references (unipolar, common average, longitudinal bipolar and transverse bipolar). For the unipolar EEG reference, the A2 channel was used as the referential electrode; for the common average EEG reference, the averaged EEG over 21 EEG channels at each time
The moving window analysis was not wellsuited to a repeated measures oneway ANOVA because of the many repeated measurements (∼ 30 windows). Thus, to apply the conventional ANOVA and posthoc analysis, each state (consciousness, anesthesia, recovery) was separated into two substates. The