Time-varying spectral analysis in neurophysiological time series using Hilbert wavelet pairs

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doi:10.1016/j.sig

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Signal Processing 85 (2005) 2065–2081

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Time-varying spectral analysis in neurophysiological timeseries using Hilbert wavelet pairs

Brandon Whitchera,�, Peter F. Craigmileb, Peter Brownc

aTranslational Medicine & Genetics, GlaxoSmithKline, Greenford Road, Greenford UB6 0HE, United KingdombDepartment of Statistics, The Ohio State University, 1958 Neil Avenue, Cockins Hall, Columbus, OH 43210-1247, United States

cSobell Department of Motor Neuroscience and Movement Disorders, Institute of Neurology, Queen Square,

London WCIN 3BG, United Kingdom

Available online 27 July 2005

Abstract

An analytic wavelet transform, based on Hilbert wavelet pairs, is applied to bivariate time-varying spectral estimation

for neurophysiological time series. Under the assumption of an underlying block stationary process, both single-trial

and ensemble studies are amenable to this method. A bootstrap procedure, which samples with replacement blocks

centered around the events of interest, is proposed to identify time points for which the event-averaged magnitude

squared coherence is non-zero. Clinical data sets are used to compare the wavelet-based technique with the classical

Fourier-based spectral measures and highlight its ability to detect time-varying coherence and phase properties.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Coherence; Electromyographic activity; Local field potential; Phase spectrum; Wavelet packet transform

1. Introduction

There is a growing belief that cognitive pro-cesses and behavioral events are reflected withinneural structures in terms of changes, not only inthe rate of neuronal firing, but also in thecooperative interplay among neurons within largeassemblies [1–3]. Much of this interplay seems to

e front matter r 2005 Elsevier B.V. All rights reserve

pro.2005.07.002

ng author. Tel.: +4420 8966 4511;

6 2757.

sses: [email protected]

[email protected] (P.F. Craigmile),

cl.ac.uk (P. Brown).

/www.image.ucar.edu/staff/whitcher/.

be underpinned by oscillatory synchronizationwithin neuronal populations. The frequency ofoscillatory synchronization varies under differentcircumstances, suggesting that activities of dissim-ilar frequency may relate to different physiologicalmechanisms [4,5]. Thus spectral measures ofsynchronization, like coherence and phase, holdan important position in human neurophysiologyas they provide insight into the extent andorganization of neuronal networks underlyingsynchronization.However, oscillatory synchronization is dyna-

mic, like the cognitive and behavioral events withwhich it is associated. Accordingly time-varying

d.

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B. Whitcher et al. / Signal Processing 85 (2005) 2065–20812066

spectral analysis techniques are an essential part ofthe neurophysiological repertoire. Most com-monly the discrete Fourier transform (DFT) isperformed on data that have been divided intoserial, often overlapping, windows. Inference isbased on these windowed Fourier analyses byaveraging across tasks at each frequency [6].Windows are necessarily kept narrow so that thesignal may be considered stationary, although theconsequence of this is poor frequency resolution.Complex demodulation is also useful when in-vestigating spectral features that vary over time,but is limited to the assessment of narrowfrequency bands [7]. Wavelet-based techniquesare receiving growing interest given their abilityto provide comparable spectral statistics that arelocal in time. Lachaux and colleagues [8–11] haveused complex Gabor wavelets to implement thecontinuous wavelet transform (CWT), looking attime-varying coherence and phase relationshipsbetween nodes. Model-based analysis has alsobeen used to estimate the spectral characteristics ofneurophysiological time series. Multivariate auto-regressive (MVAR) models have the desirableproperty of representing the characteristics of asignal with just a few coefficients, which can beused to calculate the spectral estimates. In addi-tion, MVAR spectra are continuous functions offrequency, and thus avoid the spectral resolutionproblems encountered by the DFT approach.Dynamic data series can be treated by incorporat-ing a hidden Markov model to objectively segmentsignals into different states [12] or by embedding aMVAR model into the Kalman filter [13]. MVARmodels have the disadvantage that calculatingconfidence limits is problematic and estimatedlimits are usually wider than those of DFT-derivedspectra [12].The application of Hilbert wavelet pairs

(HWPs) to time-varying spectral analysis wasoriginally proposed in [14]. Using two pairs ofconjugate quadrature filters with an approximateHilbert relation [15,16], the authors applied twonon-decimated discrete wavelet transforms (alsoknown as maximal overlap discrete wavelet trans-forms or MODWTs) to each time series of interest.To better adapt to unknown features in theunderlying spectral density function, the filtering

sequence of the MODWT may be generalized toallow any arbitrary dyadic partition of thefrequency interval using wavelet packets [17], thistransform is known as the MODHWPT. Thus,both broad- and narrow-band features may beinvestigated by varying the frequency intervalsbased on the depth of the transform. Classicalunivariate and bivariate statistics in the spectraldomain (e.g., power spectrum, coherence, phase)may be calculated from the analytic waveletcoefficients—thus producing a scale-by-scaletime-varying spectral analysis. Two types ofestimates, cyclical and evolutionary, were pro-posed to analyze an event that is either repeatingor continuously changing, respectively.Applying this to the study of time series of

neurophysiological phenomena, we demonstratethat the MODHWPT is a powerful tool forexamining different biological mechanisms.Firstly, the band-pass filtering inherent in thepacket wavelet transform allows one to zoom in onparticular frequency bands of interest. Secondly,HWPs accurately capture the time-varying var-iance and covariance of a stochastic process on aspecific wavelet scale associated with that fre-quency band. Using the fact that each waveletcoefficient can be aligned at a single time-point, wecan construct time-varying coherence and phase tocompare two different neurophysiological timeseries within a specific frequency band. This isespecially useful when performing ensemble esti-mates over a number of events that occur at fixedtime points, as we shall demonstrate. Since theMODHWT is a local transform we relax theassumption that the underlying processes arestationary; a smoothly varying spectrum aroundeach occurrence is sufficient to be able to detectlocalized frequency-specific coherence.The theory of time-varying spectral analysis

using HWPs was limited in [14] by (1) onlyconsidering fractionally differenced time-seriesmodels and (2) deriving limiting distributions forthe magnitude squared coherence (MSC) that aredifficult to compute in practice. We extend thework of [14] in two ways: by proving that theanalytic MODHWPT coefficients of a blockstationary process on a given scale are also ablock stationary process and providing a boot-

