Uiveideg

Laurea Spe ialisti ainS ienze Fisi he

a.a. 2004/2005

Chara terization of epilepti EEG's

via statisti al me hani s tools

CANDIDATO RELATORE

1 Some general informations 7

1.1 MembranePotential . . . 10

1.2 EEG . . . 13

1.3 Signal hara teristi s . . . 15

2 Inferen ing an embedded dynami s 19

2.1 Fromtoymodels... . . 19

2.2 ...toreal EEG's . . . 22

3 Diusion Entropy analysis 27

Strips . . . 31

Max-Min ounting . . . 39

4 Correlations and information 45

Minimal SpanningTree . . . 49

Transmittedinformation . . . 53

5 Epilepti EEG's 57

5.1 FirstEEG . . . 57

5.2 Se ondEEG. . . 68

6 Con lusion 73

A Bayesian Inferen e Algorithm 75

a knowledgments

The work of this thesis has been possible thanks to the pre ious

ollabora-tionofthegroupof"StellaMaris"instituteofCalambrone(LI,Italy),ledby

Prof. RenzoGuerrini. Inparti ular,manythankstoDr. Lu ioParmeggiani

who spent a long time withus, to explain thebasi sof epilepsy,to retrieve

suitable materialandto interpret theEEGfroma medi al point of view.

When this work started, when we had the rst look at an EEG re ord,

it was lear what themain goalwould have been. There was a single aim:

to getapredi tion of theepilepti seizure.

The epilepsy,inall its manifestations, ai tsabout 1% ofthe entire

popu-lation. Thisdisturbai tsoneormoreportionsofthebrainwhi h,insome

ir umstan es, un ontrollably release energy spiking on the whole erebral

mass. This disease is parti ularly serious in hildhood; in fa t, in the less

worrying ases itispossibleto liveputting upwithit(mostlywith

pharma- euti al help), but often early manifestations ause irreversible damages to

the entral nervous system, with devastating onsequen es for the learning

andthe physi al growing.

Thisphenomena is well know inits fun tional manifestation and oftenit is

possible toindividuateepileptogeni zoneinsidethebrainandtry asurgery

operation. However singleepilepti seizures are not predi table and hardly

revelablewith an automati readout. In fa t, with a hypotheti automati

devi ealertingmedi alstajustsome minutesbeforetheepilepti seizureit

wouldbepossible to inje tthesuitable pharma euti al dose.

Afun tionalmodeldes ribingthephenomenadoesnotexistsneitherwenow

iftheEEGsignalisdrivenbysomedynami alsystemregulatingthe erebral

networks a tivation. In this work we (tried to) set up one or more reading

systemindi ating presen eofsome hara terizable a tivity.

Itis di ultto retrieveEEG re ordsfromthepubli domain, andwe ould

use some of them obtained thank to the pre ious ollaboration of "Stella

EEG's:

1. a non-epilepti one, nearly 1 hour long, of a healthy 18 months old

baby,tohaveazeroreferen eonhowstatisti alme hani s toolswork.

2. an epilepti EEG, more than 1 hour long, of a 6 aged hild, withan

evident epilepti seizure, nearly 30 se onds long. It is a quite rare

re ord be auseitis aregistration withanevent thato ursrandomly

and every 2 months, inaverage.

3. another epilepti EEG, 48 hours long, of a 11 years old hild, with8

seizures. The hild hasan epilepsydiseasedueto abrainlesioninthe

o ipital partof theright hemisphere.

Westart dis ussinga seriesof possible instrumentsand weapply them rst

tothenormalEEG.Attheend,everyinstrumentsdis ussedwillbeusedon

pathologi alEEG's; inthislast part, ifanydiagnosti interrestingelements

exists, it will show up. Some words have to be spent on what the modus

operandi was adopted. "If you need to see it, you will see it": knowing

a priori where the epilepti seizures are during the EEG stream it would

have been dangerous from thepoint of viewof s ienti method. We ould

have taken the risk to spe ialize the te hniques to make them t our

ex-pe tations. To avoid the possibility of this psy hologi al risk we asked the

medi alstanottogiveusanyinformationonwhereseizureshaveo urred.

Indetails we trythe following ways:

•

Wehavetriedanansatz onthedrivingdynami oftheele tri a tivity; its oe ientshasbeeninferredwithaBayesianmethod,withthehope

of extra ting,fromthe oe ient's hanging, someinformation onthe

subground state.

•

We have developed dierent te hniques to extra t (either from indi-vidual signal, or from intera tions of them) sto hasti diusion

pro- esses. Thesepro esseshavebeenstudiedintheirstatisti alproperties,

s aling, et ...

•

Wehaveanalyzedinformationpropertiesbetweensignalssu has trans-mitted entropy,and minimalspanning tree.

Some general informations

The EEG is the most informative lab test in order to have a diagnosis of

epilepsyand to lassify thetype, theseizure and the syndrome of epilepsy.

TheEEG measuresthe dieren e inele tri al potential between twopoints

on the surfa e of the head. It tra esthe voltage u tuation re orded from

ele trodespla edoverthes alpinaspe i manner. Inthiswayu tuating

ele tri al potentials inmembranes of neuronsare represented.

The Central Nervous System (CNS): The neurons

The basi ells in the CNS are theneurons, spe ialized in ondu ting

ele -tro hemi al impulses. The neuron an re eive, elaborate and transmit

in-formationstothe adja ent ellsbya tionpotentials(spikes). Theyhave ell

bodiesthathousenu lei,fromwhi hdendritesandaxonsdepart. Dendrites

areshort bran h bers,in whi h nerve impulses aregenerated; they arethe

main re eiving bodies in the ell. The nerve impulses are then ondu ted

alongtheaxon.

Axons' fun tion is to transmit nervous impulses, that an run at a speed

up to 100 m/se . The length of some axons (that usually bran hes several

times lose to their end) is sogreat thatit is di ult to see how ell body

an ontrol them.

Manyaxonsare overedwithaglisteningfattysheath,themyelinsheath. It

isthegreatly-expandedplasmamembraneofana essory ell,theS hwann

ell. S hwann ells are spa ed at regular intervals along the axon. Their

plasmamembraneiswrappedaround andaround theaxonformingthe

my-elin sheath. Where the sheath of one S hwann ell meets that of the next,

the axonisunprote ted.

Thisregion,thenodeofRanvier,playsanimportantpartinthepropagation

ofthe nerve impulse. Thejun tion between theaxon terminalsof a neuron

andthere eiving ell is alledasynapse.

Ea haxonterminalisswollenformingasynapti knob. Thesynapti knobis

lled with membrane-bounded vesi les ontaining a neurotransmitter. The

neurotransmitter at ex itatory synapses depolarizes thepostsynapti

mem-brane As to the number of extensions originating from their ell bodies,

neurons an be lassied into:

•

Bipolarneurons,fromwhoseegg-shapedbodyoriginateadendriteand an axonof equal length.

•

Pseudounipolar neurons, that ontain a long dendrite dividing itself into two bran hes, and a smallaxon that onne ts to thespinal ord.

The impulse runsintwo oppositedire tions, to and fromthebody.

•

Multipolar neurons, that have a large number of dendrites and one axon. The impulseruns inmanydierent dire tions.

Thesignalis ondu tedthroughthenerve(dendrite)to thedorsal root

gan-glion( ellbody), thenthroughthe dorsalroot (axon) endingat thesensory

nu lei intheposterior horn ofthespinal ord.

Themaintypeofneuronslo atedinthe erebral ortexarethepyramidal

ells, (ex itatoryneurons so alled be ause of theshape of their ell body),

andthe non pyramidal ells.

