REVI EWMethods derivedfromnonlineardynamicsforanalysingheart
ratevariabilityBYANDREAS VOSS1,*, STEFFENSCHULZ1,
RICOSCHROEDER1,MATHIAS BAUMERT2ANDPERECAMINAL31DepartmentofMedical
EngineeringandBiotechnology,UniversityofAppliedSciencesJena,07745Jena,Germany2Centre
for Biomedical Engineering (CBME), University of Adelaide,
Adelaide,5005SA,Australia3Biomedical
EngineeringResearchCentre,ETSEIB,Technical
UniversityofCatalonia,08028Barcelona,SpainMethodsfrom
nonlineardynamics (NLD) haveshownnew insightsintoheart
rate(HR)variabilitychanges undervariousphysiological
andpathological conditions, providingadditional
prognosticinformationandcomplementingtraditional time-
andfrequency-domainanalyses. Inthisreview, someof
themostprominentindicesof nonlinearandfractal dynamics are
summarized and their algorithmic implementations
andapplicationsinclinical trialsarediscussed. Several of
thoseindiceshavebeenproventobeof diagnosticrelevanceor
havecontributedtoriskstratication.
Inparticular,techniquesbasedonmono-andmultifractalanalysesandsymbolicdynamicshavebeensuccessfullyappliedtoclinicalstudies.FurtheradvancesinHRvariabilityanalysisareexpected
through multidimensional and multivariate assessments. Today, the
question isnolongerabout whether ornot
methodsfromNLDshouldbeapplied; however, it isrelevant toaskwhichof
themethods shouldbeselectedandunder
whichbasicandstandardizedconditionsshouldtheybeapplied.Keywords:
nonlineardynamics; heartratevariability; fractal; chaos;
cardiology1. IntroductionThe investigation of nonlinear dynamics
(NLD) and the introduction of indices toquantify the complexity of
fractal dynamics have challenged our view onphysiological
networksregulatingheartrate(HR)andbloodpressure,
therebyenhancingour knowledge andstimulatingsignicant andinnovative
researchintocardiovasculardynamics.Duringthelastdecades,
methodsderivedfromNLDhavebeensuccessfullyapplied to many scientic
disciplines, including physics, astrophysics, chemistry,economics,
biology and medicine. However, the impact of deterministic
nonlinearPhil.Trans.R.Soc.A(2009)367,277296doi:10.1098/rsta.2008.0232Publishedonline31October2008Onecontributionof13toaThemeIssueSignalprocessinginvitalrhythmsandsigns.*
Authorforcorrespondence([email protected]).277
Thisjournalisq2008TheRoyalSocietymetrics for enhancedunderstanding
of physiologyis onlypartlyexploredtodate.
Someimportantphysiological ndingsbasedontheconceptsof
NLDarementionedin2and3.Initially the theory of NLD, developed
during the late 1970s and 1980s,generated interest among many
researchers to explore chaotic behaviour inbiological systems and
apply their ndings to biology and medicine. Later on, thefocus of
interest shifted towards explaining and accounting for
nonlinearityinthecardiovascularsystem,whichispresumablyhighdimensional.Currently,amajorapproachistorevealandcharacterizethedynamicsandcomplexityofnonlinearsystems.Theembeddingtheoremallowsonetomathematicallyreconstructanentirenonlinear
systemfromonly one observed variable, since the
reconstructeddynamics are (geometrically) similar to the original
dynamics (Takens
1981;Kaplan&Glass1995;Schumacher2004).Pioneering
workperformedbyGlass et al. (Guevara et al. 1981; Glass &Mackey
1988) introduced nonlinear approaches into heart
rhythmanalysis.Period-doubling bifurcations, in which the period of
a regular oscillation
doubles,werepredictedtheoreticallyandobservedexperimentallyintheheart
cells ofembryonicchickens. Form, qualitativechange, oscillation,
stabilityandotherimportant biological notions found inherent
expression inthenew mathematicalapproachof NLD(Garnkel 1983).