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B. Whitcher et al. / Signal Processing 85 (2005) 2065–2081 2067

strap procedure for testing the significance of thetime-varying coherence. Bootstrap methods havebeen shown to be a powerful tool in time seriesanalysis (see [18] and the references therein) andare straightforward to apply in practice. We makethe assumption that the underlying process is nearuncorrelated across events in order to implementthe bootstrap procedure. We demonstrate that thetime-varying spectral analysis is consistent withthe classical Fourier-based spectral analysis for theneurophysiological time series provided and thatthe bootstrap procedure gives meaningful results.The assumption of block stationarity, or a

smoothly varying spectrum, is common whenestimating time-varying spectral characteristics ineither the frequency or wavelet domain [19–21].When developing asymptotic theory of spectralestimators for non-stationary time series, ‘‘asymp-totic’’ is defined such that as the number ofobservations goes to infinity we gain informationabout the process on a finer grid over a fixedinterval in time (so-called infill asymptotics),rather than gathering future observations of theprocess. This theoretical framework is useful in thepresent context because the sampling rate forneurophysiological time series, typically on theorder of 200–1000Hz, ensures that there is a finegrid of sub-second observations. The assumptionof block stationarity is only imposed on scales of atenth or hundredth of a second. Of course, thesampling rate of the data implicitly imposeslimitations on what we can detect. If the data aretruly nonstationary across these scales, then theonly solution would be to acquire observations ata higher sampling rate.Implementation of the MODHWPT through a

cascade filter bank is explained in Section 2.Definitions of time-varying spectral measures(coherence and phase) are also provided alongwith a theoretical analysis of the MODHWPT ofblock stationary bivariate processes. Based on thistheory, we derive a bootstrap procedure for testingthe significance of the time-varying coherence.Neurophysiological time series from two experi-ments illustrate our methodology by (1) compar-ing it to Fourier-based spectral measures on a dataset with well-understood characteristics and (2)exploring event-specific time-varying coherence

and phase spectra in Section 3. A brief discussionof the methodology is provided in Section 4.

2. Methods

2.1. Implementation

The MODHWT was introduced in [14] toinvestigate multi-scale coherence and phase prop-erties of time-varying (non-stationary) processes.The MODHWT has a very specific band-passfiltering structure that partitions the spectrum of atime series with finer and finer detail as f ! 0. Toprovide a more flexible collection of band-passfilters, one must generalize the partitioning schemeof the MODHWT. This is easily obtained byperforming a wavelet packet transform on the timeseries [17]. We denote the wavelet packet general-ization to the MODHWT as the MODHWPT.Instead of one particular filtering sequence, theMODHWPT executes all possible filtering combi-nations to obtain a wavelet packet tree, denoted byT. Let T ¼ fðj; nÞ : j ¼ 0; . . . ; J; n ¼ 0; . . . ; 2j � 1gbe the collection of all doublets ðj; nÞ that form theindices of the nodes of a wavelet packet tree. Anorthonormal basis B � T is obtained when acollection of wavelet packet coefficients is chosen,whose ideal band-pass frequencies are disjoint andcover the frequency interval ½0; 1=2. The set B is acollection of doublets ðj; nÞ that correspond to anorthonormal basis.Designing filter coefficients that are HWPs is

thoroughly outlined in [15,16]. Briefly, a Hilbertwavelet pair is a pair of wavelet filters that aredesigned to be approximate Hilbert transforms ofone another (i.e., have a half-sample delay betweenthem). It is not possible to design a pair of waveletfilters that are exact Hilbert transforms, howeverapproximate Hilbert wavelet pairs can be designedvia spectral factorization. Given an HWP has beenselected, implementation of the MODHWPT isrelatively straightforward and closely parallels theusual non-decimated discrete wavelet packet trans-form [22]. Let ~a1;‘ ¼ a1;‘=

ffiffiffi2

pand ~b1;‘ ¼ b1;‘=

ffiffiffi2

p

denote the renormalized Hilbert wavelet filterpairs, ~a0;‘ and ~b0;‘ be the associated renormalizedscaling filters. For ‘ ¼ 0; . . . ;L � 1, we define the

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unit scale analytic wavelet (high-pass) filter to be~h‘ ¼ ~a1;‘ þ i ~b1;‘ and the unit scale analytic scaling(low-pass) filter to be ~g‘ ¼ ~a0;‘ þ i ~b0;‘. Complexnotation is convenient because the convolutionoperator does not change when using a real- orcomplex-valued filter. There is also a parallel tothe usual DFT. Here, our analytic wavelet andscaling filters are designed to be approximateHilbert transform pairs like sine and cosine waves,but with compact support.Fig. 1 displays the low-pass filter coefficients

and Fourier properties for an HWP(6,6), designedusing spectral factorization [15,16]. The firstparameter is the number of zero momentsanalogous to the property of compactly supported

0 5 10 15 20

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0

0

0

0

0

0

1

a0(n)

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

|A0(f)|, |B0(f)|

frequency

0

0

0

0

0

0

Fig. 1. Filter coefficients and Fouri

orthogonal wavelet filters [23]. The second para-meter represents the degree of approximation tothe Hilbert relation between the two filters. As thisparameter increases, the approximation improvesalong with the number of non-zero filter coeffi-cients. Because of the restriction to a finite-lengthfilter, the Hilbert relation (half-sample delay) isonly valid for frequencies f 2 ð0; 0:4. The gainfunctions, A0ðf Þ and B0ðf Þ, indicate the degree ofleakage induced by the low-pass filters whencompared to an ideal band-pass filter.There is no convenient interpretation of wavelet

packet coefficient vectors with differences atvarious scales. Instead, the wavelet packet coeffi-cient W j;n;t is associated with the frequency

.2

.0

.2

.4

.6

.8

.0

0 5 10 15 20

b0(n)

0.0 0.1 0.2 0.3 0.4 0.5

.0

.1

.2

.3

.4

.5

θ(f)/π

frequency

er properties of an HWP(6,6).

