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Brain Research Bulletin xxx (2005) xxx–xxx

Antero-posterior functional coupling at sleep onset: changesas a function of increased sleep pressure

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Luigi De Gennaroa,∗, Fabrizio Vecchiob,c, Michele Ferrarad, Giuseppe Curcioa,Paolo Maria Rossinic,e,f , Claudio Babilonib,c,f

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a Dipartimento di Psicologia, Sezione di Neuroscienze, Universit`a degli Studi di Roma “La Sapienza”, Via dei Marsi 78, 00185 Rome, Italy7b Dipartimento di Fisiologia Umana e Farmacologia, Sezione di EEG ad alta risoluzione, Universit`a degli Studi di Roma “La Sapienza”, Italy8

c AFaR-Ospedale San Giovanni Calibita Fatebenefratelli, Rome, Italy9d Dipartimento di Medicina Interna e Sanit`a Pubblica, Universit`a di L’Aquila, Italy10

e Clinica Neurologica-Universit`a di Roma “Campus Bio-medico”, Italy11f IRCCS-Centro San Giovanni di Dio, Brescia, Italy12

Received 15 July 2004; received in revised form 26 October 2004; accepted 14 December 2004

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bstract

The use of the directed transfer function (DTF), an advanced computational analysis of electroencephalogaphic (EEG) data, whin estimation of the direction of the information flow underlying cortico-cortical functional coupling, has shown that the presleepharacterized by anterior-to-posterior functional cortical coupling, while at sleep onset there is an inversion of that direction. Thupported the idea that anterior cortical areas first synchronize sleep EEG activity. The aim of the present study was to assess thunctional coupling between anterior and posterior midline cortical areas during the sleep onset process when sleep pressure is hselective slow-wave sleep (SWS) deprivation. The hypothesis was that the anterior-to-posterior direction of the cortical functionat sleep onset is enhanced by SWS deprivation. Ten normal right-handed male students slept for six consecutive nights in the ladaptation, 2: baseline, 3: baseline with awakenings, 4 and 5: SWS deprivations, 6: recovery), with standard polysomnographiche DTF was computed on data recorded during nights two and six from anterior (FzA1) and posterior (PzA1, OzA1) derivations. Resulhowed that, during the recovery night, the anterior-to-posterior direction of functional cortical coupling is already present in theeriod, indicating that SWS deprivation advances the shift to an anterior-to-posterior directionality of functional cortical coupling,s a consequence of heightened sleep pressure. These findings support the notion that a spread of synchronizing signals fromrefrontal to posterior areas play a role in the wake–sleep transition.2005 Published by Elsevier Inc.

eywords:Sleep onset; Sleep homeostasis; Recovery sleep; EEG synchronization; Directed transfer function (DTF)

. Introduction

The sleep electroencephalogaphic (EEG) exhibits broadtate-specific and frequency-specific topographical differ-nces along the antero-posterior brain axis across subsequenton-REM episodes, with a marked frontal predominance in

he delta and alpha band[15,16,39]. This anterior predomi-

∗ Corresponding author. Tel.: +39 06 49917647; fax: +39 06 4451667.E-mail address:luigi.degennaro@uniroma1.it (L.D. Gennaro).

nance is even larger in the recovery sleep that follows botal [7,18]and selective sleep deprivation[15]. Regional cerebral blood flow (rCBF), as measured by PET, is also loduring non-REM sleep than during waking[6,24, for a reviewsee 31]and a notable reduction occurs in the frontaltical association area[2,6,16,24]. Sleep deprivation furtheprompts a metabolic decrease of frontal cortices duringensuing wakefulness[13,38]. These electroencephalograpand metabolic changes have been interpreted as a signincreased local use-dependent sleep intensity, which

361-9230/$ – see front matter © 2005 Published by Elsevier Inc.oi:10.1016/j.brainresbull.2004.12.004

