EEG, Temporal Correlations, and Avalanches

INVITED REVIEW

EEG, Temporal Correlations, and Avalanches

Marc Benayoun,* Michael Kohrman,* Jack Cowan,† and Wim van Drongelen*

Abstract: Epileptiform activity in the EEG is frequently characterized byrhythmic, correlated patterns or synchronized bursts. Long-range temporalcorrelations (LRTC) are described by power law scaling of the autocorrela-tion function and have been observed in scalp and intracranial EEG record-ings. Synchronous large-amplitude bursts (also called neuronal avalanches)have been observed in local field potentials both in vitro and in vivo. Thisarticle explores the presence of neuronal avalanches in scalp and intracranialEEG in the context of LRTC. Results indicate that both scalp and intracranialEEG show LRTC, with larger scaling exponents in scalp recordings thanintracranial. A subset of analyzed recordings also show avalanche behavior,indicating that avalanches may be associated with LRTC. Artificial testsignals reveal a linear relationship between the scaling exponent measured bydetrended fluctuation analysis and the exponent of the avalanche sizedistribution. Analysis and evaluation of simulated data reveal that prepro-cessing of EEG (squaring the signal or applying a filter) affect the ability ofdetrended fluctuation analysis to reliably measure LRTC.

Key Words: Avalanches, Detrended fluctuation analysis, Power law, EEG.

(J Clin Neurophysiol 2010;27: 458–464)

Epileptiform activity is often characterized by large-amplitudedeflections in the EEG. Interictal spikes occur at irregular inter-

vals, whereas most seizure patterns show rhythmicity, and althoughthese phenomena are well described, the underlying mechanisms atthe scale of the single neuron or local networks are not wellunderstood. In general terms, large-amplitude fluctuations in theEEG are thought to arise from synchronized neuronal populations,whereas the rhythmic component is a form of temporal correlation.However, a better quantitative understanding of these neuronalprocesses across scales might help elucidate the general principlesgenerating typical EEG patterns. There is an increasing amount ofevidence that the electrical activity of the brain exhibits manyscale-invariant properties in the form of power laws (Monto et al.,2007; Parish et al., 2004; Worrell et al., 2002). These power lawshave been seen in local field potentials where they describe thedistribution of network burst sizes and in subdural and intracranialelectrode recordings where they describe the autocorrelation func-tion of these signals, indicating long-range temporal correlations(LRTC) (Beggs and Plenz, 2003; Monto et al., 2007; Parish et al.,2004). In addition, there is some evidence for a relationship betweenthese various power laws and epilepsy. A method called detrendedfluctuation analysis (DFA) has indicated the presence of LRTC in

EEG recordings of patients with epilepsy and has suggested anincrease in LRTC and a change in power law slope in recordingstaken near the seizure-onset zone (Monto et al., 2007; Parish et al.,2004). Although network bursts are certainly relevant for generatingepileptiform activity such as interictal spikes, the existence of orchanges in the power law slope of network bursts (also calledneuronal avalanches) have not been explored in the context ofepilepsy. Furthermore, recordings of electrical brain activity inawake rhesus monkeys have shown the coexistence of LRTC andneuronal avalanches (Fig. 2B in Petermann et al., 2009), suggestingthat neuronal avalanches may be associated with LRTC.

This study was designed to elucidate whether large-amplitudefluctuations of EEG could be attributed to neuronal avalanches andto assess the relationship between the presence of a power law inavalanche size distribution and LRTC. We describe acquisition andprocessing of patient data, methods of DFA used to identify thepresence of LRTC, and the algorithm for determining avalanche sizedistributions. We relate the scaling exponents determined by DFA tothose determined by avalanche size distributions in both artificiallygenerated signals and EEG data. Finally, in the Appendix, wepresent a mathematical framework to relate power law slopes de-tected with DFA and in the autocorrelation function and powerspectrum and show that these are different when calculated for theenergy and the raw signal.

