P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

Random Matrix Analysis of Human EEG Data

P. SebaDepartment of Physics, University of Hradec Kralove, Vıta Nejedleho 573, Hradec Kralove, Czech Republic

and Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, CZ - 162 53 Prague, Czech Republic(Received 26 February 2003; published 7 November 2003)

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We use random matrix theory to demonstrate the existence of generic and subject-independentfeatures of the ensemble of correlation matrices extracted from human EEG data. In particular, thespectral density as well as the level spacings was analyzed and shown to be generic and subjectindependent. We also investigate number variance distributions. In this case we show that when themeasured subject is visually stimulated the number variance displays deviations from the randommatrix prediction.

DOI: 10.1103/PhysRevLett.91.198104 PACS numbers: 87.19.Nn, 05.45.Tp

It is known that random matrix theory describes cor-rectly the spectral statistics of certain complex systems. It

elaborated concept of mutual information to extract thecross-channel correlations.

The analysis of the EEG signal has a long history.Being used as a diagnostic tool for nearly 70 years, itstill resists a strict and objective analysis and its inter-pretation remains to be mostly intuitive and heuristic. Inparticular, the cross-channel relations of the signal andits reference to the sources and hence to certain morpho-logical and/or functional brain structures are not under-stood sufficiently and are a matter of intense research [1].

Many methods have been developed with the aim ofhelping to understand the meaning of the EEG signal.They range from a visual inspection of the record by anexperienced physician to sophisticated estimations whichattempt to describe the signal using phase space methodsor to model its sources by a set of oscillating electricdipoles [2]. However none of these methods can be re-garded as fully satisfactory.

The reason is that the EEG signal—an electric activityof the brain measured by electrodes (channels) placed onthe scalp —is a superposition of electric signals which areproduced by a synchronous activity of numerous neurons.The spatial propagation of the electric signal in the com-plex brain tissue is far from being straightforward. More-over, various groups of neurons participate in various andsometimes independent tasks. All this together makes theresulting signal complex and difficult to interpret. It has tobe stressed that the morphological and functional proper-ties of the brain are not always the same. On the contrary,they can be different for different individuals. This makesthe EEG signal dependent on the measured subject. Evenif different persons perform the same tasks and the mea-surement is done under identical conditions the resultingsignals could differ.

The aim of this Letter is to demonstrate the existenceof features of the EEG signal which are universal, i.e., donot depend on the subject being measured. To this purposewe will investigate statistical properties of the cross-channel correlations of the EEG data and compare themwith the predictions of the random matrix theory.

0031-9007=03=91(19)=198104(4)$20.00

is useful, in particular, when one deals with systemswhich are chaotic but display at the same time certainwave (or coherent) properties. A typical example is quan-tum chaotic systems—see [3] for a review.

An EEG signal is a multivariate time series. Beingmeasured on the scalp, the activity of a particular cerebralarea influences the results of several EEG channels.Hence the activity of a given cortex area leads to corre-lations between the signal measured on different elec-trodes. The object to be analyzed here is the simplest onedescribing this correlation, namely, the correlation ma-trix Cl;;m�T� of the signal:

Cl;m�T� �XN2

j�N1

xl�tj�xm�tj�: (1)

Here xl�tj� denotes the EEG signal measured in thechannel l at time tj and the sum runs over j � N1, N1 �1; . . . ; N2 such that tj 2 �T; T � ��. The length � of thetime interval used to evaluate the correlation matrix willbe set to 150 ms here. We assume that during this timeinterval the activity of the electric sources in the brain canbe considered as constant. The obtained signal was notfiltered with a single exception, namely, a notched filterserving to remove the 50 Hz noise induced by the electricsupply. The signal xl�tj� used to evaluate (1) was addi-tionally preprocessed so that its mean is set to zero andthe variance equal to 1,

XN2

j�N1

xl�tj� � 0;XN2

j�N1

xl�tj�2 � 1: (2)

The importance of correlations for the signal analysiswas recognized a few years ago by Kwapien and collab-orators when they investigated the MEG (magnetoence-falography) signal of a brain auditory response [4]. Inaddition to the correlation matrix they used also a more

2003 The American Physical Society 198104-1

100 101

100.1

100.2

100.3

ξ

ρ(ξ)

FIG. 1. The eigenvalue density is plotted for data obtained bymeasuring three different people under the same conditions.The behavior for large � is similar in all three cases and showsa clear algebraic behavior (a straight line in the log-log plot).

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

Our aim is to analyze the EEG correlations using toolsof the random matrix theory. To this aim it is necessary todefine a matrix ensemble based on the EEG data. Here weuse the fact that the brain is nonstationary. This meansthat the tasks it is performing change with the time.Hence the structure of the correlation matrix (1) changes.A typical EEG measurement lasts about 15–20 min. Thistime interval can me divided into roughly 7000 not over-lapping stationary windows over which the correlationmatrix (1) is evaluated. In such a way we get a set of 7000correlation matrices. These matrices will represent theensemble we will work with.