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2 L. De Gennaro et al. / Brain Research Bulletin xxx (2005) xxx–xxx

reflect a more intense recovery process of frontal and pre-17

frontal areas[15,18]. Regional changes along the antero-18

posterior axis actually start during the wakefulness–sleep19

(W–S) transition, since frequency-specific and topographi-20

cal EEG changes have already been observed at sleep onset,21

with a general process of anteriorization of slow and middle22

EEG frequencies[10].23

To directly support the hypothesis of anterior-to-posterior24

processes during sleep onset, we recently used an advanced25

computational analysis of EEG data, called the directed trans-26

fer function (DTF), which provides an estimation of the direc-27

tion of the information flow underlying cortico-cortical func-28

tional coupling[19,20,25,27–29]. The DTF has shown that29

the presleep period is characterized by anterior-to-posterior30

functional cortical coupling, while sleep onset is reflected by31

an inversion of that direction[11]. This finding fully sup-32

ports the idea that anterior cortical areas first synchronize33

sleep EEG activity.34

The aim of the present study was to assess the functional35

coupling between anterior and posterior midline cortical ar-36

eas during the sleep onset process, when sleep pressure is37

heightened by a selective slow-wave sleep (SWS) depriva-38

tion. The hypothesis was that the anterior-to-posterior di-39

rection of the cortical functional coupling during the wake-40

to-sleep transition is enhanced by SWS deprivation. This41

leads to two non-alternative predictions regarding changes42

i se: (143

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and ended after 7.5 h of accumulated sleep. The EEG signals67

were recorded (time constant of 0.3 s and low pass filter at68

30 Hz; sampling rate of 128 Hz) from selected sites of the in-69

ternational 10–20 system (C3A2, C4A1, FpzA1, FzA1, CzA1, 70

PzA1, OzA1; electrode impedance lower than 5 k�). Standard 71

electromyographic (EMG) and electrooculographic (EOG)72

derivations were used. 73

Using an acoustic stimulation technique in nights four and74

five enabled the almost complete suppression of selective75

SWS in both deprivation nights. This caused an increase in76

stage 2 duration in both deprivation nights as well as large de-77

creases in EEG power at the frontopolar, central and parietal78

derivations encompassing the delta, theta and alpha range.79

The procedural details and results of the SWS deprivation80

procedure are reported in Ferrara et al.[15]. 81

The data reported here were collected during the second82

and sixth night, respectively, being an undisturbed baseline83

and a recovery night. Left central EEG (C3A2), EMG and hor- 84

izontal and vertical EOG were used to visually score sleep85

stages in 12-s epochs, according to the standard criteria[37]. 86

Due to the variable length of the W–S transition in different87

subjects and to the discontinuity of this process, the individ-88

ual transitional states were made comparable by selecting an89

interval of 5 min (twenty-four 12-s epochs) before (presleep) 90

and 5 min after (sleep onset) the first sleep spindle or K- 91

complex. This interval was subsequently considered for the92

q epoch93

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n DTF values: the increased sleep pressure should cautime advance of the inversion in the direction of ant

osterior functional coupling, which could already beected in presleep wakefulness; and/or (2) an increarontal-to-parietooccipital information flow after sleep ons expressed by augmented DTF values.

. Materials and methods

.1. Subjects

Ten normal right-handed male students [mean age = 2ears (S.E.M. = 0.87)] were recruited as paid volunteerhe study. The protocol of the study was approved byocal Institutional Review Board and was conducted inordance with the Declaration of Helsinki. The subjectsormal sleep duration and schedule, no daytime nap ho excessive daytime sleepiness and no other sleep, mr psychiatric disorders (clinical interview).