METHODS

Subjects and Data AcquisitionFour patients aged 1 to 23 years (1 male) with pharmacolog-

ically intractable epilepsy underwent long-term monitoring (BMSI6000 unit; Cardinal Health, Dublin, OH) for surgical evaluation(Comer Hospital, University of Chicago, Chicago, IL). Both sub-dural electrode grids (Radionics Medical Products Inc., Burlington,MA) and video-EEG (10–10 placement, American Electroencepha-lographic Society, 1994) were used to record from patients for theirentire stay at the hospital (from 5 to 10 days). The EEG is recordedwith an amplification of 6,000 and a bandwidth of 0.5 to 100 Hz.The data were digitized at 400 samples/channel/s with a 12-bit A/Dconverter. All channels use a common scalp reference (CPz). Thisstudy adheres to the guiding principles of the Institutional ReviewBoard Committee at the University of Chicago.

Visual PreprocessingSubdural and scalp EEG recordings were carefully reviewed

by the clinical neurophysiologist (M.K.) to identify regions ofseizure onset. Occasionally, interictal spikes could be seen in theEEG segments analyzed; however, these were rare and did notcontribute significantly to the results obtained. Continuous segmentsof interictal EEG of 1 hour in duration were chosen for analysis.Interictal segments were defined as continuous EEG with no sei-zures detected during the segment and no seizures detected withinthe hour before and after the segment. In addition, segments werechosen to minimize the presence of artifacts. It should also be notedthat both analytical methods (DFA and avalanche size distribution)

From the Departments of *Pediatrics and †Mathematics, The University ofChicago, Chicago, Illinois, U.S.A.

Supported by the Dr. Ralph and Marian Falk Medical Research Trust, the FrankFamily Fellowship Fund, the MSTP, and the Lynn Family.

Presented at Tools for Epilepsy Research: Tutorials and Updates, Chicago, IL,August 6–8, 2009.

Address correspondence and reprint requests to Wim van Drongelen, Departmentof Pediatrics, The University of Chicago, KCBD Room 4124, 900 E. 57th St.,Chicago, IL 60637-1470, U.S.A.; e-mail: [email protected].

Copyright © 2010 by the American Clinical Neurophysiology SocietyISSN: 0736-0258/10/2706-0458

Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010458

are relatively insensitive to artifacts so long as they represent a smallproportion of the overall fluctuations in the recordings. Ictal seg-ments were not examined because these segments are often too shortto resolve avalanches and DFA exponents accurately.

Detrended Fluctuation AnalysisThe DFA was performed on the energy of interictal EEG [as

has been done previously by Parish et al. (2004)] and artificialsignals with known scaling exponents created with the FourierFiltering Method (see Fig. 1 and Fourier Filtering Method section).The DFA algorithm begins by baseline correcting and integratingsignal Y(i) to produce the signal (Fig. 1B).

X�k� � �i�1

k

�Y�i� � �Y��. (1)

Next, the integrated signal X is divided into windows oflength �j producing N � ttotal/�j windows where ttotal is the fullduration of the signal (1 hour), and the local linear trend in eachwindow (calculated from a least squares fit to the data in eachwindow) is removed from X to produce a detrended signal Xdet (seeFig. 1C). The root mean squared fluctuation of Xdet at this scalingwindow �j is then calculated as

F��j� � �1

N�i�1

N

Xdet2 �i�. (2)

Finally, the procedure above can be repeated at multiplewindow sizes � (here window sizes ranged from 1 to 480 seconds),and the scaling of F(�) with � can be determined (see Fig. 1D).Signals that exhibit finite LRTC will show power law scaling with

F��� � ��, (3)

which yields a linear relation in bilogarithmic coordinates with slope0.5 � � � 1. The case where � � 0.5 corresponds to a completelyuncorrelated signal (white noise), and the case where � � 1corresponds to LRTC with infinite range (Parish et al., 2004).

Determination of the value of � was made by performing aleast squares fit of a line to the bilogarithmic data, and onlycorrelation coefficients with P � 0.01 are considered indicative of apower law.

