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STATISTICAL PROPERTIES OFTHEINTERBEAT INTERVALCASCADE INHUMAN SUBJECTS FatemehGhasemi,1J.Peinke,2M.RezaRahimiTabar,2,3andMuhammadSahimi4 1DepartmentofPhysics,SharifUniversityofTechnology,Tehran11365,Iran 1CarlvonOssietzkyUniversity,InstituteofPhysics,D-26111Oldenburg,Germany 3CNRSUMR6529,ObservatoiredelaCoˆted’Azur,BP4229,06304NiceCedex4,France 4DepartmentofChemicalEngineeringandMaterialsScience,UniversityofSouthernCalifornia,LosAngeles,California90089-1211,USA 6 0 Statisticalpropertiesofinterbeatintervalscascadeareevaluatedbyconsideringthejointprobabilitydistribution 0 P(∆x2,τ2;∆x1,τ1)fortwointerbeatincrements∆x1and∆x2ofdifferenttimescalesτ1andτ2.Wepresent 2 evidencethattheconditionalprobabilitydistributionP(∆x ,τ |∆x ,τ )mayobeyaChapman-Kolmogorov 2 2 1 1 equation.ThecorrespondingKramers-Moyal(KM)coefficientsareevaluated.Itisshownthatwhilethefirstand n secondKMcoefficients,i.e.,thedriftanddiffusioncoefficients,takeonwell-definedandsignificantvalues,the a J higher-ordercoefficientsintheKMexpansionareverysmall. Asaresult,thejointprobabilitydistributionsof 1 theincrementsintheinterbeatintervalsobeyaFokker-Planckequation.Themethodprovidesanoveltechnique 3 for distinguishing the two classes of subjects in terms of the drift and diffusion coefficients, which behave differentlyfortwoclassesofthesubjects,namely,healthysubjectsandthosewithcongestiveheartfailure. ] M 05.10.Gg,05.40.-a,05.45.Tp,87.19.Hh Q o. I.INTRODUCTION logicalprocesses.25−33InFigure1samplesofinterbeatsfluc- i tuations of healthy subjects and those with congestive heart b Cardiacinterbeatintervalsnormallyfluctuateinacomplex failure(CHF)areshown. - q manner.1−6 Recent studies reveal that under normal condi- [ tions, beat-to-beat fluctuations in the heart rate may display extended correlations of the type typically exhibited by dy- 1.2 1 v namical systems far from equilibrium. It has been argued,2 1 1 for example, that the various stages of sleep may be char- 5 acterized by long-range correlations of heart rates separated x(t) 0.8 0 by a largenumberof beats. The interbeatfluctuationsin the 1 0.6 heart rates belong to a much broader class of many natural, 0 aswellasman-made,phenomenathatarecharacterizedbya 06 degree of stochasticity. Turbulent flows, fluctuations in the 0.40 10000 t 20000 / stock market prices, seismic recordings, the internet traffic, o andpressurefluctuationsinpacked-bedchemicalreactorsare i 1.1 b exampleof time-dependentstochastic phenomena,while the - surfaceroughnessofmanymaterials7,8 areexamplesofsuch 1 q phenomenathatarelengthscale-dependent. : v The focus of the present paper is on the intriguing statis- x(t) 0.9 i X tical properties of interbeat interval sequences, the analysis 0.8 ofwhichhasattractedtheattentionofresearchersfromdiffer- ar entdisciplines.9−15Analysisofheartbeatfluctuationsfocused 0.7 initially on short-time oscillations associated with breath- 0 5000 10000 t 15000 20000 FIG. 1. Time series of interbeat intervals x(t) versus interval ing, blood pressure and neuroautonomic control.16,17 Stud- numbertforatypicalpersonwithcongestiveheartfailure(bottom) ies of longer heartbeat