.2. Procedure

In the present study, we analyzed data collected in a pus experiment of selective SWS deprivation[15]. The par

icipants slept for six consecutive nights in a sound-premperature-controlled room. Night 1: adaptation; nighndisturbed baseline; night 3: baseline with awakenight 4: SWS deprivation-1; night 5: SWS deprivationight 6: recovery. EEG recordings started at about 11:30

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uantitative analyses of sleep onset, defined as the firstf stage 2 sleep, in order to provide an unequivocal hallm

or the beginning of sleep[9]. Off-line, EEG segments coaminated by instrumental, eye or motion artefacts werected from the periods of interest before the spectral anafter artefact removal, the presleep was 152.4 s long (±22.2.E.), while the sleep onset period lasted 253.8 s (±12.0 S.E.)The 4-s artefact-free EEG segments were analysed

ast Fourier Transform (FFT) algorithm. The spectra fhree consecutive 4-s EEG epochs were then averagedow alignment with the visual scoring of sleep stages base2-s epochs. Power values were calculated across a 1–

requency range with a 0.25-Hz resolution. By computingean values over adjacent frequencies, the data were re

o 1 Hz bin width. The computation was made between Eata at FzA1, PzA1 and OzA1 sites covering the frontal anarietooccipital regions of interest. Indeed, previous studies had reported that these electrode sites can showEG changes during the wake–sleep transition[10,11,22].urthermore, the midline electrodes record EEG oscilla

hat have higher amplitude and thus have the most favouignal-to-noise ratio for the study of antero-posterior fuional coupling.

.3. The directed transfer function method

The DTF method is a computation method based on thalled Mvar model[3,4,25,26,29,34]. This model was usedstimate the “direction” of the information flow betweenEG data at FzA1, PzA1 and OzA1 sites covering the front

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L. De Gennaro et al. / Brain Research Bulletin xxx (2005) xxx–xxx 3

and parietooccipital regions of interest. The mathematical121

core of the algorithm was based on the ARfit programs run-122

ning on the platform (Matlab 5.3, Mathworks Inc., Natrick,123

MA). The model order was seven and the goodness of fit was124

given by the V noise matrix values of the Mvar model. In non-125

mathematical terms, the Mvar model estimates the informa-126

tion flow between electrodes A and B by computing the extent127

to which the EEG data at electrode A can be predicted, based128

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ig. 1. Estimates of the normalized directed transformation function (DTF) aefore and after sleep onset of baseline and recovery nights. To estimatereliminarily normalized, by subtracting the mean value and dividing the resu

he directed transfer function of a multivariate autoregressive (Mvar) modeifferences between fronto-posterior values and posterior-frontal ones; the palues indicate a postero-anterior direction. The upper part of the figure showf the sleep onset period. The left side reports the DTF values calculated on

he Fz–Oz (FzA1–OzA1) pair. Each box contains DTF values calculated acrosecovery night after SWS deprivation = empty circles).

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nd standard error (vertical bars) of data recorded from FzA1, PzA1and OzA1 sitesthe “direction” of the statistically significant EEG coherences, EEG signals werelt by the variance of these signals. On the normalized EEG signals, we computedl in which the order was seven (see Section2). Data are expressed as absoluteositive values denote an antero-posterior direction while, vice-versa, the negatives relative DTF values of the presleep period and the lower part shows DTFvalues

the Fz–Pz (FzA1–PzA1) electrode pair and the right side reports those calculated ons the 1–28 Hz frequency range (1-Hz resolution) (baseline night = filled circles;

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4 L. De Gennaro et al. / Brain Research Bulletin xxx (2005) xxx–xxx

on the EEG data of electrode B and vice-versa. A direction129

of the information flow from A to B is stated when that case130

is statistically more probable than directionality from B to A.131

In previous experiments, the Mvar model was successfully132

used for estimating the “direction” of the cortico-cortical and133

cortico-muscular information flow[3,4,34]. SeeAppendix A,134

for mathematical details of the model.135

It is worth noting an important property of DTF values be-136

tween couple of electrodes. These values are independent of137

the variability of the electrode reference during EEG record-138

ings and of the propagation of brain potentials to distant elec-139

trodes (i.e., head volume conduction effects). Thus, the DTF140

procedure can be applied to EEG data sets regardless of elec-141

trode reference issues and with no preliminary procedures142

for the enhancement of the EEG spatial information content143

(i.e., dipole localization, surface Laplacian, etc.).144

The DTF values were calculated at a 1 Hz frequency145

resolution (seeFig. 1). The EEG frequencies were then146

grouped within the delta/theta (1–7 Hz), alpha (8–11 Hz),147

sigma (12–15 Hz) and beta (16–28 Hz) bands, according to148

the previous EEG study in which they had been empirically149

defined by a Principal Component Analysis that grouped the150

1 Hz bins of EEG power for each antero-posterior scalp loca-151

tion [10]. These empirically defined bands, however, largely152

correspond to the traditional frequency ranges, except for the153

aggregation of delta and theta frequencies. The same pro-154

c tudy155

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The deprivation procedure caused an almost complete sup-178