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interval

lj;n ¼ �n þ 1

2jþ1;�

n

2jþ1

� �[

n

2jþ1;n þ 1

2jþ1

� �. ð1Þ

From Fig. 2 the sequence of filtering steps at theleft of the wavelet packet table involves strictlylow-pass filters, and therefore, the scaling (low-frequency) coefficients are denoted by W j;0;t ateach level.In terms of the unit scale analytic wavelet and

scaling filters, we define

~un;l ¼~gl ; if n mod 4 is 0 or 3;~hl ; if n mod 4 is 1 or 2;

((2)

to be the appropriate filter at a given node of thewavelet packet tree. The wavelet packet filters ~un;l

are defined so that on a given wavelet level j, thefrequency intervals lj;0; lj;1; . . . ; lj;2j�1 in theMODHWPT will be arranged in increasingfrequency order. Let X ¼ fX t : t ¼ 0; . . . ;N � 1gbe a finite-length vector of observations andWj;n ¼ fW j;n;t : t ¼ 0; . . . ;N � 1g denote the vectorof wavelet packet coefficients associated with thefrequency interval lj;n. Given the vector ofMODHWPT coefficients from the level above

Fig. 2. Flow diagram illustrating the decomposition of X ¼

W0;0 (the level j ¼ 0 MODHWPT coefficients) into

MODHWPT coefficientsWj;n for levels j ¼ 1; 2; 3. For exampleto obtain the level j ¼ 1 coefficients, W0;0 is filtered using the

filter eGðf Þ to obtainW1;0, and filtered using eHðf Þ to obtainW1;1.

The filtered coefficients are not subsampled. The output is then

filtered again and again to complete the wavelet packet tableT.

The execution of the filtering is done in sequency order to

preserve monotonic frequency ordering.

Wj�1;bn2c, we may compute W j;n;t recursively via

W j;n;t ¼XL�1‘¼0

~un;lW j�1;bn2c;t�2

j�1‘ mod N ,

t ¼ 0; 1; . . . ;N � 1. ð3Þ

The recursion set starts with W0;0 ¼ X. TheMODHWPT is most efficiently computed using apyramid algorithm [22]. The orthonormal basiscorresponding to the MODHWT is just one of thepossible filtering combinations available from thewavelet packet table and is given by B ¼ fðJ; 0Þg[fðj; 1Þ : j ¼ 1; . . . ; Jg. The MODHWPT providesa flexible means of isolating and investigatingdifferent frequency intervals from a univariatetime series through the choice of B and the HWP.To calculate the time shifts for an arbitrarywavelet packet vectorWj;n, in order to align eventsamong different coefficient vectors, we use a‘‘center of energy’’ argument [24]. For a real-valued filter of even length, fa‘g, we define itscenter of energy by eðaÞ ¼ ð

PL�1‘¼1 ‘a2‘ Þ=

PL�1‘¼0 a2‘ .

The time shifts for an arbitrary wavelet packetvectorWj;n are given by jtj;nj ¼ Zj;neða0Þ þ xj;neða1Þ,

where Zj;n ¼Pj�1

k¼0 ð1� cj;n;kÞ2k and xj;n ¼

Pj�1k¼0

cj;n;k2k. The length j vector cj;n is comprised of

zeros or ones where, from left to right, a one ispresent when a high-pass filtering operation wasperformed and zero when a low-pass filteringoperation was performed. See [14] for more details.

2.2. Spectral measures

Let fðW Xt ;W

Yt Þ : t 2 Zg denote the MODH-

WPTs of ðX t;Y tÞ. Their time-varying spectra canbe defined via SX ðlj;n; tÞ ¼ EjW X

j;n;tj2 and SY ðlj;n; tÞ

¼ EjW Yj;n;tj

2, respectively. The first parameter thatcharacterizes the time-varying spectrum is thefrequency interval lj;n (1), which is directly relatedto the frequency-domain attributes of the waveletfilters. The second parameter is the time index,thus isolating features from the original series bothin time and frequency.For bivariate time series, the time-varying cross

spectrum is given by

SXY ðlj;n; tÞ ¼ E½W Xj;n;tðW

Yj;n;tÞ

�

¼ CXY ðlj;n; tÞ � iQXY ðlj;n; tÞ, ð4Þ

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where

CXY ðlj;n; tÞ ¼ RfSXY ðlj;n; tÞg and

QXY ðlj;n; tÞ ¼ �IfSXY ðlj;n; tÞg ð5Þ

are the time-varying cospectrum and quadraturespectrum, respectively. The polar representation ofthe cross spectrum

SXY ðlj;n; tÞ ¼ AXY ðlj;n; tÞ expfiyXY ðlj;n; tÞg (6)

allows us to define the time-varying crossamplitude spectrum AXY ðlj;n; tÞ ¼ jSXY ðlj;n; tÞj ¼½C2

XY ðlj;n; tÞ þ Q2XY ðlj;n; tÞ

1=2, along with the time-varying phase spectrum

yXY ðlj;n; tÞ ¼ arctan�QXY ðlj;n; tÞ

CXY ðlj;n; tÞ

� �. (7)