Fourier Filtering MethodArtificial signals with characteristic scaling exponents and

LRTC were created using the Fourier Filtering Method (Peng et al.,1993). Briefly, the method works as follows: create a discrete whitenoise signal, �(t), such that the value of � at each time point is

FIGURE 1. A, 10 seconds of raw EEG (left col-umn) and 10 seconds of artificial FGN signal cre-ated with � � 0.95 (right column). Note that theartifact present in the raw EEG around 3 seconds(arrow) has little effect on the power law scalingresults of DFA (see panel D). B, Energy of signals.C, Integrated energy with linear trends using a5-second window. D, Fluctuations of detrendedintegrated signal as a function of window sizeranging from 1 to 480 seconds.

Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010 EEG, Temporal Correlations, and Avalanches

Copyright © 2010 by the American Clinical Neurophysiology Society 459

chosen independently from a standard normal distribution. There-fore, �(t) has the properties that

���t�� � 0 and ���t���t � ��� � ����. (4)

Next, we Fourier transform �(t) to obtain �̂� f � and create a new

sequence �̂� f � as

�̂� f � � �f�

2�̂� f �. (5)

Finally, we invert the transform to produce a signal in thetime domain, �(t). This signal has power spectrum

S� f � � ��̂� f ��2 � �f�. (6)

As a result, this signal also has detrended fluctuations withscaling

F��� � ��, (7)

and autocorrelation

C��� � ��� ��, (8)

with � � � 1

2and � 1 (see Appendix). This signal also

has the property that its amplitude histogram is zero mean andGaussian and is referred to as fractional Gaussian noise (FGN) orcolored Gaussian noise in the literature (Lowen and Teich, 2005;Mandelbrot, 1971). Note that if the values of �(t) are chosenindependently from some other distribution with zero mean, theabove filtering method still produces a signal �(t) with the samepower law autocorrelation, power spectrum, etc., but now theamplitude histogram is no longer Gaussian.

We usually examine the energy of the signal �2, which canbe shown to also have power law detrended fluctuations, powerspectra, and autocorrelations but with different values of � (� �2� 1), ( � 2 1), and ( � 2), respectively, andamplitude histogram corresponding to a �2 distribution with 1 df(see Appendix). These signals are referred to as fractional �2

noise (F�N).

Defining AvalanchesWe compare the presence of LRTC in real EEG signals and

artificial signals to the production of avalanches as seen in local fieldpotential recordings. We define an avalanche in EEG data similarlyto its definition in local field potentials. First, we identify negativedeflections in the EEG channel, which are �2SD below the mean ofthe signal (see Fig. 2A). These negative large-amplitude fluctuations(nLAFs) are then clustered into avalanches as follows: a sequence ofnLAFs is called an avalanche (gray boxes in Fig. 2B) if no consec-utive nLAFs in the cluster are separated by a time greater than �t.The size of an avalanche is defined as the total number of nLAFsbelonging to the sequence. Clearly, if �t is small, then avalanchesizes will be small. Indeed, in the limiting case that �t is smaller thanthe minimum time interval between any two consecutive nLAFs,each nLAF becomes its own avalanche, so all avalanches have sizeunity. Similarly, if �t is chosen large, then avalanches will be large.Again consider a limiting case, where �t is on the order of the entireEEG segment being analyzed. Then, all the nLAFs belong to asingle avalanche. We estimate an appropriate �t as the average timeinterval, �tavg, between consecutive nLAFs. More precisely, let {ti}be the ordered sequence of nLAFs in the network, then

�tavg � �{ti � 1 � ti}�. (9)

FIGURE 2. A, 10 seconds of raw intracranial EEG(left column) and FGN signal (right column). “*”denotes identification of nLAF (see text). B, Con-secutive nLAFs are clustered into avalanches basedon average inter-nLAF interval �t (see text). C, Av-alanche size distribution of nLAF and best fits ac-cording to geometric (solid line) and power law(dashed line) distributions (see text).