Pyramidal ells onstitutethemainpro essingpowerofthe ortex. They

have large dendritestrees withtypi allyfew thousandssynapses. Theyalso

have axons endingindierent regionsofthe brainor inthespinalmedulla,

withsimilar numberof synapses. Their api al dendrites go through several

orti al layers and are always perpendi ularly-oriented towards the orti al

surfa e. The pyramidal ells generate "open" elds that an add together

andberegistered bythe EEG.

Non pyramidal ells are small, star-shaped ells, that have short axons

Bioele tri al Signals

The ele tri al a tivity of the brain is measured by the ele tri al

poten-tials. Theyareprodu edbyex itatoryor inhibitorypost-synapsepotentials

(EPSP or IPSP), that, if reated in qui k su ession, add together

("sum-mation")and generate a tion potentials.

The Resting Potential

All ells(notjustex itable ells)havearestingpotential: anele tri al harge

a rossthe plasmamembrane, with theinterior of the ell negative with

re-spe ttotheexterior. Thesizeoftherestingpotentialvaries,butinex itable

ellsrunsabout -70mV.

Certain external stimuli redu e the harge a ross the plasma membrane.

Me hani al stimuli (e.g., stret hing, sound waves) a tivate

me hani ally-gated sodium hannels, ertain neurotransmitters (e.g., a etyl holine) open

ligand-gated sodium hannels. In ea h ase, the fa ilitated diusion of

so-dium into the ell redu es the resting potential at that spot on the ell

reating an ex itatory postsynapti potential or EPSP. If the potential is

redu edtothe thresholdvoltage (about -50mV inmammalianneurons), an

a tion potential is generated inthe ell.

A tion Potentials (nerve impulse)

Intherestingneuron,theinterioroftheaxonmembraneisnegatively harged

withrespe tto theexterior(Aing.1.2). Asthea tion potential passes(B

ing.1.2), the polarityis reversed. Thentheoutow ofK

+

ions qui kly

re-storesnormalpolarity(Cing.1.2). Attheinstantpi tured inthediagram,

themoving spot,whi h has tra ed these hanges on theos illos ope asthe

depolarization at a spot on the ell rea hes the threshold voltage, the

re-du ed voltage now opens up hundreds of voltage-gated sodium hannels in

thatportionoftheplasmamembrane. Duringthemillise ondthatthe

han-nels remain open, some 7000 Na

+

rush into the ell. The sudden omplete

depolarization ofthe membraneopensup moreofthevoltage-gated sodium

hannelsinadja entportionsofthemembrane. Inthisway,awaveof

depol-arizationsweepsalongthe ell. Thisisthea tion potential, thatinneurons

isalso alledthe nerve impulse.

The refra tory period

Ase ondstimulusappliedtoaneuron(ormus leber)lessthan0.001se ond

aftertherstwillnottriggeranotherimpulse. Themembraneisdepolarized

(position B of g.1.2), and theneuron is inits refra tory period. Not until

the -70 mV polarity is reestablished (position C of g.1.2) will the neuron

be ready to re again. Repolarization is rst established by the fa ilitated

diusion of potassium ions out of the ell. Only when theneuron is nally

restedarethesodiumionsthat ameinatea himpulsea tivelytransported

ba k out of the ell. In some human neurons, the refra tory period lasts

only 0.001-0.002 se onds. This means that the neuron an transmit

500-1000 impulsesperse ond.

The a tion potential isall-or-none

The strength of the a tion potential is an intrinsi property of the ell. So

long asthey an rea h the threshold of the ell, strong stimuli produ e no

stronger a tion potentials than weak ones. However, the strength of the

stimulusisen odedinthefrequen yofthea tionpotentialsthatitgenerates.

Integrating Signals

A single neuron, espe ially one in the entral nervous system, may have

thousandsofotherneuronssynapsingonit. Someofthesereleasea tivating

(depolarizing)neurotransmitters; othersreleaseinhibitory(hyperpolarizing)

neurotransmitters. The re eiving ell is ableto integrate thesesignals. The

diagram shows how this works in a motor neuron. The a tion potential is

usually generated in the axon hillo k. Having neitherex itatory nor

inhib-itory synapses of its own, it is able to evaluate the total pi ture of EPSPs

andIPSPs reated inthedendrites and ell body.

1.1 Membrane Potential

om-areimportantfortransferringinformation overlongdistan esrapidlywithin

theneuron. Chemi alsignals,ontheotherhand,aremainlyinvolved inthe

transmissionofinformationbetweenneurons. Ele tri alsignals(re eptor

po-tential, synapti potential and a tion potential) are all aused by transient

hangesinthe urrent owinto and out oftheneuron, thatdrivesthe

ele -tri al potential a ross the plasma membrane away of its resting ondition.

Every neuronhasaseparation ofele tri al hargea rossits ell membrane.

The membrane potential results from a separation of positive and negative

harges a ross the ell membrane. The relative ex ess of positive harges

outside and negative harges inside themembrane of a nerve ell at rest is

maintained be ausethelipidbilayera tsasabarriertothediusionofions,

and give rise to an ele tri al potential dieren e, whi h ranges from about

60to70mV.Thepotentiala rossthemembranewhenthe ellisatrest(i.e.

whenthere isno signaling a tivity)is knownasthe resting potential. Sin e

,by onvention, thepotential outsidethe ell is arbitrarilydened aszero,

and given the relative ex ess of negative harges inside themembrane; the

potential dieren e a ross the membrane is expressed as a negative value:

V

r

=-60 to -70mV,being

V

r

,the restingpotential. The harge separation a rossthe membrane, andtherefore the resting membrane potential, is

dis-turbedwheneverthereisanetuxofionsintooroutofthe ell. Aredu tion

ofthe hargeseparationis alleddepolarization;anin reasein harge

separ-ationis alledhyperpolarization. Transient urrent ow andthereforerapid

hangesinpotentialaremadepossiblebyion hannel, a lassofintegral

pro-teins that traverse the ell membrane. There are two types of ion hannel

inthemembrane: gated andnongated. Nongated hannels arealways open

and arenot inuen ed signi antly byextrinsi fa tors. Theyareprimarily

important inmaintaining theresting membrane potential. Gated hannels,

in ontrast, open and lose in response to spe i ele tri al, me hani al, or

hemi alsignals. Sin eion hannelsre ognizeandsele tamongspe i ions,

thea tualdistribution ofioni spe iesa rossthemembranedependsonthe

parti ular distribution ofion hannels inthe ell membrane.

Ioni spe ies are not distributed equally on the two sides of a nerve

membrane. Na and Cl are more on entrated outsidethe ell while K and

organi anions (organi a ids and proteins) are more on entrated inside.

Theoverallee t ofthisioni distributionisthe restingpotential. However,

whatpreventsthe ioni gradients frombeingdissipatedbypassive diusion

ofions a rossthe membranethrough thepassive nongated hannels.

There aretwo for esa tingonagivenioni spe ies. Thedrivingfor eof

the hemi al on entration gradient tends to move ions down this gradient

( hemi al potential). On the other hand the ele trostati for e due to the

harge separation a ross the membrane tends to move ions in a dire tion

determined by itsparti ular harge. Thus, for instan e, hlorideions whi h

negative harge inside the membrane tend to push hloride ions ba k out

of the ell. Eventually equilibrium an be rea hed so that thea tual ratio

of intra ellular and extra ellular on entration ultimately depends on the

existingmembranepotential.

The same argument applies to the potassium ions. However these two

for es a t together on ea h Na ion to drive it into the ell. First, Na is

more on entrated outside than inside and therefore tends to ow into the

ell down its on entration gradient. Se ond, Na is driven into the ell by

theele tri alpotentialdieren ea rossthemembrane. Therefore,ifthe ell

hasasteadyrestingmembrane potential, themovement of Naions into the

ell will be balan ed by the eux of K ions. Although these steady ioni

inter hange an prevent irreversible depolarization, this pro ess annot be

allowed to ontinue unopposed. Otherwise, the K pool would be depleted,

intra ellularNawouldin rease, andtheioni gradientswouldgraduallyrun

down, redu ing theresting membrane potential.