pression of slow-wave sleep and significant reductions of179

sleep EEG power in the delta–theta–alpha frequency range at180

the frontal, central and parietal derivations. Recovery sleep181

was characterized by a significant stage 4 and SWS rebound,182

by a shortening of stages 3 and 4 latencies, and by a general-183

ized increase of power in non-REM sleep encompassing the184

delta, theta and alpha bands, with a clear anterio-posterior185

gradient. Of note, the frontal derivations showed a similar186

(and the largest) delta and theta power increase. 187

4.2. DTF analysis 188

Fig. 1plots DTF values (±S.E.) for the presleep and sleep189

onset periods of the baseline and recovery nights, at the elec-190

trode pairs concerned (FzA1–PzA1, FzA1–OzA1). DTF val- 191

ues are expressed as absolute differences between the fronto-192

posterior values and the posterior-frontal ones. Hence, posi-193

tive values denote an antero-posterior direction of the func-194

tional cortical coupling, whereas negative values indicate a195

postero-anterior direction. Values close to zero indicate a bal-196

anced directionality of the functional cortical coupling. Dur-197

ing baseline sleep, the sleep onset induced an inversion of the198

information flow direction from parietooccipital-to-frontal to199

frontal-to-parietooccipital as compared to the presleep pe-200

riod. This was true in a large frequency range, the maximum201

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edure had been successfully followed in the previous showing the time evolution of the functional cortical couplong sleep onset in the baseline condition[11].

. Statistical analysis

A repeated measure analysis of variance (ANOVA) eated the differences between the “direction” of the infor

ion flow in the baseline and in the recovery night, as revey the variation of the normalized DTF values. As a depent variable of this ANOVA, the fronto-posterior DTF valu

or each EEG band were subtracted from the posterior-frnes and these absolute differences were calculated fozA1–PzA1 and FzA1–OzA1 electrode pairs. The ANOVA fa

ors were condition (baseline, recovery), time (presleep,nset), band (delta/theta, alpha, sigma and beta) and eleair (FzA1–PzA1, FzA1–OzA1). The Mauchley test evaluat

he sphericity assumption and the Greenhouse–Geissorrected the degrees of freedom when necessary. Thean test was used for post hoc comparisons (p< 0.05).

. Results

.1. Polysomnographic and quantitative-EEG effects ofWS deprivation

These results will be briefly summarized here sinceave been published previously[15].

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eing found at alpha–sigma frequencies. During the recoleep, the same phenomenon was already discernible dhe presleep interval, with no further changes after sleeet. Both nights showed a more marked fronto-posteriort the FzA1–OzA1 pair.

The ANOVA on the DTF values revealed a significwo-way interaction: time (presleep, sleep onset)× electrodeair (FzA1–PzA1, FzA1–OzA1) (F(1,9)= 10.74; MSe = 0.016< 0.01;Fig. 2). This interaction pointed to an increase

he antero-posterior information flow direction (as deny positive DTF values) during the sleep onset as comp

o the presleep interval (p= 0.0004) at the FzA1–OzA1 elec-rode pair. In contrast, the difference between presleepleep onset was not significant at the FzA1–PzA1 electrodeair (p= 0.23). During the sleep onset interval, the two e

rode pairs also showed significant differences in DTF vap= 0.002).