M. Benayoun et al. Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010

Copyright © 2010 by the American Clinical Neurophysiology Society460

Avalanche Size DistributionsWe fit two distributions to the avalanche size. First, if the

nLAFs follow a Poisson process with rate � (i.e., nLAFs areuncorrelated), then the distribution of avalanche size S is

P�S � s� � P�s consecutive ISIs �t�P�next ISI � �t�� �1 � e� �t�se� �t

(10)

which is a geometric distribution with parameter e��t. This is thesolid line in Fig. 2C. Note this probability is a conditional proba-bility that given that there is an avalanche, it is of size s. This is whyEq. 10 does not include the probability that there is an interspikeinterval (ISI) greater than �t to mark the initiation of the avalanche.

It has been shown that avalanche size distributions in localfield potential recordings are consistent with a power law distribu-tion, with the size S given by

P�S � s� � s �� (11)

for some reasonably large range of s (Beggs and Plenz, 2003; Levinaet al., 2007; Mazzoni et al., 2007). Note that this means thedistribution is linear in bilogarithmic coordinates. The best fittingpower law distribution to the avalanche size data was obtained byusing a maximum likelihood estimator for the slope of a power lawprobability distribution for discrete data (avalanche sizes are integervalues only). According to the maximum likelihood estimator, theslope � is given by the following equation (Clauset et al., 2009):

� � 1 � n��i

n

lnxi

xmin �1

2�1

, (12)

where n is the number of avalanches greater than size xmin and xi isthe size of the ith avalanche. Unless stated otherwise, we takexmin � 3. This is the dashed line in Fig. 2C. Note that this is adifferent method for obtaining slope values than the more commonlinear regression analysis (LRA) of the bilogarithmically trans-formed data. LRA is based on the assumption that the noise in thedependent variable is independent for each value of the independentvariable and normally distributed. Although this is true when thedependent variable is the probability of a certain size avalanche, itdoes not hold after the bilogarithmic transformation. The trans-formed probability distribution has log-normally distributed noise,and so a calculation of the slope from linear regression analysismethods can give spurious results (Clauset et al., 2009).

RESULTS

Artificial SignalsFGN signals with strong LRTC (scaling exponents 0.75 � � �

1 created by the Fourier Filtering Method) were simulated and DFAapplied to determine the scaling constants. Results reveal the abilityof DFA to accurately assess the scaling exponents of these signals.However, DFA fails to accurately determine the scaling exponentsof the energy of these signals, which should have scaling exponent� � 2� 1 (Figs. 3A and 3B and Appendix). DFA underestimatedthe scaling exponents of the energy of the signal for all test signalswith � � 0.85 but overestimated the scaling exponent for signalswith � � 0.8.

The relationship between neuronal avalanches and LRTC isshown in Figs. 3C and 3D. As expected, uncorrelated signals with� � 0.5 have avalanche cluster distributions following a geometricdistribution (solid line in Fig. 3C); however, signals with LRTCproduce power law avalanche size distributions (dashed line in Fig.

3C). For signals with strong LRTC (0.75 � � � 1), there is a linearrelationship between the scaling exponent � and the slope of thepower law for avalanche size distribution � for both FGN and F�Nsignals (Fig. 3D). A linear fit of the data empirically determines therelationship between these exponents for FGN and F�N signals,which is given by

� � 4.22 � 2.37� (FGN) (13)

� � 3.87 � 1.98� (F�N). (14)

The similarity of these equations over the range of � studied(see Fig. 3D) suggests a robustness of the scaling relationshipbetween DFA and neuronal avalanches with respect to the amplitudehistogram of the test signals. This finding also suggests that inaddition to the linear relationships between the scaling exponent ofDFA and those of the autocorrelation function and power spectrum(see Appendix), there is also a relationship between the avalancheexponent and the other exponents.

EEG SignalsPower law scaling of the detrended fluctuations of the energy

was observed in 510 of 512 analyzed recordings and ranged overtime scales from 1 to 480 seconds (consisting both scalp andintracranial EEG) with � � 0.76 0.13 (mean SD) (510 channelsfrom four patients), which is in agreement with previously measuredvalues (Parish et al., 2004). Significant differences in scaling expo-nents were observed between scalp and intracranial EEG recordingsin three patients (unpaired t test, P � 0.001). In these patients,