Dissipation of ioni gradients is ultimately prevented by Na-K pumps,

whi hextrudesNafromthe ellwhiletakingKin. Be ausethepumpmoves

Na and Kagainst their netele tro hemi al gradients, energy is required to

drive thesea tive transported uxes. The energy ne essaryfor this pro ess

is obtained from the hydrolysis of ATP (an energy arrying mole ule). In

addition,some ellsalsohave hloridepumpsthata tivelytransport hloride

ions toward the outside so that the ratio of extra ellular to intra ellular

on entration ofCl is greater than theratio that wouldresult from passive

1.2 EEG

AnEEGisasetofele tri signals olle tedfromele trodesinthenearbound

ofbrain. Cathodesarepla ed externally of the raniumof patient ina non

invasive manner. In normal EEG's, like the ones analyzed here, there are

usually19 athodesxedto thesurfa e ofthehead,nearlyat thesame

dis-tan e from ea h other; in this waywe olle t a grid of ele tri potential on

the surfa e of the ranial alotte. An example of geographi pla ement of

ele trodes isgiven ing.1.2.

Figure 1.3: One of the most standard s heme of ele trode pla ementof ranial

surfa eofthepatient.[12℄

Signals from hatodes are olle ted and preamplied, amplied and

re or-ded on a magneti tape. An audiovisual aid from the patient is olle ted

syn hronouslyinthesame tape;inthiswaymedi alsta isableto markon

numberstreamwhi hrepresents,atea hele trode,apotentialvaluedin

µ

V. Signalsfromele trodesarere orded against anarbitraryreferen e, and the

omputer interfa e will present the signal in any desired montages. There

aresometypi almontageused: a ommon referen esderivation,where

ele -tri dieren esofea hele trodeagainstasele ted oneofthem ispresented;

an averaged ommon referen es, where the referen e is a virtual ele trode

omposedbyanaverageofalltheothers;thereistheso alledbipolar

deriv-ation,theoneuseinthe work, wheredieren es between nearestele trodes

are al ulated. We have used bipolar derivation mainly in order to avoid

asmu h aspossibleannoying artefa ts due toheart a tivityand mus lesof

orbitalmovements. Neurologistsusedierentmontagesofbipolarderivation

(dependingonwhattheyareinterestedin); themostusedandgeneraloneis

dieren esbetween onse utivelongitudinalele trodes, onventionally alled

longitudinalmontage.

Figure 1.4: On theleft: main s heme of longitudinalmontage;yellowpoints are

a tive ele trodes, arrows onne ting ea h point indi ate the potential dieren e

extra ted from that points; ea h arrowis asignal of the EEG. On the right: an

1.3 Signal hara teristi s

Signals looked at from a non-expert eyes seem very similar to ea h other,

ex eptfor ampli ation: the potentials olle ted in the ba k of the head is

10%-30%bigger than thefront ones.

Amplitude

Theele tri potential dieren esarezero-averagesignals,ranging from

±50

µ

Vduringthesleepingtimeandlittlebigger,

±60 µ

V,duringthewaketime. Thebiggestpotentialdieren esarere ordedduringmus ulara tivity,when

theymight rea h

∼ 400 µ

V.

-200

-150

-100

-50

0

50

100

150

200

(

µ

V)

time (sec)

-200

-150

-100

-50

0

50

100

150

200

(

µ

V)

Figure 1.5: Fromtop: the1 st

signalis asleeping-timere ord; the se ond oneis

awake-timere ord, mus ulara tivityis evident. Both ofthem arefrom o ipital

regionwheredieren esfromsleepingandwakefulnessmightbesimplerinreading.

The frequen ies domain

All theEEGre ords presented aresampledat 200 Hz, sotheNyquistlimit

of the readable spe trum is 100 Hz; this is not a limitation sin e themost

interresting part of the spe trum is far below 50 Hz. At high frequen ies,

in fa t, what is re orded is not brain a tivity but mainly dierent sour es

of noise. In medi al literature dierent rhythms are well know in dierent

0

1

2

3

4

5

6

7

8

9

10

amplitude (a.s.)

frequency (Hz)

Figure 1.6: A Fourierspe trumevaluatedduring anhourlyexample ofEEG

re- ording.Thepatientisa7years hildrenduringamixedsleepingandwakefulness

time. Thespe trum omes fromasignallteredwithahighpassinglterof.5Hz

andalowpassinglterof70Hz.

- Theta: 4.4-8Hz. Typi alin hildhood andadoles en e.

- Alpha: 8.5-12Hz. Chara teristi ofarelaxed,alertstateof

ons ious-ness.

- Beta: over 12 Hz. It is asso iated with a tive on entration or busy

thinking.

- Gamma: over 40 Hz. This rhythm might appear in higher mental

a tivity,in ludingper eptionand ons iousness.

In g.1.6 a typi al Fourier spe trum is presented averaged over 1 hour of

De orrelation time

From adynami alpoint ofview itisinteresting to al ulate the

auto orrel-ationfun tionofthesignalsevaluatingthede orrelationtime. The

auto or-relation fun tionis simply

C(τ ) =

hx(t + τ ) x(t)i

t

hx(t + τ )i

t

hx(t)i

t

(1.1)

Ingeneralitgivesarude quanti ationofthememoryofthedynami al

sys-tem. Ing.1.7 the auto orrelationfun tionsofseveralsignalsarepresented.

All ofthem areanti orrelated after

∼

0.4.

0

0.5

1

1.5

2

2.5

3

C(t) (.1)

time (sec)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

5

6

C(t)

time (sec)

Figure1.7: Auto orrelationfun tionofsomesignalsfromthesameEEG.Onthe

left: auto orrelation of 8signals from the right hemisphere during sleeping time.

Ontheright: auto orrelationof8signalsfromtherighthemisphereduringanalert

Inferen ing an embedded

dynami s

2.1 From toy models...

Weknowfrommedi alliteraturethatagreatnumberofbiologi alsignals an

be hara terized byadynami alsysteminterms ofa fewsimple equations.

It has be shown from authors of [4℄, even in ase of haoti ma ros opi

signals (the blood pressure signal) thatit might be possible to put forward

a valid model and test it. We know better and better how a single neuron

workandfrombio hemistryweknowtheequationsregulatingthemembrane

voltage readout. From a dynami al point ofview, the rst question we are

going to address in an EEG is: does it exist a set of simple equations that

drivethesystemdynami sfromanembeddedlayer overedbynoise? Letus

onsiderea h ele trode signal as oming fromthespa ial degree offreedom

ofan os illator. Insu h awaywe an imagine theentire setof

N

signals of EEGasa

N

-os illatorssystem:



































˙

X

1

(t)

= F

1

X

1

, ...X

N

(t)
+ σ

1

ξ

1

(t)

. . .

˙

X

i

(t)

= F

i

X

1

, ...X

N

(t)
+ σ

i

ξ

i

(t)

. . .

˙

X

N

(t) = F

N

X

1

, ...X

N

(t)
+ σ

N

ξ

N

(t)

(2.1) or inshorter:

˙

X(t) = f X(t)
+ σξ

i

(t)

(2.2)

In equations 2.2 we are for ed to take ina ount multiple sour es of noise.