More crucially, the significant two-way interactiF(1,9)= 10.61; MSe = 0.032;p< 0.01) between the conditiobaseline, recovery) and time (presleep, sleep onset) faointed to a different directional pattern between basnd recovery nights (Fig. 3). The interaction was clearly elained by the different direction of the antero-posterior fu

ional coupling before sleep onset. During the presleep ial, the DTF values of the recovery night already shofrontal-to-parietooccipital direction of functional corti

oupling. On the other hand, the DTF values of the basight were characterized by a clear antero-to-posterior d

ion of functional coupling only after sleep onset. In statisterms, the DTF values of the recovery night did not cha

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L. De Gennaro et al. / Brain Research Bulletin xxx (2005) xxx–xxx 5

Fig. 2. Mean relative DTF values and standard error (vertical bars) of thetime (presleep, sleep onset)× electrode pair (FzA1–PzA1, FzA1–OzA1) in-teraction. The data are expressed as absolute differences between fronto-posterior values and posterior-frontal ones, as inFig. 1 (FzA1–PzA1

pair = filled triangles; FzA1–OzA1 pair = empty triangles). At the FzA1–OzA1

electrode pair there was a significant (p= 0.0004) increase of the antero-posterior information flow direction during the sleep onset as compared tothe presleep interval. Moreover, during the sleep onset interval the two elec-trode pairs (FzA1–PzA1, FzA1–OzA1) differed significantly (p= 0.002).

across the two 5-min intervals (p= 0.77), while this differ-232

ence was significant during the baseline night (p= 0.001).233

Furthermore, the recovery and baseline nights did not signif-234

icantly differ during the sleep onset interval (p= 0.92), while235

Fig. 3. Mean relative DTF values and standard error (vertical bars) of thesignificant (p< 0.01) condition (baseline, recovery)× time (presleep, sleeponset) interaction. While during the baseline night there was a significantdifference (p= 0.001) between presleep and sleep onset in the direction ofthe functional coupling, this difference disappeared in the recovery night,w ep pe-r sleepi tweenfn

their difference was significant during the presleep interval236

(p= 0.001). 237

These effects also changed as a function of the EEG238

frequency band. In fact, the ANOVA revealed a significant239

three-way interaction (F(3,27)= 3.26; MSe=0.0078;p< 0.05) 240

between the condition (baseline, recovery), time (presleep,241

sleep onset) and band (delta/theta, alpha, sigma, beta) fac-242

tors (Fig. 4). For the baseline night,Fig. 4 shows a promi- 243

nent inversion of direction in the delta/theta (p< 0.0001) and 244

alpha (p< 0.0001) bands during the W–S transition (i.e.,245

comparing the presleep versus sleep onset intervals) from246

postero-anterior (negative values) to antero-posterior (posi-247

tive values). Instead, the sigma (p< 0.003) and beta (p< 0.01) 248

bands showed a significant prevalence of the frontal-to-249

parietooccipital direction only after sleep onset, i.e., values250

close to zero change to positive ones. During the recovery251

night, the delta/theta (p< 0.02) band only showed a signif-252

icant prevalence of the frontal-to-parietooccipital direction253

after sleep onset as compared to the presleep interval (again,254

values close to zero change to positive ones). Moreover, the255

difference between the presleep values of the baseline and256

recovery nights was significant (p= 0.02). 257

The positive presleep DTF values of the alpha, sigma and258

beta bands indicate that the direction of information flow259

does not change during the recovery night, being frontal-to-260

parietooccipital across the whole W–S transition. Of note,261

t ween262

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Ohen the antero-posterior coupling was already present in the presleiod. Moreover, the recovery and baseline nights differed only in the prenterval (p= 0.001). The data are expressed as absolute differences beronto-posterior values and posterior-frontal ones, as inFig. 1 (baseline

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he presleep period showed significant differences bethe baseline and recovery flow direction in all the EEG badelta/theta:p= 0.02; alpha:p= 0.00002; sigma:p= 0.0003eta:p= 0.00003).

. Discussion

The present study using the DTF technique showedaseline and recovery sleep are characterized by differen

erns of functional cortical coupling during the W–S tranion. In fact, in the recovery night following SWS deprivate found that the shift to an anterior-to-posterior directlity of functional cortical coupling is already present durresleep wakefulness.