The time-varying MSC

KXY ðlj;n; tÞ ¼A2

XY ðlj;n; tÞ

SX ðlj;n; tÞSY ðlj;n; tÞ(8)

is a normalized and squared version of the time-varying cross spectrum.Two types of spectral estimates can be com-

puted after applying the MODHWPT to the timeseries of interest. The first is a time-varying spec-trum that follows the original series in time. This isequivalent to single-trial techniques in neurocog-nitive studies, since it does not require block repe-titions of events. The second is a cyclical spectrumthat is applied to repeated trials and then avera-ged. This is equivalent to ensemble methods whereresults from each trial are averaged together.Let X and Y be finite length vectors of obser-

vations covering the same interval of time and letWX and WY denote their MODHWPTs. Assumewe are interested in estimating the time-varyingspectrum averaged over K disjoint time intervals oflength T þ 1, with each interval centered at timepoint ek, for k ¼ 1; . . . ;K . The cyclical autospectraassociated with the frequency interval lj;n for Xand Y can be estimated via, respectively,

bSX ðlj;n; tÞ ¼1

K

XK

k¼1

jW Xj;n;ekþtj

2 and

bSY ðlj;n; tÞ ¼1

K

XK

k¼1

jW Yj;n;ekþtj

2 ð9Þ

for t ¼ �T=2; . . . ;T=2. The cyclical cross spec-trum associated with lj;n between X and Y can beestimated via

bSXY ðlj;n; tÞ ¼1

K

XK

k¼1

W Xj;n;ekþtðW

Yj;n;ekþtÞ

�

for t ¼ �T=2; . . . ;T=2 ð10Þ

and from this are the estimated cospectrumbCXY ðlj;n; tÞ and quadrature spectrum bQXY ðlj;n; tÞ.The cyclical phase spectrum yXY ðlj;n; tÞ and cycli-cal MSC cKXY ðlj;n; tÞ can be obtained from theestimated cospectrum and quadrature spectrumusing (7) and (8), respectively.Estimating the time-varying spectra does not

provide the convenient averaging found whenestimating cyclical spectra, and hence, we mustsmooth the data over time in order to produceinterpretable results. This problem is similar towhat one experiences in traditional bivariate timeseries analysis, that is, constructing estimates ofthe MSC using the periodogram results in anestimate that is equal to one for all frequencies.There are not enough degrees of freedom toestimate both the univariate and bivariate spectralquantities. We propose to apply a simple two-sided moving average (length M þ 1) on theanalytic wavelet coefficients yielding

SX ðlj;n; tÞ ¼1

M þ 1

XtþM=2

t0¼t�M=2

jW Xj;n;t0 j

2, (11)

SY ðlj;n; tÞ ¼1

M þ 1

XtþM=2

t0¼t�M=2

jW Yj;n;t0 j

2, (12)

CXY ðlj;n; tÞ

¼1

M þ 1

XtþM=2

t0¼t�M=2

RfW Xj;n;t0 ðW

Yj;n;t0 Þ

�g, ð13Þ

QXY ðlj;n; tÞ

¼1

M þ 1

XtþM=2

t0¼t�M=2

�IfW Xj;n;t0 ðW

Yj;n;t0 Þ

�g ð14Þ

for t ¼ 1; . . . ;N. Estimates of the time-varyingphase spectrum yXY ðlj;n; tÞ and MSC KXY ðlj;n; tÞ

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can now be constructed by plugging in thesesmoothed quantities into (7) and (8), respectively.

2.3. A bivariate block-stationary time series model

Suppose that we observe a bivariate timeseries model, fðX t;Y tÞ : t 2 Zg. To study theMODHWPTs of this process we re-express W X

j;n;t

as given by (3) in terms of a filtering of the originalprocess X t [22, Chapter 6]. First define eUj;nðf Þ tobe the Fourier transform of the filter eun;l . Let Lj ¼

ð2j � 1ÞðL � 1Þ þ 1 so that the filter euj;n;l (l ¼ 0; . . . ;Lj � 1) at scale lj;n can be defined in terms of theinverse Fourier transform of the transfer function,

eUj;nðf Þ ¼Yj�1k¼0

eU2k ð2kf Þ. (15)

Then we can obtain the wavelet coefficients atfrequency interval lj;n by the convolution

W Xj;n;t ¼

XLj�1

l¼0

~uj;n;lX t�l mod N ,

t ¼ 0; . . . ;N � 1. ð16Þ

Without loss of generality we consider thestatistical properties of the MODHWPT definedby

W Xj;n;t ¼

XLj�1

l¼0

~uj;n;lX t�l . (17)

Similarly consider the MODHWPT of fY tg.With a suitable adjustment for the length L of

the wavelet filter used, we will assume that thebivariate process is stationary in each eventwindow of at least length T þ 1, centered aroundeach event ek, k ¼ 1; . . . ;K . In particular for aMODHWPT at frequency interval lj;n we willassume that fðX t;Y tÞ : t 2 ½ek � T=2� Lj ; ek þ

T=2g is a bivariate stationary process with meanl ¼ ðmX ;mY Þ and spectral density matrix

Sðf Þ ¼SX ðf Þ SXY ðf Þ

ðSXY ðf ÞÞ� SY ðf Þ

" #; jf jp1=2 (18)

for each k. The next result shows that mostMODHWPTs of X t and Y t are bivariate sta-

tionary processes, with a known mean andcovariance structure.

Theorem 2.1. For each j and n ¼ 1; . . . ; 2j � 1,fðW X

j;n;t;WYj;n;tÞ : t 2 ½ek � T=2; ek þ T=2g is a com-

plex-valued bivariate stationary process with

EfW Xj;n;tg ¼ EfW Y

j;n;tg ¼ 0 and spectral density ma-

trix Sj;nðf Þ ¼ j eUj;nðf Þj2Sðf Þ, for jf jp1=2.