FIGURE 3. A, Comparison of the DFA measured scalingexponent against a target value. B, The scaling exponent �of the energy of the signal. The dashed line indicates thebest fit line to the data, whereas the solid line indicates theexpected value based on theory (see Appendix). C, Ava-lanche size distribution for two FGN signals with known scal-ing exponents. � represents the uncorrelated signal (� �0.5) and � represents a signal with LRTC (� � 0.9). Thesolid line is a geometric distribution, and the dashed line isthe best-fit power law with � � 2.04 (see Avalanche SizeDistributions section). D, Scatter plot of the avalanche sizepower law exponent � against the DFA scaling exponent �for FGN and F�N signals. Linear fits show that for both FGNand F�N signals, there is a linear relationship between theavalanche exponent and DFA scaling exponent.

Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010 EEG, Temporal Correlations, and Avalanches

Copyright © 2010 by the American Clinical Neurophysiology Society 461

scaling exponents were larger for scalp recordings than intracranial(see Table 1).

Despite this clear presence of power law scaling as measuredby DFA, avalanche size distributions were difficult to resolve inEEG data as a result of the low detection rate of avalanches in therecordings. The reduced rate of avalanche detection in these signalsoccurs because the amplitude histogram of typical EEG data showsa much lower concentration of amplitude values above the 2SD cutoff for defining nLAFs. With so few nLAFs to cluster, it wasdifficult to resolve the size distribution; however, some signals (29of 512 recordings analyzed) clearly produced avalanche size distri-butions resembling a power law as determined by the methods ofBeggs and Plenz (2003). Estimates of � from these signals showed� to be between 2 and 3, which corroborates results found inartificial signals, whereas estimates of � from all recordings (512total) show a range from 1.42 to 3.45, suggesting that many of theunresolved size distributions may exhibit power law scaling withinthe range of 1.8 to 2.5 as determined by artificial signals.

DISCUSSIONIn this article, we examined scalp and intracranial EEG

recordings for the presence of LRTC and neuronal avalanches. Weshowed that in four patients, scalp and intracranial EEG energyexhibit LRTC ranging from seconds to minutes with scaling expo-nents � ranging from 0.55 to 1.29 (recall that � characterizes apositive slope), similar to previously reported results (between 0.58and 1.0) (Parish et al., 2004). Within each patient, there was anapparent difference in scaling exponents between intracranial andscalp EEG, with scalp EEG exhibiting larger scaling exponentsindicative of increased LRTC (statistically significant in three offour patients). Perhaps, this is because scalp EEG electrodes samplefrom a larger network as compared with intracranial ones. Thesimultaneous identification of neuronal avalanches in scalp andintracranial EEG recordings was impeded by the infrequent occur-rence of nLAFs, which prevented quantitative analysis of avalanchesize distribution in most recordings. In 29 of 510 recordings,avalanche size distributions following a power law were observedwith exponent � ranging from 2 � � � 3 (recall that � is defined asthe magnitude of the negative slope, see Eq. 11). This low fractionof recordings showing sufficient nLAFs is not surprising becauseour signals represent compound activity of huge networks, and

network fluctuations are attenuated by a factor of 1/�N (N is thenetwork size) (Benayoun et al., 2010).

To further explore the possible relationship between LRTCand neuronal avalanches, artificial signals with known scaling ex-ponents � were created and avalanche size distributions determined.Results with these signals clearly indicated the presence of a powerlaw size distribution of avalanches with � linearly related to � forsignals with strong LRTC. Results indicate that for � between 0.75and 1, � should lie between 2 and 3, as seen in 85% of the EEG