Ritzenberget al. (1984) weretherst toprovide evidence of nonlinear
behaviour inthe electrocardiogram(ECG) andarterial blood pressure
traces of a dog that had been injected with
noradrenaline.SincetheoriginalreportsbyWolfetal.(1978)andKleigeretal.(1987),theanalysis
of spontaneous variations of beat-to-beat intervals (BBI) has
becomeanimportantclinicaltool,familiartocardiologists(Lombardietal.2000).The
rst approaches of the HR variability (HRV) analyses based on
nonlinearfractal dynamics were performed by Goldberger & West
(1987). It was suggestedthat self-similar (fractal) scaling may
underlie the 1/f-like spectra (Kobayashi
&Musha1982)seeninmultiplesystems(e.g. interbeatinterval
variability, dailyneutrophil uctuations).
Theyproposedthatthisfractal
scaleinvariancemayprovideamechanismfortheconstrainedrandomnessunderlyingphysiologicalvariability
and adaptability. Later, Goldberger et al. (1988) reported
thatpatients prone to high risk of sudden cardiac death showed
evidence of nonlinearHRdynamics, includingabrupt spectral changes
andsustainedlowfrequency(LF) oscillations. At a later date, they
suggested that a loss of complexphysiological
variabilitycouldoccurundercertainpathological
conditionssuchasreducedHRdynamicsbeforesuddendeathandageing(Goldberger1991).Babloyantz
&Destexhe (1988) performed the rst multivariate
nonlinearanalysisofHRV.WiththehelpofseveralindependentmethodsforquantifyingNLD,
such as phase portrait, Poincare section, correlation dimension,
Lyapunovexponent and Kolmogorov entropy, the ECGs of four normal
human hearts werestudied qualitatively and quantitatively. They
demonstrated that the variabilityunderlying interbeat intervals is
not random, but exhibits
short-rangecorrelationsgovernedbydeterministiclaws.Techniques of
phase-space reconstruction and dimensional analysis
wereappliedtoHRtracesobtainedfromscalpelectrodes in12normal
foetusesbyChafnetal.(1991).A.Vossetal.
278Phil.Trans.R.Soc.A(2009)To estimate the complexity of
cardiovascular dynamics, Pincus (1991)modiedthe original
correlationdimensionandKolmogoroventropynotions(Grassberger&Procaccia1983a,b;
Eckmann&Ruelle1985), creatingtheapp-roximate entropy (ApEn).
This technique was later improved and termedsampleentropy (SampEn)
byRichman&Moorman(2000) andreduces
thesuperimposedbiaswithintheoriginalmethod.As a further milestone,
Novak et al. (1993) provided evidence of a
closenonlinearcouplingbetweentherespiratoryandcardiovascularsystems.Pengetal.
(1995)applieddetrendeductuationanalysis(DFA)toquantifythefractal
structureof theHR, whichwaslatervalidatedin1999(Ma
kikallioetal.1999).Also,Kurthsetal.(1995)introducedsymbolicdynamics(SDyn)totheHRVanalysisandfurtherdemonstrateditspowerforriskstraticationofsuddencardiacdeathbasedonthemultivariateapproaches(Vosset
al. 1996,1998). Themethodof SDynwas further developedbyPortaet al.
(2001) forapplicationonshort-termHRtimeseries (Guzzetti et al.
2005; Maestri et al.2006). The discovery of themultifractal nature
of HR dynamics by Ivanov et al.(1999) showed that the heartbeat
modulation is even more complex thanpreviously suspected, requiring
multiple scaling exponents for its character-ization.
Averypromisingwaytoquantifycomplexityovermultiplescaleswasrecently
introduced by Costa et al. (2002, 2005). The apparent loss of
multiscalecomplexityinlife-threateningconditions(Norrisetal.
2008)suggestsaclinicalimportanceofthismultiscalecomplexitymeasure.Toinvestigate
the interactions andcouplings
betweenHRandrespirationandHRandbloodpressure,respectively,avarietyofmethodsfromNLDhavebeendevelopedandapplied(e.g.Paratietal.1988;Pompeetal.1998;Baumertetal.2002;Schwabetal.2006).Themethodologicalwealthwithinthissubareaofresearchdeservesaseparatereviewandshallnotbefurtherdiscussedwithinthiscontribution.Variousattemptshavebeenmadetoemploynonlinearapproachestomodelparts
of the cardiovascular system(Vinet et al. 1990; Christini et al.