We recently demonstrated that a similar shift does indccur also in baseline sleep, but only after sleep onsett the beginning of stage 2)[11]. Given that the recoveleep following total[7,18] and selective sleep deprivati15] is characterized by an increased frontal predominn the 1–8 Hz range, we hypothesized that the heightleep pressure should also affect the antero-posteriorectivity at sleep onset. More specifically, it was poss

o put forward two non-alternative hypotheses: the increomeostatic sleep pressure could have led to, (1) the ant

o-posterior functional coupling being brought forward dng the presleep period and/or (2) an increased anterioosterior functional coupling during the sleep onset per

The results were mainly in favour of the first hypothes a matter of fact, there was a clear posterior-to-anterio

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6 L. De Gennaro et al. / Brain Research Bulletin xxx (2005) xxx–xxx

Fig. 4. Mean relative DTF values (positive values = antero-posterior direction; negative values = postero-anterior direction) and standard error (vertical bars) ofthe condition (baseline, recovery)× time (presleep, sleep onset)× band (delta/theta, alpha, sigma, beta) interaction (p< 0.05). Each box reports the relative DTFvalues calculated across the delta/theta (1–7 Hz), alpha (8–11 Hz), sigma (12–15 Hz) and beta (16–28 Hz) frequency bands, and represents a different patternof information flow direction between the baseline and recovery nights (baseline night = filled circles; recovery night after SWS deprivation = empty circles).In the baseline night, there was an inversion of direction in the delta/theta (p< 0.0001) and alpha (p< 0.0001) bands during the wake–sleep transition frompostero-anterior (negative values) to antero-posterior (positive values). The sigma (p< 0.003) and beta (p< 0.01) bands only showed a significant prevalenceof the frontal-to-parietooccipital direction after sleep onset. During the recovery night, the delta/theta band showed a significant (p< 0.02) prevalence of thefrontal-to-parietooccipital direction after sleep onset. Moreover, the difference between the presleep values of the baseline and recovery nights was significant(p= 0.02). Notably, the presleep period showed significant differences between the baseline and recovery flow direction in all EEG bands (delta/theta:p= 0.02;alpha:p= 0.00002; sigma:p= 0.0003; beta:p= 0.00003). For further details, see text.

rection of the functional cortical coupling during the presleep289

period, with no further change after sleep onset (Fig. 3).290

This phenomenon encompassed the 8–28 Hz EEG frequency291

range and was less evident in the slower EEG frequency range292

(Fig. 4). In the baseline condition, the delta–theta presleep293

values of the DTF showed a prevalent postero-anterior di-294

rection of the functional cortical coupling. This directional-295

ity was significantly less marked in the recovery condition.296

During the sleep onset period, the frontal-to-parietooccipital297

direction of the functional cortical coupling was comparable298

in the recovery and in the baseline conditions (Fig. 4).299

The present results are in line with previous evidence300

showing that EEG changes during recovery sleep may pre-301

cede sleep onset. It has been reported that changes of sleep302

EEG slow-wave activity (SWA) in response to sleep manipu-303

lations concern not only its overall trend, i.e., power density304

values across non-REM sleep episodes, but also its build-up305

within non-REM episodes[1,5,12,21,30]. Total sleep depri-306

vation also induced a more pronounced rise in SWA in the307

frontal areas as compared to parietal and occipital ones dur-308

ing the first 36 min after sleep onset[7]. Finally, significant 309

differences in the intercept of the curves of time courses of310

SWA during the first 40 min after sleep onset in the post-311

deprivation versus baseline nights have been reported. Dur-312

ing the recovery sleep, the SWA curve started from a higher313

level compared to the baseline due to an increased slow-wave314

sleep pressure[15]. These results highlight that recovery pro-315

cesses following manipulations of sleep homeostasis affect316

both the amount and the build-up of SWA. They could also be317

related to the well-known EEG changes characterizing pro-318

longed wakefulness, i.e., the increase of theta power mainly319

on the anterior derivations[7,17]. Moreover, the present re- 320

sults indicate that, during the presleep onset period, bringing321

forward the anterior-to-posterior direction of the functional322

coupling concerned not only the delta and theta EEG fre-323

quencies previously affected by SWS deprivation, but also324

the higher frequencies. This was not surprising, given that325

total sleep deprivation and selective SWS deprivation affects326

the power of a large range of frequencies (up to 10 Hz) during327

the recovery night[15,18]. 328

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Previous studies[11,25] interpreted the occipital-frontal329