recordings where a power law in avalanche size was detectable.Empirical determination of this scaling relation is robust to theamplitude distribution of the artificial signal (both for Gaussian and�2 distributions). Signals with amplitude distributions that are con-centrated close to the mean will have fewer nLAFs than signalswhose amplitude distribution spreads out beyond 2SDs, making thedetection of avalanches more difficult. However, determination ofthe scaling exponent is sufficient to predict the exponent of theavalanche size distribution in these artificial signals and can becalculated with shorter time segments than are needed to sufficientlydetermine the slope of the avalanche size distribution directly.Therefore, we hypothesize that despite our inability to resolveavalanches in the EEG with quantitative precision in most cases, theempirical relationship determined for the test signals and the obser-vation of power law avalanche size distribution in a subset ofrecordings suggest the presence of avalanches in EEG, with expo-nent � linearly related to the scaling exponent �. For both types ofartificial signals examined (FGN and F�N), increased LRTC (in-creased �) decreases the exponent �. Because previous work hassuggested that seizure-onset zones exhibit increased LRTC, wewould expect that seizure zones would also show avalanche behav-ior with decreased �, indicating an increased probability of largeravalanches and reiterating the increase in correlated activity. If wesuppose that the ictal period is a further exaggeration of this effect,then this period would be marked by EEG showing a much largerfrequency of occurrence of large-amplitude fluctuations within ashort time span (high probability of large avalanches) as is oftenobserved. In a more general sense, the presence of a power law inavalanche size distribution may indicate that neuronal avalanches, aform of synchrony in neuronal networks (Beggs and Plenz, 2003),may play a key role in the generation of large-amplitude fluctuationsin scalp and subdural EEG, which may be particularly relevant forthe formation of interictal spikes and bursts.

Finally, the inability of DFA to accurately determine thescaling of the energy of the test signals has implications for com-paring results obtained in different studies. It is very common topreprocess EEG before obtaining the real signal of interest forfurther study. The most common forms of preprocessing are (1)filtering of the EEG to remove frequencies above or below a certainthreshold or to remove artifacts such as 60 Hz environmental noiseand (2) analyzing nonlinear transforms of the raw EEG such as thesquare of the signal (i.e., the energy). DFA has been applied underdifferent preprocessing conditions, e.g., narrow band filtering (ornot) and squaring of the signal, versus using wavelet methods toisolate the amplitude envelope of the signal (without squaring)(Linkenkaer-Hansen et al., 2001; Parish et al., 2004). Filtering of thesignal alters the power spectrum, in particular the scaling exponent (Appendix), which then influences the scaling exponent measuredby DFA, which agrees with the previous findings (Valencia et al.,2008). Previous studies to examine the effects of filtering andnonlinear transformations of test signals (including squaring) on thescaling exponent � determined by DFA have been carried out beforebut under the false assumption that squaring the signal should haveno influence on the expected scaling exponent, which is incorrect(Chen et al., 2005) (see Appendix).

APPENDIXThis appendix calculates analytically the relationships between

the power law exponents for DFA, the autocorrelation function, and thepower spectrum of signals exhibiting LRTC. In addition, it shows theeffect on these exponents resulting from squaring the signal to examinescaling in the energy of the signal.

Suppose we have a signal �(t) that has amplitudes distributedaccording to a standard normal distribution and a power spectrum

TABLE 1. Summary Statistics of Scaling Exponent in Scalpand Intracranial EEG

Patient � Scalp EEG � Intracranial EEG

1* 1.10 0.09 0.69 0.13

2 0.66 0.04 0.72 0.07

3* 0.74 0.10 0.72 0.07

4* 0.89 0.07 0.82 0.10

*Statistically significant difference (P � 0.001, unpaired t test) between scalp andintracranial scaling exponent.

M. Benayoun et al. Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010

Copyright © 2010 by the American Clinical Neurophysiology Society462

given by S���� � ���, where � is radial frequency and is aconstant. In the literature, such a signal is referred to as colorednoise or FGN (Lowen and Teich, 2005; Mandelbrot, 1971). A signal�(t) with the same power spectrum but with amplitude distributiondrawn from some other distribution, a �2 distribution for example,would be referred to as F�N. The autocorrelation function of eithersuch signals C�(�) is given by the inverse Fourier transform of thepower spectrum

C���� �1

2���

�

S����ei�� d� (A1)

�1

2���

�

��� � ei�� d� (A2)

�1

2���

�

��� � �cos���� � i sin����� d� (A3)

�1

2���

�

��� � cos���� d� (A4)

�1

��0

�

� �cos���� d�. (A5)

From Eq. A3 to Eq. A4, we have taken advantage of the factthat ��� is even, so the integral involving the odd sin functionvanishes. If we let u � ��, then du � �d�, we get

C���� �1

�� � �1��

0

�

ucos�u� du (A6)

and we can see that

C���� � � � �1 � �, (A7)

whenever the integral converges. It can be shown that this integral

is equal to ��1 � � cos��

2�1 � ��, where � is the (complete)

gamma function (Arfken and Weber, 1995) and therefore convergesfor � (0, 1).