The proof of this theorem appears in AppendixA. For n ¼ 0, a similar result holds, except that themean of the MODHWPT process will not be zero(the proof of this fact is similar to derivationsgiven in [22]).

2.4. Time-varying spectral analysis of ensembles

We extend the results in [14] to provide large-sample properties of ensemble estimates given by(9), (10), and the associated time-varying cospec-trum, quadrature spectrum, and MSC. Insteadof averaging spectral estimates over fixed inter-vals of time (corresponding to a cycle or season),here the events have different time spans asdictated by the ek values and are thus irregu-larly spaced throughout the experiment. Wepropose a bootstrap test for zero MSC thatincorporates the intrinsic correlation structurepresent around each event. We begin by assu-ming that covfW X

j;n;ekþt;WXj;n;ek0 þt0 g, covfW X

j;n;ekþt;

W Yj;n;ek0þt0 g, and covfW

Yj;n;ekþt;W

Yj;n;ek0 þt0 g are close to

zero for each k, k0, t 2 ½ek � T=2; ek þ T=2 andt0 2 ½ek0 � T=2; ek0 þ T=2.

Let Ck;t ¼ W Xj;n;ek�t and CKþk;t ¼ W Y

j;n;ek�t. The

bootstrap test based on B bootstrap samples is asfollows. For b ¼ 1; . . . ;B:

(1)
Sample with replacement a sample of 2K
integers from the integers 1; . . . ; 2K . Letfm1; . . . ;m2kg denote this sample.

(2)
Let WX
j;n;tþek¼ Cmk ;t for each k ¼ 1; . . . ;K

and t.
(3) Let W
Y

j;n;tþek¼CmKþk ;t for each k¼1; . . . ;K and t.

(4)
Calculate the time-varying estimate of the
MSC, cKb

XY ðlj;n; tÞ for each t 2 ½�T=2þ

1;T=2 based on the ensemble estimate usingW

X

j;n;tþekand W

Y

j;n;tþek.

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A point-wise rejection region for the bootstrap testof approximate significance level a is obtained bycalculating the ð1� aÞth quantile of the sample

½cK1

XY ðlj;n; tÞ; . . . ;cKB

XY ðlj;n; tÞ. The bootstrap ap-

proximation improves with increasing B. Thisbootstrap test is valid under the assumption madeabove that the time series around event ek areapproximately uncorrelated with all other events

ek0 (kak0). We sample from the null hypothesis byrandomly sampling with replacement from theMODHWPT coefficients of either X t or Y t. Thecorrelation structure is preserved around eachevent by sampling blocks of coefficients.

3. Results

We first compared the coherence and phasespectra derived using Hilbert wavelet pairs withthose estimated using the DFT, which is mostcommonly used to analyze static neurophysiologi-cal data series. To this end we examined a wellcharacterized system with established phase rela-tionships [5]. Microtremor can be picked up withan accelerometer on the dorsum of the hand as thewrist is slowly flexed and extended. This micro-tremor is driven by the activity, particularlysynchronized activity, of motor units innervatingthe active muscles in the forearm [25,26]. Duringsuch slow wrist movements synchronization isdriven by the corticospinal system and preferen-tially occurs in the so-called beta frequency(13–30Hz) band [27]. Moreover, mechanical ac-tivity follows electromyographic activity (EMG)by an interval determined by electromechanicalcoupling [28].Fig. 3 compares the coherence spectra between

hand microtremor measured by the accelerometerand a forearm extensor EMG. Two Fourier-basedspectral estimates are provided. The first is the so-called periodogram estimates of autospectra, MSCand phase. Non-overlapping blocks of length 512were averaged together to provide valid estimates.The second is a multitaper spectral estimatorwhere several orthogonal tapers are applied tothe time series and each (direct) spectral estimate isthen averaged together [29,30]. Although there is a

slight loss of frequency resolution, multitaperspectral estimates do not suffer from bias as isknown to occur in the periodogram for spectraldensity functions with a large dynamic range. Thisbias is apparent in the periodogram estimators forboth autospectra and manifests itself in anelevated MSC especially across the 13–30Hzfrequency interval. As expected, the time-inte-grated estimates using HWPs more closely mimicsthe results from the multitaper spectral estimates,indicating that these wavelet-based estimators donot suffer from bias. All three estimators show theexpected peak at 20Hz. The respective phasespectra are shown in Fig. 3, but only for themultitaper and HWP estimators. Both are con-sistent with EMG driving muscle around 20Hz.This experiment shows that the time-integratedspectra derived with Hilbert wavelet pairs showessentially the same features as those derived usingFourier-based methods and are in line with apriori expectations.In the above example we assumed that the

biological situation could be modeled as astationary process. Does time-varying spectralanalysis, using Hilbert wavelet pairs, also givephysiologically plausible results? To address thisquestion we analyzed time series drawn from arecently characterized biological system [31,32].The local field potential (LFP) picked up from thesubthalamic nucleus in the course of functionalneurosurgery for Parkinson’s disease is coherentwith electroencephalographic activity (EEG) re-corded over motor areas of the cerebral cortex. Inuntreated parkinsonian patients this couplingpreferentially occurs at around 25Hz and issuppressed by movement, but following treatmentwith the dopamine prodrug levodopa, this syn-chronization between LFP and EEG is attenuatedand coupling preferentially occurs at around 70Hzand increases with movement [32].Fig. 4 illustrates the univariate and bivariate

spectral estimates using Fourier and Hilbertwavelet pairs for the LFP of the subthalamicnucleus and EEG signals recorded in a patientwith Parkinson’s disease following treatmentwith levodopa. The data from the whole recordhave been averaged, irrespective of whether thepatient was resting or making a movement. Both

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Fig. 3. Spectral analysis of two neurophysiological time series: acceleration of the wrist and EMG from the forearm extensors.