propagation during the presleep interval as reflecting the in-330

formation flow of the occipitofrontal and superior longitudi-331

nal fasciculi along postero-anterior projections, sub-serving332

a passive bottom-up transfer of information from visual sen-333

sory to prefrontal association areas[11,27]. Recent high-334

density (256 channels) EEG recordings during sleep onset335

provide an independent support to this hypothesis, since slow336

(<1 Hz) EEG oscillations behave as a travelling wave orig-337

inating more frequently in prefrontal-orbitofrontal regions338

and propagating in an anteroposterior direction, putatively339

reflecting cortico-cortical connectivity[32]. The present find-340

ings are in line with the above-mentioned results, supporting341

the notion that, in the recovery condition, a spread of syn-342

chronizing “top–down” signals from associative prefrontal343

to posterior areas play a role in the wake–sleep transition.344

Finally, it may be speculated that the anterior-posterior in-345

formation flow could explain some formal and sensorial fea-346

tures of hypnagogic imagery during these transitional states347

[23] as well as the behavioural worsening or lapses during the348

normal W–S transition[e.g., 8]and during sleep deprivation349

[e.g., 14].350

Uncited references351

352

A353

he354

d ision.355

A356

357

∑358

w359

c r360

L sti-361

m var362

m e363

z364

X365

w as366

c367

H368

and369

z−i = exp(−i2πfdt) 370

To transform it to the frequency domain, thez-transformed 371

Mvar model was formulated as372

X(f ) = H(f )E(f ) 373

This formulation of the Mvar model allowed the computa-374

tion of the spectral matrixS(f) of the EEG signal. This was 375

performed by 376

S(f ) = X(f )X(f )∗ = H(f )VH(f )∗ 377

whereV is the residual white noise. TheH(f) matrix of the 378

Mvar model served to estimate the DTF. The DTF from the379

‘ jth’ to ‘ ith’ EEG channel was defined as the square of the380

element of theH(f) matrix divided by the squared sum of all381

the elements belonging to its ‘ith’ row. This was obtained by 382383

DTFij(f ) =∣∣Hij

∣∣2∑L

m=1|Him(f )|2 384

where ‘i’ is the row of theH(f) matrix and ‘j’ and ‘m’ are its 385

columns. As a result, the DTF is a normalized value ranging386

from zero to one. In the present case, the difference between387

D r-388

m ly 389

t 390

R 391

vity392

pine393

394

ilja,395

e, B.396

lized397

. 18398

399

ci, 400

ini,401

wer402

ehav.403

404

la,405

con-406

tion407

7. 408

wave409

they410

1995)411

412

rga,413

ebral414

tudy,415

416

tive417

ss in418

419

UN

CO

RR

EC

T

[33,35,36].

cknowledgements

Thanks to Dr. Fabrizio Anitori for his contribution to tata analysis and to Frank Amodeo for the language rev

ppendix A

The Mvar model was defined asp

j=0

AjXt−j = Et

hereXt is the L-dimensional vector representing theL-hannel signal at timet, Et is white noise andAj stands fo×L matrices of the model coefficients. In order to eate the spectral properties of the EEG signal, this Model was preliminarily subjected to az-transformation. Th

-transformed Mvar model was expressed by

(z) = H(z)E(z)

hereH(z) is the transfer function of the system, which walculated by

(z) =

p∑j=0

Ajz−j

−1

PR

O

BRB 6928 1–8

TFij (f) and DTFji (f) was an index of an asymmetric infoation flow between the ‘jth’ and ‘ith’ EEG channels, name

he “direction” of this flow.

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