In ase ofadditive stationarywhitenoise:

The fun tions

f

are unknown and possibly neither linear nor simple. F ol-lowingthe authorsof[5℄,we supposethatat xedtimes

{t

k

; k = 0, 1, ...}

we readaposition

Y = {y

k

≡ y(t

k

)}

. Thispositionisareadoutoftheunknown state

X = {x

k

≡ x(t

k

)}

whi h is to be inferred, and it is generated from a probability density fun tion

p

0

(Y|X )

. A blo k of data

Y

give us some information about thefun tions

f

,thatis, alling

M

thesetof parameters that ontrols

f

,and

p

pr

(M)

beingthe

prior

probabilityof

M

,itgivesanew probability inspa eof parameter(Bayestheorem):

p

post

(M|Y) =

L(Y|M)p

pr

(M)

R L(Y|M)p

pr

(M)dM

(2.3)

Clearly

L(Y|M)

isthe

likelihood

fun tion. In[4℄isshownhowto paramet-erize the unknown ve tor eld

f

(x)

to onstru t an algorithm to infer the bestparameterset

M

. InappendixAthe ompletealgorithm isshown. We parameterize the nonlinear ve tor eld

f

(x)

to make it linear withrespe t to a parameters ve tor

c

. We hoose an appropriate set of base fun tions

φ

b

(x)

insu h awaythat

f

(x)

maybe written as:

f

(x) = ˆ

U(x)c ≡ f (x; c)

(2.4)

where

ˆ

U

is onstitutedby

B

diagonal blo ksofsize

N × N

and

M = BN

is the numberofparameters to be inferredfromthedynami :

ˆ

U

=





















φ

1

0

. . .

0

0

φ

1

. . .

0

. . . . . . . . . . . .

0

0

. . . φ

1





















φ

2

0

. . .

0

0

φ

2

. . .

0

. . . . . . . . . . . .

0

0

. . . φ

2











. . .











φ

B

0

. . .

0

0

φ

B

. . .

0

. . . . . . . . . . . .

0

0

. . . φ

B





















To give an idea on some examples, tests on toy models will be presented.

Letus onsider5 noisy oupledVan DerPolos illators:

˙x

i

= y

i

˙y

i

= m

ij

y

j

+ α

ij

x

j

+ ǫ

i

x

2

i

y

i

+ η

i

(t)

(2.5)

withwhitegaussiannoises:

hη

i

(t)i = 0

hη

i

(t)η

j

(t

′

)i = δ(t − t

′

)δ

ij

D

ˆ

We use this as a model for the simulation be ause is the one we will use

for real EEG, as we will see innext se tion. The model is ontrolled by25

parameter oe ients for thematrix

m

,25 parameters for

α

, 5 parameters for the ve tor

ǫ

and 25 more oe ients for the diusion matrix

ˆ

D

. The

parameterizationof (2.4) now reads:

φ

1

= y

1

,

φ

2

= y

2

,

φ

3

= y

3

,

φ

4

= y

4

,

φ

5

= y

5

,

φ

6

= x

1

,

φ

7

= x

2

,

φ

8

= x

3

,

φ

9

= x

4

,

φ

10

= x

5

,

φ

11

= x

2

1

y

1

,

φ

12

= x

2

2

y

2

,

φ

13

= x

2

3

y

3

,

φ

14

= x

2

4

y

4

,

φ

15

= x

2

5

y

5

(2.6)

-15

-10

-5

0

5

10

15

-4

-3

-2

-1

0

1

2

3

4

y

x

0

1

2

3

4

5

6

7

8

9

spectrum a.s.

frequency (Hz)

Figure2.1: AnexampleofaVanDerPolos illator ouppledwithother4os illator

whose dinami s is des ribedin eq.(2.5). The oe ientsof the model areshown

in table (2.1), the rst line. Despite the white gaussian noise whi h makes the

spe trum (right) quite high in a large window of frequen ies, the hara teristi

limit y leof theVanDerPolisstillvisible inthephase-spa e(left).

Withthis hoi e of fun tions

φ

k

the

c

oe ient ve tor isgiven by:

c

T

≡



m

11

, m

21

, . . . , m

51

, m

12

, m

22

, . . . , . . . , m

55

, α

11

, α

21

, . . .

, α

51

, α

12

, α

22

, . . . , . . . , α

55

, ǫ

1

, ǫ

2

, . . . , ǫ

5

,



(2.7)

Wethenrunasimulationwiththearbitrary oe ientsshownintable(2.1).

Ing.2.1 anexampleofaphasespa eandofaspe trumareplotted

respe t-ively. After the generation of

10000−

steps long stream of signals we use it to feedthe algorithm presentedinappendixA. The resultsofinferred

m

i1

m

i2

m

i3

m

i4

m

i5

α

i1

α

i2

α

i3

α

i4

α

i5

ǫ

i

1

st 0.84 -0.08 0.02 0.01 0.21 -0.08 0.02 0.01 0.21 -9.86 0.93 0.86 -0.09 0.04 -0.01 0.18 -0.09 0.04 -0.01 0.18 -9.81 0.93

2

nd -0.14 1.01 -0.00 -0.06 0.00 1.01 -0.00 -0.06 0.00 0.01 1.02 -0.07 1.04 -0.01 -0.07 -0.04 1.04 -0.01 -0.07 -0.04 0.17 1.01

3

rd 0.03 0.05 0.87 0.13 -0.01 0.05 0.87 0.13 -0.01 -0.03 0.95 0.05 0.06 0.88 0.12 -0.03 0.06 0.88 0.12 -0.03 -0.11 0.95

4

th 0.13 0.11 0.16 0.84 -0.01 0.11 0.16 0.84 -0.01 0.08 0.97 0.14 0.13 0.09 0.91 -0.00 0.13 0.09 0.91 -0.00 -0.06 0.96

5

th -0.02 -0.07 0.11 -0.10 0.90 -0.07 0.11 -0.10 0.90 0.02 0.91 -0.05 -0.07 0.10 -0.10 0.94 -0.07 0.10 -0.10 0.94 0.08 0.90

Table 2.1: Coe ients of a simulated 5- oupled Van Der Pol Os illators of

modeldes ribedineq.(2.5). Onea h elltherearetwonumbers: therstone

isthe true oe ientgivento thesimulator,these ondoneisthe oe ient

returned from the Bayesian algorithm des ribed in appendix A. As it an

be seentherelative errorsareremarkably little.

2.2 ...to real EEG's

Inorderto nd asuitable dynami alsystemon whi hwe an inferits

oef- ient we need to have:

•

Abasi allysimpleos illatorwithfew oe ients, but"elasti "enough to simulate very dierent dinami s, depending on whi h set of

para-meters isprovided.

•

Enough os illators to " over" the dierent ele trodes, but not too many, otherwise it would be impossible to manage dynami ally

hun-dreds of oe ients.

TheVanDerPoltstherequirementsoftheformer: itis ontrolledbyonly

3parameters andittakesinto a ountvis osity,itisanharmoni os illators

withdissipationinthelimitof

ǫ = 0

,itmighthavealimit y leornot,with anunstableor stableorigin, independingonsign of

ǫ

,thelatterpositiveor negative. For the se ond point, we have made this hoi e: we have at our

disposal a 55-minutes EEG re ording of a non-epilepti healthy 18-months

oldbabyandwewillusethesamesystemofeq.(2.5)applyingitto4adja ent

signals oming from dieren esof 5neighboring same-sided ele trodes. For

example, onsidering theright o ipital region, we will onsider signal

[C4-P4℄,[P4-O2℄, [T4-T6℄,[T6-O2℄(g.2.2).

Figure 2.2: S hemeoftheinferred signalsin theo ipitalregion.

the oe ients onverge. In g.2.3 a typi al attempt is shown. The initial

onditions provided are:

α

ij

=

(

−1

if

(i = j)

−0.1

if

(i 6= j)

m

ij

= 0.1

ǫ

i

= 0.01

.

The inferen ing ma hine seemsto try to suppressas mu h as possible the

non mat hing oe ients, i.e. every

α

ij

→ −∞

; that means the errors are ompensated by the noise diusion matrix and there is no eviden e of

onvergen e of the oe ients whi h ould have made the model reliable.