Autospectra are plotted on the left, while the magnitude squared coherence and phase spectrum are on the right. Periodogram

estimates (thin lines), multitaper estimates (thin dashed lines) and wavelet-based estimates (thick lines) are plotted on identical vertical/

horizontal scales. Vertical lines denote the interval 13–30Hz. The range of the abscissa is ð�2p; 2pÞ to minimize the influence of

discontinuities around the boundary of �p.


periodogram and multitaper spectral estimateswere computed, but in this case no substantialdifference was observed between the two. There isa peak in coherence seen with all three techniquesat about 70Hz. The observed peak at 50Hz is dueto noise.Fig. 5 shows the MSC between LFP and EEG

estimated using Hilbert wavelet pairs and averagedaround the time of movement (t ¼ 0) for the samedata set. The frequency intervals match those inFig. 4 and were selected because they provided agood balance between frequency resolution andtemporal resolution in the data. Thresholds to testfor MSC40, at a ¼ 0:05, were calculated using thebootstrap procedure outlined in Section 2.4. Thatis, sustained time intervals where the MSC isgreater than the threshold indicate significant

coherence between the two time series. Theevent-averaged MSC exhibits a significant increasearound 70Hz, particularly in the 69–75Hz bandthat begins just prior to movement, in agreementwith the dynamic MVAR of similar data [31]. Thesignificant coherence observed at almost all timepoints in the 62–69Hz band agrees nicely with thebroad-band peak in Fig. 4. In Fig. 6 we observephase destabilization in the 19–25Hz and25–31Hz bands, and phase stabilization in the69–75Hz band upon movement (t ¼ 0).Figs. 7 and 8 give the single-trial MSC and

phase spectra between LFP and EEG that are usedto construct the event-averaged MSC and phasespectra in Figs. 5 and 6 for the 69–75Hz band. Alocal average of M ¼ 100 was used to produceconsistent spectral estimates (M ¼ 100 being

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Fig. 4. Spectral analysis of two neurophysiological time series: the right subthalamic contact and midline cortical EEG (Cz-FCz).

Autospectra are plotted on the left, while the magnitude squared coherence and phase spectrum are on the right. Periodogram

estimates (thin lines), multitaper estimates (thin dashed lines) and wavelet-based estimates (thick lines) are provided. Vertical lines

denote 20 and 70Hz. The range of the abscissa is ð�2p; 2pÞ to minimize the influence of discontinuities around the boundary of �p.


roughly four times the length of the waveletfilters). On the whole, coherence increases aroundthe time of movement but it is not uniform acrossevents. If the increase occurred at the same timepoint across events, then the event-averagedcoherence would look stable. However, in thiscase the increases are not uniformly centeredaround each event and the event-averaged coher-ence is noisy and highly variable.

4. Discussion

The wavelet method presented here, usingHilbert wavelet pairs, provides an intuitive, power-ful and computationally efficient way to estimatetime-varying features between pairs of neurophy-siological time series, provided signals are locally

stationary across the duration of the fitted wave-lets. By computationally efficient, we mean thatfor a time series of length N and filter length L theMODHWPT is an OðLN2Þ calculation if wecalculate the MODHWPT to log2 N levels, com-pared to the OðN2 logNÞ calculation for a movingwindow spectral analysis. The methodology uti-lizes the discrete wavelet transform, rather thanthe continuous wavelet transform, so estimates ofcoherence and phase are associated with a discreteset of frequency intervals. By selecting the depth ofthe transform, and thus varying the frequencyintervals associated with each node of the waveletpacket tree, both broad- and narrow-band featuresmay be investigated. Currently, a fixed level j ofthe MODHWPT is used so that the frequencyinterval is broken into fixed-width intervals, butthis is not strictly necessary. Using adaptive

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50 to 56 Hz

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75 to 81 Hz

Time (seconds)

Fig. 5. Event-averaged MSC between the right subthalamic contact and midline cortical EEG (Cz-FCz). Frequency intervals

associated with the MODHWPT ðJ ¼ 4Þ are provided in Hz. The bootstrap threshold for MSC40 is in grey. Self-paced movement of

the left hand occurs at time zero.


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50 to 56 Hz

Time (seconds)

56 to 62 Hz

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62 to 69 Hz

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69 to 75 Hz

Time (seconds)

75 to 81 Hz

Time (seconds)

Fig. 6. Event-averaged phase spectra between the right subthalamic contact and midline cortical EEG (Cz-FCz). Frequency intervals

associated with the MODHWPT ðJ ¼ 4Þ are provided in Hz. The range of the abscissa is ð�2p; 2pÞ to minimize the influence of

discontinuities around the boundary of �p. Self-paced movement of the left hand occurs at time zero.


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Fig. 7. Single-trial MSC between the right subthalamic contact and midline cortical EEG (Cz-FCz) for the frequency interval

69–75Hz. Self-paced movement of the left hand occurs at time zero.


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Fig. 8. Single-trial phase spectrum between the right subthalamic contact and midline cortical EEG (Cz-FCz) for the frequency

interval 69–75Hz. Self-paced movement of the left hand occurs at time zero.