records, however, revealed 1/f−like andahealthysubject(top). behavior.18,19 Recent analysis of very long time series indi- cates that under healthy conditions, interbeat intervals may Recently, Friedrich and Peinke were able34 to derive a exhibitpower-lawanticorrelations,20followuniversalscaling Fokker-Planck (FP) equation for describing the evolution of intheirdistributions,21andarecharacterizedbyabroadmul- the probability distribution function of stochastic properties tifractal spectrum.22 Such scaling features change with the of turbulent free jets, in terms of the relevant length scale. diseaseandadvancedage.23 Thepossibleexistenceofscale- Theypointedoutthattheconditionalprobabilitydensityofthe invariantpropertiesin theseeminglynoisyheartbeatfluctua- incrementsof a stochastic field (forexample, the increments tionsismaybeattributedtohighlycomplex,nonlinearmecha- inthevelocityfieldinturbulentflow)satisfiestheChapman- nismsofphysiologicalcontrol,24asitisknownthatcircadian Kolmogorov(CK)equation,eventhoughthethevelocityfield rhythmsare associated with periodicchangesin key physio- 1 itself contains long-range, nondecaying correlations. As is well-known, satisfying the CK equation is a necessary con- ditionforanyfluctuatingdatatobeaMarkovianprocessover 0.7 therelevantlength(ortime)scales.35Hence,onehasawayof analyzing stochastic phenomena in terms of the correspond- 0.6 ingFPandCKequations. Inthispaperthemethodproposed by Friedrich and Peinke is used to compute the Kramers- 0.5 Moyal (KM) coefficients for the increments of interbeat in- tervalsfluctuatations,∆x(τ)=x(t+τ)−x(t). Here,∆xis ) x10.4 theinterbeatincrementswhich,forallthesamples,isdefined ∆ | as,∆x ≡ ∆x/στ,whereστ isthestandarddeviationsofthe ∆x20.3 increments in the interbeats data. It is shown that the first P( and second KM coefficients representing, respectively, the driftanddiffusioncoefficientsintheFPequation,havewell- 0.2 defined values, while the third- and fourth-orderKM coeffi- cientsaresmall. Therefore,a FPevolutionequation35 isde- 0.1 velopedfortheprobabilitydensityfunction(PDF)P(∆x,τ) which, in turn, is used to gain information on changing the shapeofPDFasafunctionofthetimescaleτ36(seealsoRef. ∆x 2 [37]foranotherinterestingandcarefully-analyzedexampleof FIG. 2. Test of Chapman-Kolmogorov equation for theapplicationoftheCKequationtostochasticphenomena). ∆x = −0.42, ∆x = 0and ∆x = 0.42. Thesolid and open The plan of this paper is as follows. In Section 2 we de- 1 1 1 symbolsrepresent, respectively, thedirectly-evaluatedPDFandthe scribetheFriedrich-PeinkemethodintermsofaKM expan- oneobtainedfromEq. (1). ThePDFsareshiftedinthehorizontal sionandtheFPequation.WethenapplythemethodinSection directionsforclarity.Valuesof∆xaremeasuredinunitsofthestan- 3totheanalysisoftheincrementsintheinterbeatfluctuations. darddeviationoftheincrements. Thetimescalesτ ,τ andτ are 1 2 3 10,30,and20,respectively. II.THEKRAMERS-MOYALEXPANSIONAND It is well-known that the CK equation yields an evolution FOKKER-PLANCKEQUATION equation for the distribution function P(∆x,τ) across the scales τ. The CK equation, when formulated in differential A completecharacterizationof the statistical propertiesof