It ould be asked if by looking at small windows a lo al simulation ould

work. Not even in this ase a onvin ing solution appears. A 6-se onds

window (

12000

ti s) has been used to s an theentire EEG onsisting of 55 minutes. At ea hin rement we shift thewindow andthe oe ient

c

post of

thepreviouswindowisusedasapriori oe ients

c

pr

fornextone. Wewere

for edtotrunktheiterationuptothe12 th

steps,otherwiseweobservedthat

one or more oe ients started to drift to innity. In g.2.4 the oe ient

for the signal [T6-O2℄, and all

ǫ

i

are shown. Although we had to limit the number of iterations, and at best the oe ients an be trusted on a

qualitative basisonly,adierentbehavior anbeseeninthe entral partof

theEEGin ontrast withthe beginning andtheend of there ords.

This is be ause in the rst 7 minutes and in the last 10 ones the baby is

awake,whileduring the entral 30 minutes hesleeps. During thesleep time

the oe ients are (when they do onverge) mu h slower and at the end

of the 12 th

iteration they still smaller in modulus that the orresponding

-2

-1

0

1

2

3

4

5

0

10

20

30

40

50

60

iterations

-12

-10

-8

-6

-4

-2

0

2

0

10

20

30

40

50

60

iterations

-2

0

2

4

6

8

10

12

14

0

10

20

30

40

50

60

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0

10 20 30 40 50 60

Figure 2.3: Coe ientof the inferen ingdynami s after ea hsteps ofre ursion.

Re ursion formulas aregivenin appendix A. On the top(from left to right) are

presentedtheevolutionof

m

23

and

ǫ

1

; onthebottom (fromleftto right)

α

14

and

-250

-200

-150

-100

-50

0

50

100

150

200

250

0

10

20

30

40

50

60

time (minutes)

α

coefficient of T6-O2

C4-P4 coupling

P4-O2 coupling

T4-T6 coupling

eigenfrequency

-60

-40

-20

0

20

40

60

0

10

20

30

40

50

60

time (minutes)

m coefficient of T6-O2

C4-P4 crossdissipation

P4-O2 crossdissipation

T4-T6 crossdissipation

dissipation

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0

10

20

30

40

50

60

time (minutes)

ε

coefficient of T6-O2

ε

₁

ε

₂

ε

₃

ε

₄

Figure 2.4: Coe ient of the inferen ing dynami s as a ontinuous fun tion of

anEEG re ord. From top,in therst andse ond frame oe ients

α

1j

and

m

1j

respe tivelyarepresented,in thethirdoneall

ǫ

i

areprinted.

0

2

4

6

8

10

Spectrum a.s.

frequency (Hz)

Infered Spectrum

Real Spectrum

Figure2.5: Spe trafromanEEGsignal(P4-O2)andfromitsinferred oe ients.

atoy modelto ompare the Fourier spe tra. Unfortunately it isdi ult to

nd a time point where all the oe ients are suitable for a simulation. In

fa tifonly oneofthe

ǫ

i

onstantsisnegative itleads toanunstablesystem; ifonlyoneofthe

α

ii

ispositiveitprodu esdivergentsolutions. Bytrials,we managed to nd some instan es the o urren es where inferred oe ients

leadto limitedsolutions. The orrespondingpower spe trumofone ofthose

isshowed ing.2.5and ompared withtheEEG'sone. Asit anbeseenthe

two spe traarereallydierent anditisthenaleviden ethatthisBayesian

Diusion Entropy analysis

Oneofthe mainobje tiveswhen onsidering anexpli itlynon deterministi

time series is to nd out if the system is driven by a deterministi and

noisy dynami , or if it is haoti , or if it is intrinsi ally sto hasti . Pure

statisti al instruments fail in su h sort of lassi ation. What it is done,

when dealing with biologi al, so iologi al or nan ial series,it isto extra t

fromthe seriesa diusion pro ess, andthen study its s alingproperties. A

diusionpro essisdes ribedbyadistribution

p(x, t)

ofitsdiusingvariable

x

,whi h isexpe ted to t thes alingproperty

p(x, t) =

1

t

δ

F



x

t

δ



(3.1)

where

δ

is a onstant whi h ontrols the s aling. For a standard browian motion

δ = 0.5

;anydeviationfromthisvalueissymptomati ofthepresen e of an anomalous s aling sour e: this means that the standard deviation

σ

l

ofa signalof size

l

grows faster than the squaredroot of

l

:

σ

l

∝ l

δ

,

δ > 0.5

. (3.2)

A stationarydis rete diusion pro ess of arbitrary size

N

is given. For example,assumethat itisa sequen eof oating number

ξ

k

≡ ξ(k)

where

k

is a time-like variable

k ≡ {0, 1, 2, ..., N }

. Inorder to give an estimation of theanomalous s aling

δ

we need to onsidera statisti alensemble of

l

-long diusion walkers, for any size of

l

. In this way

p(x, t)

an be onstru ted analyti ally. Themost e ient wayto have a fast and reliable dete tion of

δ

is explainedin[1℄. Consider an integer number

l

varyingfrom

1 < l < N

. We dene the

i

-th traje tory as:

ξ

(i)

_l

≡ {ξ

i

, ξ

i+1

, ξ

i+2

, ..., ξ

i+l

}

with

i + l 6 N

. (3.3)

We maythink of

ξ

asofa velo ityeldofpointlike parti lesina unidimen-sional

k

-stepsizegrid. Its orrespondingdiusion pro ess(orits position) is

given bytheintegralofthevelo ity overthedis rete time oordinate:

x

(i)

≡

i+l

X

k=i

ξ

k

=

X

ξ

_l

(i)

. (3.4)

We an extra t froman

N

-sizesequen eup to

N − l + 1

dierent but over-lapping traje tories of length

l

. In this onstru tion, two neighboring tra-je tories are not independent at all, but have

l − 1

points in ommon. In the denition of the Kolmogorov Sinai (KS) entropy, in fa t, every single

ombinations of symbols from a sour e stream are olle tionable, and

de-penden iesbetweenthemarenotrelevant. Allofthisholdsonlyinthelimit

N − l + 1 ≫ 1

. Forany

l

, onsidernowthisensembleofwalkers: theentropy ofthe parent distribution fun tion

p(x, l)

an be evaluated as:

S(l) = −

Z

∞

−∞

p(x, l) log p(x, l)
dx

. (3.5)

Inordertogive thebestapproximationof

p(x, l)

,the

x

-axishasbeen parti-tionedinto ellsofsize

ǫ(l)

. Ahistogram isthenlled: we ount thenumber

N

of diusionpro essesdroppedinea h ell;normalized ountsgive a stat-isti alapproximationof

p(x, l)

:

p

j

(l)

≡

N

j

(l)

(N − l + 1)

∼

Z

(j+1)ǫ(l)

jǫ(l)

p(x, l)

. (3.6)

Estimationof the entropyisthen

S(l) = −

X

j

p

j

(l) log p

j

(l)

. (3.7)

If

δ

isafun tionoftime,itishazardoustoinfers alingpropertiesofeq.(3.1). We ansafelyrestorethe ansatz ifweassumethat

δ

hangesslowlyintime. Finally,putting eq.(3.1)into eq.(3.5), aftersome simple algebra,we get:

S(t) = A + δ(t) log(t)

(3.8)

where

A ≡ −

Z

∞

−∞

F (y) log F (y)
dy

. (3.9)

Itis learhowtoextra tthes alingparameter

δ

fromsequen es. Wesuppose that we have a parti le undergoing random motion; let the parti le have a

superdiusive motion, supposing that the probability density

p(x, t)

s ales asin(3.1). Asitasbeen shownin[1℄,ananalogous wayto thinkaboutthis

motionis to give a probability ofthe timeneeded bythe parti le to es ape

1

10

100

1000 10000 100000

DE (a.s.)

size l (tics)

10

100

1000

10000

normalized coutings

time differences (tics)