1995;Amaraletal.1999;Gomesetal.2000;Lin&Hughson2001;Tulppoetal.2005;Baselli
etal. 2006; Khoo2008). Again,
thistopiccannotbediscussedindetailwithinthispaper.Thisreviewfocusesonthesignicanceof
NLDincardiovascularvariabilityanalysis, to explore dynamic and
structural features of cardiovascular regulationand its clinical
relevance. Some of the most commonly used HRV indices,
derivedfromNLDwithprovenrelevancetoclinical research, will
besummarizedandimportantfeaturesrelatingtotheirpracticalapplicationsdiscussed.2.
IndicesofHRVderivedfromNLDDuringthe1980s,
therewasmuchanticipationthat manyof thecomplicatedsystems observed
in nature could be described by a few nonlinear coupled modes.The
properties of those systems could then be characterized by
fractaldimensions, Lyapunov exponents or KolmogorovSinai entropy.
However,currently it is evident that such a lowdimensionality is
exhibited only byrather coherent phenomena. Physiological data, as
discussed here, have amuch more complex structure (gure 1).
Considering the variety of factors279
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)inuencing
HR, e.g. respiration or mental load (within dashed boxes in gure
1),it becomes apparent that HRregulationis one of the most
complexsystemsinhumans.This simpliedmodel of
HRregulation(adaptedfromHejjel &Ga l 2001)shows the sinus node,
generating the heartbeat as the primaryphysiologicalpacemaker that
is innervated by sympathetic and parasympathetic efferents
andaffectedbyseveral humoral factors. Thesinusnodeactsasthenal
summingelementofsympatheticallyandparasympatheticallymediatedstimuliandtheirrelationisreectedintheactual
interbeatinterval.
Theregulatorysubsystemsresultinascale-invariantcardiaccontrol
acrossdifferenttimescales,
showinglong-rangecorrelationswithatypical scalingbehaviour.
Therefore,
toextracttherelevantpropertiesofNLDsystems,classicallinearsignalanalysismethodsare
often inadequate. Most physiological systems, such as HR
generation, exhibita very complex behaviour, which is far froma
simple periodicity. Suchphysical exercisemovementsmental
loadpersonalityrespirationcortexmedullaparasympatheticgangliasympatheticgangliahumoral
effectspreloadstroke volumesinus nodeheart ratearterial toneblood
pressurebaroreceptorsarrhythmiaposturehypo-thalamusFigure 1. A
simplied model of HR regulation (adapted from Hejjel & Gal
2001). Additional
factorsinuencingconsiderablytheHRareshownwithinthedashedboxes.A.Vossetal.
280Phil.Trans.R.Soc.A(2009)a complexity within the obtained
biosignals (here, the HRtime series) iscaused by different
components of the intrinsic systems dynamics and
especiallybythenonlinearinterplayof differentphysiological control
loopsasillustratedingure1.Possiblesources proposedfor thenonlinear
behaviour of biological
systemsanddifferentdegreesofbiologicalcomplexityareasfollows.Different
subsystems (control loops) acting in a network with
feedbackinteractions to help constantly adapt the system to its
physiological needs andrequirements.Inthe case of a
pathophysiological process andageing, the adaptationofa subsystemto
changed basic conditions/needs of the total
system(e.g.changingoftheoperationalpoints).Inthecaseofseverepathophysiologicaldevelopments,thecompensationofadisturbedorfailingsubsystembyoneormoreotherinteractingsubsystems.This
leads to the objective of quantifying the complexity of
physiologicaldynamics with the help of different indices. However,
there is no general
accepteddenitionforcomplexity.Intheanalysisofsignalsfrombiologicalsystems,thetermor
concept of complexityis mostlyusedwithregardtotheir
dynamicaland/orstructuralexpression,includingimportantfeaturessuchasnonlinearity,time