form,yieldsa masterequation,whichtakesonthe formofa theinterbeatfluctuationrequiresevaluationofthejointPDFs, FPequation:35 P (∆x ,τ ,···,∆x ,τ ), for an arbitrary N, the number N 1 1 N N of data points. If the phenomenon is a Markov process, an d P(∆x,τ)= importantsimplificationarisesinthat,theN-pointjointPDF dτ P is generatedby the productof the conditionalprobabili- N ∂ ∂2 tiesP(∆xi+1,τi+1|∆xi,τi),fori = 1,···,N −1. Thus,as − D(1)(∆x,τ)+ D(2)(∆x,τ) P(∆x,τ). (2) (cid:20) ∂∆x ∂∆x2 (cid:21) the first step of analyzing a stochastic time series, we check whethertheincrementsinthedatafollowaMarkovchain.As The drift and diffusion coefficients, D(1)(∆x,τ) and mentionedabove,anecessaryconditionforastochasticphe- D(2)(∆x,τ),areestimateddirectlyfromthedataandthemo- nomenontobeaMarkovprocessisthattheCKequation,34 mentsM(k) oftheconditionalprobabilitydistributions: P(∆x2,τ2|∆x1,τ1)= D(k)(∆x,τ)= 1 lim M(k) , (3) k!∆τ→0 d(∆x )P(∆x ,τ |∆x ,τ )P(∆x ,τ |∆x ,τ ), (1) 3 2 2 3 3 3 3 1 1 Z 1 M(k) = d∆x′ (∆x′−∆τ)kP(∆x′,τ +∆τ|∆x,τ) . shouldholdforanyvalueofτ , intheintervalτ < τ < ∆τ Z 3 2 3 τ .35Therefore,wecheckthevalidityoftheCKequationfor 1 (4) describingthedatausingmanyvaluesofthe∆x triplets,by 1 comparingthe directly-evaluatedconditionalprobabilitydis- The coefficients D(k)(∆x,τ) are known as the Kramers- tributions P(∆x ,τ |∆x ,τ ) with those calculated accord- Moyal(KM)coefficients. 2 2 1 1 ing to right-hand side of Eq. (1). In Fig. 2, the directly- computed PDF is compared with the one obtained from Eq. (1). Allowingforastatisticalerroroftheorderofthesquare A.ApplicationtoAnalyzingHeartbeatData rootofthenumberofeventsineachbin,wefindthatthePDFs arestatisticallyidentical. As an application of the method, we analyzed both day- time(12:00pmto18:00pm)andnighttime(12:00amto6:00 2 am) heartbeat time series of healthy subjects, and the day- timerecordsofpatientswithCHF.Ourdatabaseincludes10 D(2)(∆x,τ)= healthysubjects(7femalesand3maleswithagesbetween20 and50,andanaverageageof34.3years),and12subjectswith 0.11 0.287 CHF, with 3 femalesand 9 males with ages between 22 and 0.01+ (∆x)2+ 0.057+ , (5) (cid:18) τ (cid:19) (cid:18) τ (cid:19) 71,andanaverageageof60.8years). Theresultingdriftand diffusioncoefficients,D(1)andD(2),aredisplayedinFigures whereasforthepatientswithCHFweobtain, 3and4. ItturnsoutthatthedriftcoefficientD(1) is alinear function of ∆x, whereas the diffusivity D(2) is quadratic in D(1)(∆x,τ)=−0.013∆x−0.0018, ∆x. Estimatesofthesecoefficientsarelessaccurateforlarge valuesof∆xand,thus,theuncertaintiesincrease. Usingthe D(2)(∆x,τ)= datasetforthehealthysubjectswefindthat, 0.005 0.066 0.005+ (∆x)2+ 0.013+ . (6) (cid:18) τ (cid:19) (cid:18) τ (cid:19) τ=10 WealsocomputedtheaverageofthecoefficientsD(1) and 0.06 τ=15 D(2)fortheentiresetofthehealthysubjects,aswellasthose τ=20 with CHF. According to the Pawula‘s theorem,34,37 the KM 0.04 expansionistruncatedafterthesecondterm,providedthatthe fourth-order coefficient D(4)(∆x,τ) vanishes. For the data 0.02 that we analyze the coefficient D(4) is about 110D(2) for the (1)∆D(x) 0 hheeaaEllqtthhuyyatssiuoubnbjsjee(cc5ttss),aaannnddd(ta6hb)oossuteattw2e10ittDhha(Ct2t)HhfFeordhrtaihvfotescetohewefifisthacmiCeneHtsoFr.fdoerrthoef