Figure 3.1: On the right, waiting time distribution extra ted from asimulation

where the oe ients of (3.10) are:

µ = 2.2 T = 1.1

. Theoreti al expe tations (3.11) forthe diusion parameter

δ

leadsto

δ = 1/(µ − 1) = 0.8¯

3

. On theright, DEgrowthas afun tionofthelittlewindowsize

l

. Inlog-logplotitrea hesquite easily alinear regime in agreement with (3.8). The slope of the tting urve is

δ

exp

= 0.824 ± 0.008

whi h ompletely fulllsexpe tations.

andanotherone, wemayimagineto have apowerlawdistributionfor these

times:

ψ(t) = (µ − 1)

T

µ−1

(t + T )

µ

(3.10)

For example it is possible to have a walker that jumps a step always in

the same dire tion (Asymmetri Jump Model, AJM); at this point we an

hara terize itsdiusion pro essbystudyingan ensemble oftraje toriesfor

anytime length

l

. Theauthorsof[1℄demonstrated therelationbetween the powerlaw oe ient

µ

and the diusions ale oe ient

δ

:

δ =











(µ − 1)

if

1 < µ < 2

1/(µ − 1)

if

2 < µ < 3

0.5

if

µ > 3

(3.11)

We will illustrate the extra tionof the parameter

δ

from a simulation. We have taken a Monte Carlo generator of the distribution (3.10) and we

ex-tra ted

10

7

time dieren es

∆τ

i

roundedinintegers(presented ontheright ofg.3.1). Then we onstru tthediusionpro ess: abinarystream islled

with all 0 and 1. 1-points represent jumps of the walker, where distan es

between them arethe rounded

∆τ

i

. We then pro edeasexplained on page 27. We show theresultsof the simulation ing.3.1. Thetted valueof

δ

is inagreement withthetheoreti al ondition(3.11).

Going ba kto the EEGsignal, weneed to perform a sortof ourse graining

inorder to extra t a diusion pro ess from thevoltage series. The authors

of [2℄[3 ℄ did it on a timesequen e represented by time-frequen ies of heart

beating by onsidering an AJM jumping ea h time the signal en ountered

a given horizontal line. Following their example we divide the

y

-s ale rep-resenting the voltage of thesignal in several strips ofequal width; in other

words we have written thesignalina 'multi-pentagram'. We an all event

whentheEEG signal rossa line. The main waiting timegenerator onsist

ingiving the intervals between an event and another. Inthis way, we have

dened spa e intervals and we ount how long the time of permanen y in

thesame regionis.

Here is a summary of the algorithmi steps we need to onstru t su h a

−80

−60

−40

−20

0

20

40

60

80

µ

V

time (sec)

Figure 3.2: An exampleof asignalstripping. The stripis 18

µ

V wide. Verti al linesarevisualmarkersontimestream,markingwheretheEEG rossesastrip.

timegeneratorto feed DE al ulus:

•

Choose a xed big windows of size

N

, big enough to ontain a good statisti sample of the signal; small enough to assume that diusion

pro essesdo not hange their statisti al behaviour.

•

Choose a proper strip width

s

; for automation, a fra tion of

σ

, the standard deviation ofthe signalof length

N

.

•

Stripthesignal: inwords, onvertthevoltagesignalinaseriesofonly

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

20

40

60

80

100

120

140

160

normalized counting

time (tics)

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

100

Figure 3.3: Experimental distribution of waiting times generated by the strip

methodona55minutesEEGre ording. Somesignals(mixedfromo ipital,

pari-etal and front region) are presented. The strips are

0.5 × σ

wide, were

σ

is the standard deviation of ea h series. Logarithmi s ale; double logarithmi in the

smallwindow. 1ti =0.05se .

•

Feed the AJM withthe

0/1

streamand perform theDE al ulus. Inordertondasuitable

N

,werememberthat

N

should ontainmanylittle windows ofsize

l

,

l

should be ranginginmorethan one orderofmagnitude, we obtainthatan a eptable

N

shouldbe,at least,intheorderof

10

5

ti s.

We anhardlyaordit. Infa t,

10

5

ti smeans,at200Hz,500se onds,and

so8minutes. Ifwewishtoobtainapredi torindex ofsomething anomalous

inbrain dynami s we maywant toll abuer not longerthan 2-3minutes.

That'swhy inalmostevery algorithms wehave hosen

N = 40.000

.

Before anyfurther detail, it may be interesting to have an eyeball

over-viewonthe experimentaldistributionof thewaitingtimesgenerated bythe

strip method. We are interested in a distribution whi h possibly,

behav-ing dierently from an ordinary s aling, generates a diusion pro ess with

some deviation from an exponential distribution of random ounting. This

is the ase. It an be shown that for some signals (espe ially those in the

posteriorlead)the ountingpro essdepartfromaPoissonpro ess. Ing.3.3

t

-distributionsre onstru tedfromanensembleofsignalsareshown;theyare not linearin log-log s ale, suggesting the distribution might des ribed by a

stret hed exponential. Ing.3.4 thet is performed on one of these

distri-bution. As it an be seen, thestret hed exponential fun tion 1

exp (t/τ )

c

a

1e-05

1e-04

0.001

0.01

0.1

1

20

40

60

80

100 120 140 160 180 200

normalized counting

time (tics)

1e-05

1e-04

0.001

0.01

0.1

1

10

100

Figure 3.4: Waiting times generated bythe strip method on a55 minutes EEG

re ordingfromaT6-O2signal: itsdistributiontsperfe tlyastret hedexponential

fun tion with

c = 0.69 ± 0.01

;a tually to takea ountof ati jump intrinsi ally involvedinthealgorithmgeneratorthestret hexponentialhasbeen orre tedbya

shiftonthe

x

axis,

x

′

_{= x + δx}

. Logarithmi s ale;doublelogarithmi inthesmall

window. 1ti =0.05se .

very goodt.

Now we an pro ede showing the rst results of the DE algorithm: in

g.3.5 DE values arepresented asa fun tionof traje tories length

l

. Inthe graphmany results are extra ted from dierent striplengths and the same

time series. As it is evident, verti al slopesof DE does not depend on the

strip width

s

, and the rst and the last results are tted almost with the same slope

δ

. The dierent onstant

A

's areperfe tly justied: from (3.9),

A

represents the entropy to the PDF

p(x, l)

, and the width of this fun -tion depends on the mean number of jumps in the traje tory spa e, given

a length

l

: thatis, howmanystrip hanges areen ountered within

l

ti s of time. When

l ≪ 200

, there are not enough points to ll a histogram from whi ha meaningful al ulus ofDE ouldbe arriedout (inotherwords, we

arefar awayfrom onsidering thelimit

l → ∞

); when

l

is omparable with

40000

, the statisti al behaviorof theoverlapping windows is lost: there are toofewindependenttraje tories,andwe ansaythattheDEmethodresults

"saturate". All previous gures refer to a "frozen" 200 se onds (40000 ti )

long EEG stream. In a real EEG, we an repeat su h analysis even every

se ond usinga buer withthe lastfew minutes ofdata. It is what we have

doneing.3.6onT6-O2signalofthenonepilepti 18monthsbaby. The

2

3

4

5

6

7

8

100

1000

10000

DE (a.s.)

size l (tics)

Figure3.5: DEisgrowingifthewindowsize

l

getbiggerandbigger. Thisgraphi istheanalogousof thenumeri al al ulationof g.3.1. Ea h oloredfun tion isa

dierentattemptwithadierentstripsize

s

. Frombottom(red olor)

s

=1

µ

V,to top(other olors),

s

isgrowingbysteps of1

µ

V until 31

µ

V. Fit isperformedin themorelinearregion(

200 < l < 800

)bythelog-linear

A + d × log(l)

.

d

oe ient dierslessthan1%: theyare,respe tivelyfrombottom totop,

d = 0.866 ± 0.004

and

d = .871 ± 0.004

.

nalbrownian diusions an be extra ted. When thebabyis awake theDE

slope variesranging between

0.7 < δ < 0.8

whi h indi ate presen eof some sort of anomale s aling. It might be argued that the slope is intrinsi ally

onne ted withtheamplitude of thesignal, but,rst we notethatthestrip

widths aleswithamplitude foralgorithm onstru tion, se ond(asshownin

g.3.5)theslopeshouldnotdependonhowmu hstripsare ompressed

om-paredto the signal. Howevervisual evoked potentials andmus ulara tivity

aresuperimposeddynami s onthe mainbaselinewhi ha tually disturbthe

orre treadoutof

δ

. Ing.3.7 and3.8thediusion oe ient

δ

isshownfor allsignals fromtheleft hemisphere ofthesame baby.