irreversibility, fractality and long-range correlations. Complex
physiologicalsignalsaretypicallynon-stationary,butnotrandom.Prominentnonlinearmeasures,
withemphasisontheirmainpropertiesandapplicability,willbediscussedforthefamiliesAD(table1).(a
) FamilyA:fractal measuresConcept: to assess self-afnity of
heartbeat uctuations over multiple time scales.(i)
Power-lawcorrelation(scalingexponentb)Kobayashi &Musha(1982)
rst reportedthefrequencydependenceof thepower spectrumof
RR-interval uctuations. The slope of the regressionlineof the
log(power) versus log(frequency) relation (1/f ), usually
calculated inthe 10K410K2Hz frequency range corresponds to the
negative scaling
exponentbandprovidesanindexforlong-termscalingcharacteristics(Sauletal.1987).Thisbroadbandspectrum,characterizingmainlyslowHRuctuationsindicatesafractal-likeprocesswithalong-termdependence(Lombardi2000).Sauletal.(1987)foundthatbissimilarto
K1inhealthyyoungmen.Biggeretal.(1996)reportedanalteredregressionline(bzK1.15)inpatientsafterMI.Limitations:
stationarity, periodicity and the need for large datasets
arerequired;artefactsandpatientmovementinuencespectralcomponents.(ii)
Detrendeductuationanalysis(indicesa1anda2)This method is based on a
modied random walk analysis and was introducedandappliedto
physiological time series by Peng et al. (1995). It
quantiesthepresenceorabsenceof
fractalcorrelationpropertiesinnon-stationarytime-seriesdata.DFAusuallyinvolvestheestimationofashort-termfractalscaling281
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)exponent
a1 over the range of 4%n%16 heartbeats anda
long-termscalingexponent a2 over the range of 16%n%64 heartbeats
(Peng et al. 1995). DFA wasdeveloped to quantify the uctuations
onmulti-length scales. The self-similarityoccurring over a large
range of time scales can be dened for a selected time
scalewiththismethod(Ma kikallioet al. 1999).
Healthysubjectsrevealedascalingexponent of approximately1,
indicatingfractal-like behaviour. Patients withcardiovascular
diseaseshowedreducedscalingexponents, suggestingaloss
offractal-likeHRdynamics (a1!0.85, Ma kikallioet al. 1999; a1!0.75,
Huikurietal.2000).Limitations: atleast8000datapointsshouldbeused;
monofractal method;normal-to-normal interbeat intervals are
required; dependency on editingectopicbeats.Table 1. Summaryof some
important features of the selectednonlinear indices. (F,
familyofnonlinearmeasures;forabbreviationofindices,see2.)F
descriptor indicesshorttermlongterm correlatedpartlywithA
power-lawcorrelationscalingexponentb X
frequencycomponents:(UVLF,VLF,LF)A
detrendeductuationanalysisa1(shortterm) X
LFn,HFn,LF/HF,HF/P,LF/(HFCLF),SD1/SD2a2(longterm) X LF,VLF/(HFCLF)A
multifractalanalysisD(h)withlocalexponenthX X notyetknownB
approximateentropyApEn X
indexesdescribingvagalmodulationofheartrate(rmssd,pNN50,HFpower)B
sampleentropy SampEn X
negativelywithLFnandLF/HF;naturallogar-ithm(ln)ofthetotalpower;lnLFandlnLF/HFB
multiscaleentropyMSE X notyetknownB compressionentropyCE X X
sdNN,rmssd,wpsum02,plvar,forbwordsC symbolicdynamicsShannonandRe
nyientro-pies,forbiddenwords,wpsum02,wpsum13,phvar,plvar,0V,1V,2LV,2UV,0V%,1V%,2LV%,2UV%X
X cvNN,sdNN,rmssd,pNN50,SD2D Poincare plot
SD1(shortterm),SD2(longterm),SD1/SD2X X
SD1:rmssd,mainlywithHF,lesserwithLF;SD2:sdNN,LFandHFpowerA.Vossetal.
282Phil.Trans.R.Soc.A(2009)(iii) Multifractal
analysisMultifractalanalysisdescribessignalsthataremorecomplexthanthosefullycharacterizedbyamonofractal
model. Ivanovet al. (1999)demonstratedthathealthyHRVis
evenmorecomplexthanpreviouslysuspectedandrequires amultifractal
representation, usingalargenumberof local
scalingexponentstofullycharacterizethescalingproperties.