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techniques the orthonormal basis may be acollection of variable-width frequency intervals,providing they are disjoint and cover ½0; 1=2. TheMODHWPT is therefore able to capture higherand lower frequency local oscillations contained inthe signals.Both single-trial and ensemble results are easily

derived using our HWP-based methodology. Wehave extended the theoretical results in [14] toblock stationary time series models and provided abootstrap test for significant coherence betweentime series. These improvements should makeHWP-based analysis of non-stationary time seriesaccessible to a wider audience. Software toimplement these routines can be found in wave-slim, a package written for the R statisticalprogramming environment [33].There are a number of time-varying spectral

analysis techniques that allow the assessment ofautospectra, coherence and phase over time. Eachhas its advantages and disadvantages. Mostcommonly the DFT is utilized, keeping windowsnarrow so that signals can be considered station-ary. However, the consequence of this is poorfrequency resolution and this is compounded bythe fact that the period of local stationarity insignals such as EEG can well vary inversely withfrequency so that a fixed window can not beoptimally tuned to stationarity across the spec-trum. Analysis using the Hilbert wavelet pairsapproach can provide a finer temporal resolutionthat also increases as frequency increases. Anothermajor problem with the DFT is that it involvesover-fitting of the data so that averaging isrequired to give reliable estimates of coherence.This is possible where ensemble results of multipleevent-related data are desired but leads to poortemporal resolution when blocks of windows,albeit overlapping, are used to provide dynamicestimates of coherence and phase within singlerecords. The use of multitaper spectral estimatorsmay also be required in order to overcome bias,with an additional loss in frequency resolution.Time-varying spectral analysis using Hilbert

wavelet pairs is not without its limitations. Likethe DFT and complex demodulation, it onlyallows the analysis of bivariate data sets. Whenintegrated across time, to allow direct comparison

with the DFT, its frequency resolution is limited bythe depth of the transform. Thus, spectral estimatesdo not have as good a frequency resolution as theDFT (e.g., Fig. 4) and ambiguities in theirinterpretation may be present [34]. On the otherhand, the HWP-based estimates do not overfit thedata and therefore provide unbiased and consistentestimates of the true spectral characteristics.MVAR models may be used to follow spectralchanges over time [12,13]. Multivariate signals maybe considered in the same model and frequencyresolution is very good. MVAR models have thedisadvantage that the calculation of confidencelimits is problematic and they are computationallydemanding. They provide high frequency resolu-tion but at the cost of making assumptions aboutthe physical process that are difficult to verify.Complex Gabor wavelets can be used to implementthe CWT and produce a highly redundant collec-tion of wavelet coefficients across the time-frequency plane. The CWT, as applied to neuronalsynchrony, was compared with the direct applica-tion of the Hilbert transform in [11]. The twomethods were found to differ only slightly with theconclusion that they were fundamentally equiva-lent for the study of neuroelectrical signals. SinceHilbert wavelet pairs are filters specifically designedto be orthogonal, finite impulse response filterswith compact support and satisfy the Hilberttransform over a specific range of frequencies, webelieve that they can provide the same level ofinformation as previous complex wavelet andHilbert transform methods, but with much lessredundancy in their representation. Of course, acomparison study between all three methods wouldbe required to make definitive conclusions.Ultimately, the choice of analysis technique

must be dictated by the nature of the data setand the scientific emphasis put on differentspectral aspects. We have shown that HWP-basedmethods are a useful tool with the ability tocontribute in the field of cognitive neuroscience.

Acknowledgements

The authors would like to thank the GuestEditor and two anonymous reviewers for

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constructive comments that led to an improvedmanuscript.

Appendix A. Proof of the Theorem

Fix a value for j. When n40, the transferfunction for the wavelet filter eUj;nðf Þ containseHð2kf Þ for some k ¼ 0; . . . ; j � 1. Since eHð�Þ

contains an inherent differencing filter in boththe real and complex parts [15], it follows thatEfW X

j;n;tg ¼ EfW Yj;n;tg ¼ 0. We now proceed with

the covariance function. For an integer h and t 2

½ek � T=2þ 1; ek þ T=2 we have the following:

covfW Xj;n;t;W

Yj;n;tþhg

¼XLj�1

l¼0

XLj�1

l0¼0

Ef ~uj;n;lX t�lð ~uj;n;l0Y tþh�l0 Þ�g ðA:1Þ

¼XLj�1

l¼0

XLj�1

l0¼0

~uj;n;l ~u�j;n;l0EfX t�lY tþh�l0 g ðA:2Þ

¼XLj�1

l¼0

XLj�1

l0¼0

~uj;n;l ~u�j;n;l0

Z 1=2

�1=2ei2pf ðh�l0þlÞ

�SXY ðf Þdf ðA:3Þ

¼

Z 1=2

�1=2ei2pfhj eUj;nðf Þj

2SXY ðf Þdf , ðA:4Þ

by definition of the transfer function for aMODHWPT filter associated with the frequencyinterval lj;n. A similar derivation holds forcovfW X

j;n;t;WXj;n;tþhg and covfW Y

j;n;t;WYj;n;tþhg. Thus,

the MODHWPT wavelet coefficients of X t and Y t

on each scale form a bivariate stationary processwith the stated spectral density matrix.

References

[1] A.K. Engel, W. Singer, Temporal binding and the neural

correlates of sensory awareness, Trends Cognitive Sci. 5

(2001) 16–25.

[2] A.K. Engel, P. Fries, W. Singer, Dynamic predictions:

oscillations and synchrony in top-down processing, Nat.

Rev. Neurosci. 2 (2001) 704–716.

[3] B. Pesaran, J.S. Pezaris, M. Sahani, P.P. Mitra, R.A.

Andersen, Temporal structure in neuronal activity during

working memory in macaque parietal cortex, Nat.

Neurosci. 5 (2002) 805–811.

[4] G. Pfurtscheller, F.H.L. da Silva, Event-related EEG/

MEG synchronization and desynchronization: basic prin-

ciples, Clin. Neurophysiol. 110 (1999) 1842–1857.

[5] P. Brown, The Piper rhythm and related activities in man,

Progr. Neurobiol. 60 (2000) 97–108.

[6] C. Andrew, G. Pfurtsheller, Event-related coherence

as a tool for studying dynamic interaction of brain

regions, Electroencephalogr. Clin. Neurophysiol. 98

(1996) 144–148.