Next, we turn to the DFA method. Let x�t� � �0

t��t� dt,

which is the continuous-time equivalent to Eq. 1, then

�x�t�2� � �0

t

�0

t

���t���t��� dtdt�, (A8)

but by defining �̂��� as the Fourier transform of �(t), we can statethat

���t���t��� �1

�2��2��

�

��

�

��̂����̂*����ei��t�t��d�d�

(A9)

�1

�2��2��

�

��

�

��� � [��� � ��

� ��� � ��]ei��t � �t��d�d� (A10)

�1

�2��2��

�

���[ei��t � t�� � ei��t � t��]d�,

�t � t�� is part of the exponent [as in original] (A11)

where the last equation comes from integrating over d� and *denotes the complex conjugate. Therefore,

�x�t�2� �1

�2��2�0

t

�0

t

��

�

���{ei��t�t�� � ei��tt��}d�dtdt� (A12)

�1

�2��2��

�

���{�0

t

ei�tdt�0

t

�ei�t� � e � i�t��dt�}d�

(A13)

�1

�2��2��

�

���{�0

t

ei�tdt�0

t

�2cos�t��dt�}d� (A14)

Taking advantage of the fact that

�0

t

e i�tdt � �e i�t

i�and �

0

t

2 cos �t�dt� �2 sin �t

�,

we can show that

�x�t�2� �1

2�2 ��

�

��� � {ei�t

i�

sin��t�

� }d�

(A15)

�1

2�2 ��

�

��� �sin2��t�

�2d� (A16)

�1

�2�0

�

�� � � 2��1 � cos�2�t�� d� (A17)

If we let u � t�, then du � td�, and

�x�t�2� �1

�2t�1�

0

�

u��2��1 � cos�2u��du. (A18)

Although this integral diverges for arbitrarily large u, wenever consider arbitrarily large time windows, so the integral con-verges for any finite time interval �. Finally, note that

F�t� � ��x�t�2� � t � 1

2 , (A19)

so

F��� � �� (A20)

Journal of Clinical Neurophysiology • Volume 27, Number 6, December 2010 EEG, Temporal Correlations, and Avalanches

Copyright © 2010 by the American Clinical Neurophysiology Society 463

where � � � 1

2and � is the length of a time window.

Next, we explore the autocorrelation of the energy of a FGNsignal, �2, and assume that � has DFA exponent between 0.75 and 1.Because the amplitude distribution of �(t) is Gaussian, the energyhas amplitudes distributed according to a �2 distribution with 1 dfwith mean 1 and variance 2 (assuming � is standard normallydistributed) (Arfken and Weber, 1995). By definition (Van Kampen,2007), the autocorrelation of �2 is

���t�2��t � ��2� � 2����t���t � ����2 � ����t�2��2 (A21)

� 2C����2 � ����t�2��2 (A22)

� K�� 2�1 � � � 1, (A23)

and K �2

�2�2�1 � � cos2��

2�1 � ��.

From Eq. A21 to Eq. A22, we have used Wick’s theorem toreduce the fourth moment of the Gaussian signal � to products oflower moments (Zinn-Justin, 2002). Therefore, the energy has anautocorrelation described by a power law with exponent

� 2�1 � � � 2. (A24)

Similarly, the exponent of the power spectrum of the energyis given in terms of the power spectrum of the original signal by

� 1 � � 1 � 2 � 2 � 1, (A25)

and the DFA exponent is given by

� � 2� � 1. (A26)

Therefore, we do not expect the DFA exponent of the EEGsignal to be the same as the DFA exponent calculated from theenergy. In fact, because � � 1, the DFA exponent of the energy willalways be less than the DFA exponent of the original signal as hasbeen observed (Chen et al., 2005).

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