Multifractalityinheartbeatdynamicsindicates the involvement of
coupledcascades of feedbackloops inasystemoperating far
fromequilibrium. Ivanov et al. (1999) found a loss in
HRVmultifractalityinpatientssufferingfromcongestiveheartfailure(CHF).Limitations:
requiresmanylocal
andtheoreticallyinniteexponentstofullycharacterizetheirscalingproperties.(b
) FamilyB:entropymeasuresConcept: to assess the
regularity/irregularity or randomness of heartbeatuctuations.(i)
Approximateentropy/sampleentropyThe ApEn represents a simple index
for the overall complexity andpredictabilityof
timeseries(Pincus1991). ApEnquantiesthelikelihoodthatruns of
patterns, whichare close, remainsimilar for subsequent
incrementalcomparisons (Ho et al. 1997). High values of ApEn
indicate high irregularity andcomplexityintime-seriesdata.
Forhealthysubjects, ApEnvaluesrangefromapproximately 1.0 to 1.2 and
for post-infarction patients ApEn values areapproximately1.2(Ma
kikallioetal.1996;Hoetal.1997).Limitationsof ApEn:
stationarityandnoise-freedataarerequired; inherentbias exists;
counting self-matches; dependency on the record length;
lacksrelative consistency; evaluates regularity on one scale only;
outliers (missed
beatdetections,artefacts)mayaffecttheentropyvalues.SampEn,
improvingApEn, quanties the conditional probabilitythat
twosequencesofmconsecutivedatapointsthataresimilartoeachother(withinagiventolerancer)
will
remainsimilarwhenoneconsecutivepointisincluded.Self-matchesarenotincludedincalculatingtheprobability.
Lakeetal. (2002)described a reduction in SampEn of neonatal HR
prior to the clinical diagnosis
ofsepsisandsepsis-likeillness.TheSampEnwasfoundtobesignicantlyreducedbeforetheonsetofatrialbrillation(Tuzcuetal.2006).Limitations
of SampEn: stationarity is required; higher pattern length
requiresanincreasednumber of data points; evaluates regularity
onone scale
only;outliers(missedbeats,artefacts)mayaffecttheentropyvalues.(ii)
MultiscaleentropyBiological systems are likely to present
structures on multiple spatio-temporalscales. Multiscale
entropy(MSE) assesses multiple time scales to measure asystems
complexity. The main advantage of MSEis its ability to
measurecomplexity according to its denition a meaningful structural
richness and beingapplicable to signals of nite length (Costa et
al. 2005). The MSEmethoddemonstrated that healthy HRV is more
complex than pathological HRV. Costaetal.
(2002)foundthatpathological
dynamicsassociatedwitheitherincreased283
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)regularity/decreasedvariabilityor
withincreasedvariabilityarebothcharac-terized by a reduction in
complexity due to the loss of correlation properties. Costaet al.
(2002)reportedthebestdiscriminationbetweenpathological
(CHF)andhealthy HR signals on scale 5. MSE analysis revealed
signicantly lower SampEnvalues in young patients with diabetes
mellitus on scale 3 (Javorka et al. 2008).Limitations: stationarity
is required; outliers (missed beat detections,artefacts) may affect
the entropy values; the consistency of MSE will
beprogressivelylostasthenumberofdatapointsdecreases.(iii)
CompressionentropyThe entropy of a given text is dened as the
smallest algorithm that is capableof generatingthetext (Li
&Vitnyi 1997). Ziv&Lempel (1977) introducedauniversal
algorithm for lossless data compression (CE), using string matching
ona sliding window. With some modications, this algorithm can be
applied for theanalysis of heartbeat time series (Baumert et al.
2004, 2005). Here, thecompressionentropyquanties the extent to
whichthe datafromheartbeattimeseries canbecompressed, i.e.
repetitivesequences occur. Reducedshort-term uctuations of HRV
result in an increased compression. Entropy reductionappears to
reect a change in sympathetic/parasympathetic HR
control(Baumertetal. 2005). Baumertetal.
(2004)investigatedCHFpatientsbeforethe onset of ventricular
tachyarrhythmia, and showed reduced CE valuescomparedwithpatients
during normal sinus rhythm. Truebner et al.