Because of the individuality of the EEG signal and alsodue to the problems related to the technique of attachingthe electrodes to the scalp we cannot mix together dataobtained during different measurement sessions. Thismeans that even for one person we get different measure-ments for different matrix ensembles. There is one moredrawback which is related to the above definition of the150 ms wide stationary window: to get enough samplingpoints inside a particular window we have to use mea-surements performed with a high sampling rate (we willuse data obtained with the sampling rate of 1012 Hz).

Random matrix theory deals with statistical propertiesof the eigenvalues �n of the correlation matrix C,

Cj�ni � �nj�ni; (3)

with j�ni being the corresponding eigenvectors. The sim-plest property of the eigenvalues family related to amatrix ensemble is the eigenvalue density ����. It countsthe mean number of eigenvalues contained in an interval�a; b�, i.e.,

f�n; �n 2 �a; b�g �Z b

a����d�: (4)

Assuming that the time series xl�tj� is random andGaussian for all l, i.e., that for each l the values xl�tj�are random, normally distributed with the variance equalto 1 and not correlated, we get an ensemble of randommatrices which was mathematically studied by Mar-chenko and Pastur [5,6]. In particular, the density ����of this ensemble is known and given by the formula [7]

���� �

����������������������������������������������max ���� �min�

p2�Q�

; (5)

where Q � M=N with M being the number of channels, Nis the number of the sampling points tj, N > M, and�max � 1�Q� 2

����Q

p; �min � 1�Q 2

����Q

p.

The spectral density is known to be dependent on theunderlying data. Since EEG is not a random signal, but onthe contrary, it is synchronized and correlated by thecorresponding brain activity, we might expect that thespectral density will not follow the Marchenko-Pasturprediction and will be subject dependent. This is, to oursurprise, not true at all. We have investigated the EEG

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records of 90 people and evaluated the correspondinglevel density function. The result is plotted in Fig. 1 (tokeep the plot simple we displayed densities evaluated forthree different subjects as a typical example).

For small eigenvalues the spectral density indeed de-pends on the measured subject. It displays nevertheless aprofound and subject-independent algebraic tail for large�. This is a surprise since one would expect just the op-posite behavior. Specifically, one might expect that smalleigenvalues of the correlation matrix feel the influence ofthe system noise. The spectral density should thereforedisplay universal behavior for small eigenvalues similarto that of the Marchenko-Pastur ensemble. The largeeigenvalues on the other hand contain the informationabout significant correlations and hence about processesin the brain. Consequently, one would expect a nonuni-versal and subject-dependent behavior.

Instead of being described by the Marchenko-Pasturensemble the spectral density obtained with the help ofthe EEG signal seems to fit better into the predictionsvalid for Wishard matrices and chiral ensembles [3]. Forthese ensembles, which are related to the covariancematrices of the signal, the spectral density has an alge-braic singularity at the origin, just like the density de-rived from the EEG data.

The universal behavior of the spectral density for large� points to the fact that the seemingly individual brainactivity contains some common level of synchronizationwhich is reflected in the generic large tail behavior of thespectral density. This fact remains valid even when thesubjects are stimulated (we tried acoustic and opticalstimulations). The algebraic character of the tail is alsoof importance. A similar behavior (universal and alge-braiclike tail of the spectral density) has been observed

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0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

s

P(s

)

FIG. 2. A typical level spacing distribution obtained for aresting subject (eyes open, no stimulation) (crosses) and astimulated subject (squares). The result is compared with for-mula (8) (solid line).

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also for correlation matrices of systems with local fluc-tuations governed by Levy flights [8]. A typical exampledisplaying this type of behavior is a time series describ-ing the price fluctuations of individual stocks at an assetmarket. It has been argued by Burda and collaborators [9]that the existence of an algebraic tail in the spectraldensity is a sign for critical behavior of the underlyingsystem. Using MEG measurements, signs of critical be-havior of the brain have been found also in the powerspectra of the related signal which has a ‘‘1=f’’ typebehavior [10]. This was interpreted as an indication forself-organized critical behavior of the brain which isgenerated by a fractal (scale-invariant) avalanchelikemechanism.