-50

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V / DE slope x100

minutes

DEslope x 100

Figure3.6: Ingreen,ontheba kground,theT6-O2signalfromthenonepilepti 18

monthsbaby. Duringtherst7minutesheisawake,thenhesleepsfor35minutes

andthenhewakesup,remainingawaketilltheendofre ord. Inred,theDEslope

is shown; it hasbeen al ulatedwith

N = 40000

,

200 < l < 500

,

s = 0.5 × σ

. It hasadelayof200se onds, i.e. thereadingvaluerefersto thelast 3.5minutes.

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

time (minutes)

Fp1-F3

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

C3-P3

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

Fp1-F7

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

T3-T5

DEslope x 100

Figure3.7: Asinforg.3.6butforthelefthemispherereadoutofnonepilepti 18

months baby. Signalareindi atedontopofea hpi ture. TheDEslopehasbeen

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

time (minutes)

F3-C3

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

P3-O1

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

F7-T3

DEslope x 100

0

50

100

0

5

10

15

20

25

30

35

40

45

50

55

µ

V

T5-O1

DEslope x 100

Figure3.8: Asinforg.3.6butforthelefthemispherereadoutofnonepilepti 18

monthsbaby. Signalare indi atedontopofea hpi ture. TheDE slopehasbeen

In order to perform another ourse graining to infer dierent diusion

pro- esses, we will onsider maxima and minima from the voltage stream. We

onsider an event ea h time the EEG signal rea h a lo al maximum or a

lo al minimum; an example is showed in g.3.9. It is ne essaryto lowpass

thesignalbeforeanyfurther analysis,otherwise maxima and minimao ur

within everyti . This method hasbeen onsidered for tworeasons:

- Given that the potential waves whi h are olle ted on the s alp are

given by an average of millions of neurons intera ting syn hronously

orsemi-syn hronously,itmight ontaininformationonsub orti al

fre-quen ies,andit might be seenasamemory reset ofthe orrelationat

themeso olumnars ale.

- Itis unbiased from ampli ationor varian e hanging.

- Itdoesnotneed anyglobal hoi elike stripwidth,andjumping times

aredire tlydetermined.

−50

−40

−30

−20

−10

0

10

20

30

40

µ

V

time (tics)

Figure3.9: Anexampleofalowpassedsignal. Verti allinesarevisualmarkerson

thetimestreamwhi hhavebeentra kedattheo urren eofalo allmaximumor

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

0

10

20

30

40

50

60

70

normalized counting

time (tics)

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

1

10

100

Figure3.10: Experimentaldistributionsofthewaitingtimesgeneratedbythe

max-minmethodona80minutesEEG re ording. Somesignals(mixed fromo ipital,

parietalandfrontregion) arepresented. Logarithmi s ale;double logarithmi in

thesmallwindow. 1ti =0.05se .

Again, the pro edure des ribed in hapter 3 has been be repeated, but a

littlesimplied be ause we do notneed to sele t astripwidth:

•

Consider a lowpassed signal.

•

Chooseof thebigwindowof size

N

.

•

Build a binary stream onsidering positive events only inthose point werethe lteredsignalhasa stationarypoint.

•

Feed the AJM withthe

0/1

streamand perform theDE al ulus. Followingthesamepro eduredis ussedin hapter3,thewaitingtimebetween

maxima and minima are shownin g.3.10 for the same signals. The

distri-butions seem, in this ase as well, power law for the rst few steps, then

theyturnonamoredepressedregime, andalmostall ofthemaretrun ated

before 60 ti s. In this ase the stret hed exponential t has been used 2

,

too. Max-min distributions aremore depressed than thedistributions

om-ingfromthestrip-method. Inthe aseofstripping, hangingthestripwidth

hasan immediate ee t on the mean waiting time. Inthis asethere is no

han e to set ne tuning of a parameter to ontrol the distribution; infa t

the only parameter that an be onsidered is the frequen y ut-o, but it

ae tsonlythehighfrequen ystatisti ,beinguselesstogivetimeseparation

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

20

30

40

50

60

70

normalized counting

time (tics)

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

100

c=0.64

Figure 3.11: Waiting times generated by themax-min method ona 80 minutes

EEGre ordingfromdierentsignals. T6-O2signalhasbeentted: itsdistribution

tsastret hed exponentialfun tion with

c = 0.64 ± 0.01

; just likeg.3.4, totake a ountofati jump intrinsi allyinvolvedin thealgorithmgeneratorthestret h

exponentialhasbeen orre tedbyashiftonthe

x

axis,

x

′

_{= x + δx}

. Logarithmi

s ale;doublelogarithmi inthesmallwindow. 1ti =0.05se .

rareevents,whi h,inthis ase,aretimedieren esfromsemi-globalmaxima

andminima. Ingures3.10and 3.11itmightbenoteda uriousslight

lus-tering of events aroundtime points3 ti distant from ea h other, a learly

ai tionofthe low-passingoperation. Ing.3.12 DEvaluesareshownasa

fun tionoftraje torieslength

l

. Inthegraph,threedierentportionsofthe same signal arepresented. Verti al slopes are denable sin e

l

is less than

∼ 1000

. As it an be seen there is a quite large portion of semi-linearity, biggerthan inthe aseofstripmethod(g.3.5). Thatmakesitalittle more

easy to automati allyperform the omputation oftheangular oe ient in

thelog-plot. As for the hoise of thelength

N

ofthe big window, thesame argumentsof hapter 3apply, andwe have hosen again

N = 40000

. When a ontinuoustimeanalysisofEEG'sstreamsisperformedweobtaing.3.13

whi h isto be ompared withg.3.6. The fun tionsdierfrom ea h other.

During the rst 7 minutes of sleeping-time the

δ

from strip-method and

δ

from max-min-method areup to

0.80

approximately the same, and during sleeping time is onstantly around0.5. It onrmsthat during thesleeping

timenoanomalous s alingsarepresent. But, whenthebabyisawakeagain,

while with strip method

δ

jumps again to higher values,

δ

from max-min-method still os illating. Although this might suggest that the onne tion

2.5

3

3.5

4

4.5

5

5.5

6

100

1000

10000

DE (a.s.)

size l (tics)

Figure 3.12: DE against window size

l

for a big window of size

N = 40000

. From bottom to top (respe tively in green, blue, red) three dierent portions of

thesameEEG ofa18-months oldbabyand from thesame signalT6-O2. A

log-linear

A + d × log(l)

t hasbeenperformedinthemostlinearregion(

l < 800

).

d

oe ientsare,respe tively,

d = 0.54 ± 0.01, d = 0.65 ± 0.02, d = 0.71 ± 0.01

.