(2006)foundsignicantdifferencesbetweenthehigh-
andlow-riskCHFpatientsandBaretal.(2007)foundsignicantlyreducedcomplexity(CE)ofHRtimeseriesinpatientswithacuteschizophreniaincomparisonwithhealthycontrols.Limitations:
dependency on sampling rate, the window length and
thelookaheadbuffersize;integernumbersrequired.(c )
FamilyC:symbolicdynamicsmeasuresConcept: to assess the
coarse-grained dynamics of HRuctuations basedonsymbolization.(i)
Symbolicdynamics(entropiesandprobabilities)SDyn was introduced by
Hadamard (1898) and allows a simple description of asystems
dynamics with a limited amount of symbols. SDyn are suitable
todescribetheglobal short-timedynamics of beat-to-beat
variability(Voss et al.1993, 1996, 1998; Kurths et al. 1995). At
rst, time series were transformed into asymbolsequenceof four
symbols with the alphabetAZ{0,1, 2, 3}to classifythedynamic changes
within that time series. Three successive symbols fromthealphabets
were used to characterize the symbolstringswhereby 64 different
wordtypes (bins) were obtained. The resulting histogramcontains the
probabilitydistribution ofeach singleword within a word sequence.
SDyn
investigatesshort-termuctuations.Theseshort-termuctuationsaremainlycausedbyvagalandbaroreexactivities.
Wedifferentiatewords consistingof
alternating/constant/increasing/decreasing symbol strings reecting
especially vagal/reduced vagal(increasedsympathetic)/bradycardic
baroreex/tachycardic baroreexactivity.Porta et al. (2001)
introduced a modied procedure of SDyn. Here, the amount
ofA.Vossetal.
284Phil.Trans.R.Soc.A(2009)theRRintervalswaslimitedto300beats.
Thefull rangeof thesequenceswasuniformly spread on six levels (05),
and patterns of length LZ3 were constructed(Guzzetti et al. 2005).
All patterns with LZ3 were grouped, without any loss,
intofourfamilies.Thesewere:(i)patternswithzerovariation0V,(ii)patternswithone
variation1V, (iii) patterns with two like variations2LV, and (iv)
patternswith two unlike variations2UV. The rates of occurrence of
these patterns will
beindicatedas0V,1V,2LVand2UV%(Portaetal.2007).Limitations: detailed
information will be lost; outliers (ectopic beats
andnoise)inuencesymbolstrings.(d ) FamilyD:Poincare
plotrepresentationConcept: to assess the heartbeat dynamics based
on a simplied phase-spaceembedding.(i) Poincare
plots(SD1andSD2)ThePoincare plot analysis
(PPA)isaquantitativevisual technique,wherebytheshapeof theplot is
categorizedintofunctional classes (Weisset al. 1994;Kamenet al.
1996; Brennanet al. 2002) andprovides detailed
beat-to-beatinformation on the behaviour of the heart. Usually,
Poincare plots are applied fora two-dimensional graphical and
quantitative representation (scatter plots),where RRnis plotted
against RRnC1. Most commonly, three indices arecalculated
fromPoincare plots: the standard deviation of the
short-termRR-interval variability(minoraxisof thecloud, SD1),
thestandarddeviationofthelong-termRR-intervalvariability(majoraxisofthecloud,SD2)andtheaxes
ratio(SD1/SD2) (Kamen&Tonkin1995; Brennanet al. 2002). For
thehealthy heart, PPA shows a cigar-shaped cloud of points oriented
along the line ofidentity. These indices are correlatedwithlinear
indices. Laitioet al. (2002)showed that an increased SD1/SD2 ratio
was the most powerful predictor of post-operativeischaemia. Ma
kikallio(1998)foundSD2z125 msinhealthysubjectsandSD2z85
msinpost-infarctionpatientswithventriculartachyarrhythmia.Limitations:SD1,SD2dependentonothertime-domainmeasures.Open
source computer software versions of DFA, multifractal
analysis,correlation dimension, Lyapunov exponent, SampEn and
MSEanalysis areavailableatwww.physionet.org.3.