magnitude,whereasthediffusioncoefficientsforgivenτ and -0.02 ∆xaredifferentbyaboutoneorderofmagnitude.Thispoints toarelativelysimplewayofdistinguishingthetwoclassesof -0.04 thesubjects. Moreover,theτ-dependenceofthediffusionco- efficientforthehealthysubjectsisstrongerthanthatofthose -0.06 withCHF (inthe sense thatthe numericalcoefficientsof the -1 -0.5 0 0.5 1 τ−1 are largerfor the healthysubjects). These are shown in ∆x Figures3and4. Thestrongτ−dependenceofthediffusioncoefficientD(2) forthehealthysubjectsindicatesthatthenatureofthePDFof their increments∆x for given τ, i.e., P(∆x,τ), is intermit- τ=3 τ=5 tent,andthatitsshapeshouldchangestronglywithτ. How- τ=15 ever,forthesubjectswithCHFthePDFisnotsosensitiveto 0.4 thechangeofthetimescaleτ,henceindicatingthattheincre- ments’fluctuationsforthesubjectswithCHFisnotintermit- tent. Theseresultsarecompletelycompatiblewiththerecent 0.3 discoveriesthattheinterbeatfluctuationsforhealthysubjects x) and those with CHF have fractal and multifractalproperties, ∆ (2)D(0.2 respectively.22 III.SUMMARY 0.1 Wehaveshownthattheprobabilitydensityoftheinterbeat 0 intervalincrementssatisfiesaFokker-Planckequation,which encodestheMarkoviannatureoftheincrements’fluctuations. -3 -2 -1 0 1 2 ∆x WehavebeenabletocomputereliablythefirsttwoKramers- Moyalcoefficientsforthestochasticprocesses∆x-thedrift FIG. 3. The drift and diffusion coefficients D(1)(∆x) and anddiffusioncoefficientsintheFPrepresentation-and,using D(2)(∆x),estimatedfromEq. (5)forahealthysubject,followlin- the polynomialansatz,34 obtain simple expressionsfor them earandquadraticbehavior,respectively. intermsof∆xandthetimescaleτ. Wehaveshownthatthe driftanddiffusioncoefficientsofthe incrementsintheinter- D(1)(∆x,τ)=−0.03∆x−0.0046, beat fluctuations of healthy subjects and patients with CHF 3 havedifferentbehavior,whenanalyzedbythemethodweuse ianmotionorothertypesofstochasticprocessesthatgiverise in this paper. Hence, they help one to distinguish the two tosuchcorrelations.Inthatmethodonedistinguisheshealthy groups of the subjects. Moreover, one can obtain the form subjectsfromthosewithCHFintermsofthetypeofthecor- ofthepathprobabilityfunctionaloftheincrementsinthein- relationsthatmightexistinthedata. Forexample,ifthedata terbeat intervals in the time scale, which naturally encodes followafractionalBrownianmotion,thenthecorresponding thescaledependenceoftheprobabilitydensity. This,inturn, Hurst exponent H is used to distinguish the two classes of providesaclearphysicalpictureoftheintermittentnatureof thesubjects,asH <0.5(>0.5)indicatesnegative(positive) interbeatintervalsfluctuations. correlationsinthedata, whileH = 0.5indicatesthatthein- crements in the data follow Brownian motion. The method proposed in the present paper is different from such analy- ses in that, the increments in the data are analyzed in terms of Markov processes. This is not in contradiction with the 0.06 τ=2 τ=5 previousanalyses.Ouranalysisdoesindicatetheexistenceof τ=15 correlationsintheincrements,but,asiswell-knowninthethe- 0.04 oryofMarkovprocesses,suchcorrelations,thoughextended, eventually decay. 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