In the principal-component method the eigenvaluesand eigenvectors of the correlation matrix are used toidentify the most important sources of the signal. Smalleigenvalues are usually neglected and the related corre-lations are regarded as unimportant. But the fact thatsome eigenvalue is small does not necessarily mean thatthe related signals are not important. The ‘‘smallness’’ ofthe eigenvalue can be a consequence of external influencewhich screens a part of the signal coming from a certaindomain. To investigate the spectral statistics of the corre-lation matrices more closely we have to first put all theeigenvalues on the same footing. This is done by unfold-ing the spectra, i.e., by a spectral mapping �n ! ~��n that‘‘unifies’’ the system-dependent spectral densities in sucha way that the resulting eigenvalue density of ~��n is con-stant,

~�� n � N��n� (6)

with N�E� being the spectral counting function

N�E� �Z E

0����d�: (7)

After the unfolding we get ��~��n� � 1.In the remaining part of this Letter we will work with

unfolded eigenvalues without mentioning them explicitlyand we will omit the tilde in the notation for the sake ofsimplicity. The random matrix theory predicts universalstatistical properties of the unfolded spectra provided theunderlying matrix ensemble is large enough to suffi-ciently fill the space of all matrices with a given symme-try. This generic behavior is observed, for instance, in thespectra of chaotic quantum systems. To demonstrate that asimilar universal property is observed also for correlationmatrices resulting from the EEG measurements we willfocus on two statistical distributions widely used in thefield of quantum chaos.

The simplest one, which takes into account only corre-lations between neighboring eigenvalues, deals with spac-ings sn � �n�1 �n (here we assume the eigenvalues �nto be ordered with respect to the magnitude, i.e., �n �n�1). The probability density of finding a given spacing s

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in the spectra is for random real symmetric matrices wellapproximated by the Wigner formula

P�s� ��2s exp

�

�4s2�: (8)

Another, more subtle distribution that takes into accountsimultaneous correlation between a group of L subsequenteigenvalues is the number variance �2 defined as

�2�L� � h�f�n < Lg L�2i (9)

(here f�n < Lg denotes the number of eigenvalues smallerthen L).

In this case the theory of random matrices predicts

�2�L� �2

�2

�log�2�L� � 1:5772

�2

8

�: (10)

These distributions are of universal validity. This hasbeen confirmed, for instance, for spectra of chaotic wavesystems that display this distribution independently oftheir physical character. A similar observation has beendone also for a time series which characterizes the evo-lution of atmospheric quantities [11] or the price fluctua-tions on an asset marker [12].

As we have already mentioned the EEG signal is sub-ject dependent. Nevertheless, the spectral statistical prop-erties of the correlation matrices are universal. Inparticular, the behavior of the level spacing distributionand number variance obtained from the EEG data ofhealthy persons are well described by Eqs. (8) and (10),respectively.

We have investigated the data of 90 people which weremeasured without an external stimulation. A part of thedata was obtained by a standard clinical EEG device with

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0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

L

Σ2 (L)

FIG. 3. A typical number variance obtained for a restingsubject (eyes open, no stimulation) (crosses) and a stimulatedsubject (squares). The result is compared with formula (10)(solid line).

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

19 channels and a part with the help of an experimentaldevice having 15 channels which was placed inside aFaraday trap to avoid external influence. Before the analy-sis the data were visually inspected and parts corruptedwith artifacts (eyes blinking, a motion of the subject, etc.)were removed. Finally, the resulting spectral data wereunfolded. As a result we obtained a convincing universalbehavior which is in accordance with the predictions ofthe random matrix theory.

Typical results for the level spacing and number vari-ance distributions are shown in Figs. 2 and 3. The datawere obtained under the circumstances that volunteerswere measured first under resting conditions (no stimula-tion) and then again when these persons were visuallystimulated by a periodically moving square. The stimu-lation is expected to change the correlation pattern of thedata due to its large response in the visual cortex. Never-theless, it turned out that these changes are too subtle toinfluence the level spacing distribution. In all cases wefind a very good agreement with the random matrixtheory prediction. The standard deviation between thevalues given by the Wigner formula and the results ob-tained from the EEG data is equal to 0.026 for the restingsubject and 0.027 for the stimulated case.

The number variance is, however, quite sensitive andchanges when the subject is visually stimulated. Therandom matrix prediction is in this case valid for non-stimulated data only, where the standard deviation be-tween the predicted and measured values is equal to 0.05.

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(The measurement with a stimulation leads to the stan-dard deviation as large as 0.6.) This can be easily under-stood. A visual stimulation changes the activity of thevisual cortex which proceeds the visual input. As a con-sequence, its relative correlations with the remainingparts of the cortex are lowered which leads finally tothe observed deviation of �2�L� for larger L.

This result is of interest since the number variance maychange not only due to an external stimulation but alsowhen the correlation ensemble is influenced, for instance,by some pathological process. It has to be stressed at thispoint, however, that number variance is also quite sensi-tive to artifacts. To obtain a result which is in agreementwith the random matrix theory, all artifacts have to becarefully removed. We mean by artifacts those changes inthe EEG signal that lead within the correlation window toa variance which is 2 times larger than the mean varianceevaluated over all correlation windows. Such windowswere rejected and were not used in further considerations.

This work was supported by the Grant No. 202/02/0088 of the Czech Grant Agency. A cooperation with theInstitute of Pathological Physiology in Hradec Kraloveand the neurological department in Rychnov nadKneznou is also gratefully acknowledged.

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[5] V. A. Marchenko and L. A. Pastur, Mat. Sb. 72, 507(1967).

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