therstone: inthelast15 minutes ofEEGthebabyhasbeen stressed with

ash of lights; this ondition probably plays an important role and might

10

20

30

40

50

60

70

80

90

0

5

10

15

20

25

30

35

40

45

50

55

µ

V / DE slope x100

minutes

DEslope x 100

Figure3.13: Ingreen,ontheba kground,theT6-O2signalfrom thenonepilepti

18 months baby. During the rst 7 minutes he is awake, then he sleeps for 35

minutesandthenhewakesup,remainingawaketilltheendofre ord. Inred,the

DEslopefrom max-minmethod isshown;ithasbeen al ulatedwith

N = 40000

,

100 < l < 400

. It hasa delayof 200se onds, i.e. the reading valuerefersto the

Correlations and information

When dealing with a set of signals is often useful to dene a topologi al

distan e between them. The orrelation matrix between signals oers su h

a "distan e" parameter: it is a rude instrument but is well dened, solid

under many ir umstan es and limitedbetween

−1

and

1

. Letus onsider the orrelationmatrix between

M

signals dened as:

g

ij

(t) ≡=

hx

i

(t) x

j

(t)i − hx

i

(t)ihx

j

(t)i

q

hx

2

i

(t) − hx

i

(t)i

2

i hx

2

j

(t) − hx

j

(t)i

2

i

(4.1)

where

h i

standsfortimeaveraging. Theeigenvaluesandeigenve torsofthis matrixmight give pre iousinformationon howthesystemisauto orrelated

and quanties how the signals intera t with ea h other. During epilepti

seizure, the epileptogeni portion of the brain drives the whole dynami of

the network and makes potential dieren es intera t withea h other. The

wholebrain isspiking, not randomly,but on ertain frequen ies. The main

ideaisto studythe orrelationmatrix

g

onaEEGre ordingtryingtograsp information on hypotheti al pre-seizureinstauration of asort of orrelation

amongintera tingnetworks. We anforexample, hooseawindowoflength

K

on whi h al ulate

g

ij

(K)

. A s alar quantity to exanimate this on-tinuously updating matrix is its eigenvalues. In g.4.1 the evolution of the

biggesteigenvaluesisshownonthesameEEGofg.3.6rea hesamaximum.

Be ausewe do not need to ll longstreams for statisti al purposeasinthe

ase of the DE method, we an aord to payattention to justfew se onds

ofEEG, and build

g

ij

(K)

withextremely lo alsignal. Ing.4.1

g

ij

(K)

has been al ulatedonlyonthelast5se onds(1000ti s)ofre ording. Thisleads

to a global and semi-instantaneous reading index. Asit an be seen during

thesleeping timethe eigenvalue is at its minima. During wakefulness there

are frequently jumps; they orrespond to mus ular artefa ts, when signals

aremore syn hronous. During those jumps there is also adramati hange

of omponentsoftherelatedeigenve tor. Onemayaskif,givensu ha

0

50

100

150

200

250

300

350

400

450

500

550

10

20

30

40

50

60

Biggest eigenvalue

minutes

15

30

45

60

75

20

21

22

Figure4.1: Inred,thebiggesteigenvalue

λ

ofthe orrelationmatrixofleft hemi-sphereof thesame1hourEEGofthe18 monthsold baby;it hasbeen al ulated

with

K = 1000

,that means

λ

isrefreshedti sbyti sand al ulatedonthelast5

se onds. Inthesmallwindowinside,amagni ationofthebigone.

it. Wenotethatthisindex in ludeinformation onintera tions ofallsignals,

so,somehow, itmightbe onsidereda"global"index ofnetworka tivity. So

anyfurtherstatisti isreferredtothe "wholeme hanism", not toa lo al

in-formationasin aseofstripsandmax-minmethodswhi hhavebeenapplied

onalo alvoltagereadout. Inpra ti e,thisevolutionof

λ

onsistsona oat-ing numbers stream. This fun tion might be treated just like the oating

streamfrom theEEGsignal. Asin hapter3 we apply the same pro edure

not dire tlyon EEGsignal, but onthe biggesteigenfun tion evolution: We

"write" the evolution of

λ

in a multi-pentagram of strips and we onsider timedieren esbetweenevents,whereevent meansthatthefun tionof

λ(t)

has rossed a strip line. Again the s aling of the exstra ted diusive

pro- ess is onsidered. As itmight be seenon g.4.2, the hange ofthe biggest

eigenvalue

λ

is ranging between a nearly brownian motion (

δ

=0.68) and a typi al non-brownian motion (

δ ∼ 0.9

). Thisisan indi atoron whetherthe resetofthememory ofthesystemisapoissonianornon-poissonianpro ess;

i.e., expe ially during the wakefulness time, the evolution of

λ

has a non exponentiallyde ay of thememory. We will dis uss on hapter 5details of

50

60

70

80

90

100

0

10

20

30

40

50

microV / DE slope X 100

time (minutes)

Figure 4.2: In green,theEEG signalin

µ

V;in red, DE slopefrom stripmethod applied totheevolutionofthebiggesteigenvalueof orrelationmatrix

λ

of whole hemisphere (16-signals) ofthe same55 minutes EEG of the18 monthsold baby;

onblue andon purple, thesameindex but extra ted from respe tively right and

left hemisphere. Theinde eshavebeen al ulatedwiththeseries of g.4.1. The

To follow the biggest eigenvalue evolution it is not the only way to

on-dense the information en losed in the orrelation matrix. The orrelation

matrix

g

ij

give an intrinsi ally denition of distan es between its ompon-ents. For example, we an dene distan es between two elements

i

and

j

d

i,j

≡

p2 − 2g

ij

,thatis,themore orrelatedtwosignalsare,themorelittle thedistan efromea hother. Thispointofviewhasemergedinlastyearsas

the fundamental approa h to nan ial series to evaluate their "proximity".

Letus onsideran ultrametri spa eweredistan es aredened as:

d(i, j) = 0 ⇔ i = j

d(i, j) = d(j, i)

d(i, k) 6 max d(i, j), d(j, k)

∀j

(4.2)

Thisnaturally denesaminimalspanningtreeon allthepossiblespanning

trees onne ting the vertex(signals). In other words ifwe have a omplete

onne tinggraphbetweenea hpoint,therearealotofspanningtrees(whi h

are in luded in the omplete graph) that an rea h all the points without

loops. Ifweassignaweighttoea hedgeproportionaltothedistan ebetween

two onne tingpoints, this algorithm automati ally givesa unique solution

to the problemof minimal spanning tree,whi h is thespanning tree whose

sumof all thedistan es of its edgesisminimal.

Given adistan ematrix,analgorithm (Kruskal'salgorithm [8 ℄)todrawthe

minimalspanning treeis thefollowing:

- Sortall distan es in in reasing order (we will have a listof ouples of

elements).

- Drawan edgebetween therst ouple.

- Keep drawing edges between any ouple of elements only if that

ele-mentsare not yet onne ted throughsome other path (do not permit

any loop)

Ifweknowonlyaboutthe minimalspanningtree onstru ted withamatrix

ofdistan ewedonotknowthedetailsofthematrix,butwehavea learidea

0.1

1

10

100

1000

10000

1

10

100

1000

normalized counting

time (tics)

right

all

c=0.48

c=0.59

Figure4.3: Waiting timedistribution forminimalspanning tree hanges. Inred,

the distribution is omputed from the orrelation matrix al ulated only on the

righthemisphere(8signals);ingreenthedistribution omputedwithall16signals.

Ex ellent t with a stret hed exponential is show; note the ex ellent agreement

between data and a best t done with a stret hed exponential. Parameter

c

is respe tively

c = 0.48 ± 0.01

,and

c = .56 ± 0.01

. Meantimes

τ = 4.65 ± 0.09

and

τ = 4.17 ± 0.08

.

100

1000

10000

DE (a.s.)

size l (tics)

Figure4.4: DE againstdiusionpro essoflength

l

,inbigawindowsofsize

N =

40000

. Indierent olors, 4dierent portionsof the whole 0/1 stream extra ted

fromtheminimalspanningtreesoftherighthemisphere. Thelinearregimeislost