[7] R.W. Thatcher, Tomographic electroencephalography/

magnetoencephalography: dynamics of human neural

network switching, Amer. Soc. Neuroimaging 5 (1)

(1995) 35–45.

[8] J.-P. Lachaux, E. Rodriguez, J. Martinerie, F.J. Varela,

Measuring phase synchrony in brain signals, Human Brain

Mapping 8 (1999) 194–208.

[9] J.-P. Lachaux, E. Rodriguez, M. Le Van Quyen, A. Lutz,

J. Martinerie, F.J. Varela, Studying single-trials of phase

synchronous activity in the brain, Internat. J. Bifurcation

Chaos 10 (10) (2000) 2429–2439.

[10] J.-P. Lachaux, A. Lutz, D. Rudrauf, D. Cosmelli, M. Le

Van Quyen, J. Martinerie, F. Varela, Estimating the time-

course of coherence between single-trial brain signals: an

introduction to wavelet coherence, Clin. Neurophysiol. 32

(3) (2002) 157–174.

[11] M. Le Van Quyen, J. Foucher, J.-P. Lachaux, E.

Rodriguez, A. Lutz, J. Martinerie, F.J. Varela, Compar-

ison of Hilbert transform and wavelet methods for the

analysis of neuronal synchrony, J. Neurosci. Methods 111

(2001) 83–98.

[12] M.J. Cassidy, P. Brown, Hidden Markov based autore-

gressive analysis of stationary and non-stationary electro-

physiological signals for functional coupling studies, J.

Neurosci. Methods 116 (2002) 35–53.

[13] M. Cassidy, W.D. Penny, Bayesian nonstation-

ary autoregressive models for biomedical signal

analysis, IEEE Trans. Biomed. Eng. 49 (10) (2002)

1142–1152.

[14] B. Whitcher, P.F. Craigmile, Multivariate spectral analysis

using Hilbert wavelet pairs, Int. J. Wavelets, Multiresolu-

tion Inform. Process. 2 (4) (2004) 567–587.

[15] I.W. Selesnick, The design of approximate Hilbert trans-

form pairs of wavelet bases, IEEE Trans. Signal Process.

50 (5) (2002) 1144–1152.

[16] I.W. Selesnick, Hilbert transform pairs of wavelet bases,

IEEE Signal Process. Lett. 8 (6) (2001) 170–173.

[17] M.V. Wickerhauser, Adapted Wavelet Analysis from

Theory to Software, Wellesley Massachusetts, 1994 (edited

by A.K. Peters).

[18] D.N. Politis, The impact of bootstrap methods on time

series analysis, Statist. Sci. 18 (2003) 219–230.

[19] M.B. Priestley, Spectral Analysis and Time Series, Aca-

demic Press, London, 1981.

[20] R. Dahlhaus, Fitting time series models to nonstationary

processes, Ann. Statist. 25 (1) (1997) 1–37.

[21] A. Serroukh, A.T. Walden, D.B. Percival, Statistical

properties of the wavelet variance estimator for

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non-Gaussian/non-linear time series, J. Amer. Statist.

Assoc. 95 (449) (2000) 184–196.

[22] D.B. Percival, A.T. Walden, Wavelet Methods for Time

Series Analysis, Cambridge University Press, Cambridge,

England, 2000.

[23] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF

Regional Conference Series in Applied Mathematics, vol.

61, Society for Industrial and Applied Mathematics,

Philadelphia, 1992.

[24] N. Hess-Nielsen, M.V. Wickerhauser, Wavelets and time-

frequency analysis, Proc. IEEE 84 (4) (1996) 523–540.

[25] P. Brown, Muscle sounds in Parkinson’s disease, Lancet

349 (1997) 533–535.

[26] D.M. Halliday, B.A. Conway, S.F. Farmer, J.R. Rosen-

berg, Load-independent contributions from motor-unit

synchronization to human physiological tremor, J. Neu-

rophysiol. 82 (1999) 664–675.

[27] P. Brown, S. Salenius, J.C. Rothwell, R. Hari, The cortical

correlate of the Piper rhythm in man, J. Neurophysiol. 80

(1998) 2911–2917.

[28] J.H. McAuley, J.C. Rothwell, C.D. Marsden, Frequency

peaks of tremor, muscle vibration and electromyographic

activity at 10, 20 and 40Hz during human finger muscle

contraction may reflect rhythmicities of central neural

firing, Exp. Brain Res. 114 (1997) 525–541.

[29] D.J. Thomson, Spectrum estimation and harmonic analy-

sis, IEEE Proc. 70 (9) (1982) 1055–1096.

[30] D.B. Percival, A.T. Walden, Spectral Analysis for Physical

Applications: Multitaper and Conventional Univariate

Techniques, Cambridge University Press, Cambridge,

England, 1993.

[31] M. Cassidy, P. Mazzone, A. Oliviero, A. Insola, P. Tonali,

V. Di Lazzaro, P. Brown, Movement-related changes in

synchronization in the human basal ganglia, Brain 125

(2002) 1235–1246.

[32] P. Brown, The oscillatory nature of human basal ganglia

activity: relationship to the pathophysiology of Parkin-

son’s disease, Movement Disorders 18 (2003) 357–363.

[33] R Development Core Team, R: A language and environ-

ment for statistical computing, R Foundation for Statis-

tical Computing, Vienna, Austria, ISBN 3-900051-00-3,

2004, URL: http://www.R-project.org

[34] J. Gotman, Measurement of small time differences

between EEG channels: method and application to

epileptic seizure propagation, Electroencephalogr. Clin.

Neurophysiol. 56 (1983) 501–514.

http://www.R-project.org

Time-varying spectral analysis in neurophysiological time series using Hilbert wavelet pairs

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