ApplicationofmethodsfromNLDforHRVanalysis(a )
HRVinhealthyconditionsHealthyHRuctuationsshowacomplextypeofvariability,embeddingfractalself-similaructuationsontimescalesrangingfromsecondstohours,andthusgeneratinglong-rangepower-lawcorrelations(Goldbergeret
al. 2002). Resultsfromseveral studies indicatethat
greatercomplexity(irregularity) appearsinhealthy systems. The
common hypothesis is that the organismis a highlycomplex adaptive
system, and that the complexity of its behaviour allows for
thebroadest rangeof adaptiveresponses duetodifferent levels of
input withinaphysiologicalrange.285
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)(i)
AgeeffectsCardiovascular structures and functions change with age,
increasing the risk
ofdevelopingcardiovasculardisease(Oxenham&Sharpe2003).
Effectsof ageingon HRV have been observed with linear as well as
nonlinear complexity measures(Kaplan et al. 1991; Ryan et al.
1994), and are apparent in short-term records of30
min.Todemonstratethisageingeffect,wecomparedHRVofyounghealthymale
subjects (32G8 years, nZ10) withthat of older healthymale
subjects(56G5years, nZ10)underrestingconditionsinthesupineposition.
HRVwasquantied with indices fromlinear domains and NLD(see 2 for a
moredetaileddescription). Table2(grouptest T1) shows thenumerical
results ofunivariate statistics based on the MannWhitney U-test,
and gure 2(i, ii)
showsthetachograms(timeseriesofbeat-to-beatintervals)andworddistributionsof64differentthree-letterwordtypes(SDyn)fromayoungandanolderhealthymale.
HRVwasgenerallyreducedinthegroupofoldersubjects.
Inparticular,thecomplexityindicesof SDynrevealedalossof
complexityinoldersubjects.This is in accordance with the previous
studies (Peng et al. 1995; Voss et al. 1996;Goldberger et al.
2002), where larger dimensions and entropies implied a
greaterTable2.Resultsofthegroupcomparisons.(T1younghealthymalesversusolderhealthymalesand
T2older healthy females versus older healthy males (10 subjects in
every group). Signicancevalue: n.s., not
signicant;p!0.05,p!0.01,p!0.001; TD, time-domain indices;
FD,frequency-domain indices; A, fractal measures; B, entropy
measures; C, symbolic dynamicsmeasures; D, Poincare plot
representation. For standard HRV parameters in TD and FD, see
TaskForce(1996); forparametersinA, BandD, see2; forparametersinC,
seeVossetal. (1996).CE, compression entropy; pW110, pW021 and
pW321single word type probabilities from SDyn;,permille.)grouptests
meanvalueGs.d.parameter T1 T2 youngmales
oldermalesolderfemalesage(years)n.s. 32.43G8.04 55.90G5.36
56.40G2.46TD meanNN(ms) n.s. n.s. 819.5G120.7 894.5G127.9
899.6G129.1sdNN(ms) n.s. n.s. 47.2G13.1 40.0G9.4 43.4G15.2rmssd(ms)
n.s. n.s. 29.4G13.8 22.2G12.3 34.0G17.1FD LF/HF(arb.units) n.s.
n.s. 3.79G2.49 3.86G2.34 2.40G1.57LFn(arb.units) n.s. n.s.
0.74G0.13 0.71G0.21 0.64G0.17HFn(arb.units) n.s. n.s. 0.26G0.13
0.29G0.21 0.36G0.17A a1(arb.units) n.s. n.s. 1.02G0.18 1.18G0.23
1.02G0.23a2(arb.units) n.s. n.s. 0.90G0.20 0.97G0.13 0.98G0.12B
CE(arb.units) n.s. n.s. 0.64G0.08 0.58G0.06 0.60G0.11C
Shannon(bit)n.s. 3.12G0.32 2.73G0.38 3.10G0.49Forbword(arb.units)
n.s.24.63G9.52 33.80G9.53 20.60G14.83Renyi025(bit) n.s.3.57G0.27
3.30G0.26 3.63G0.32pW110(arb.units, )n.s. 38.5G11.1 24.0G12.0
21.0G12.5pW021(arb.units, ) n.s.1.6G2.2 1.6G4.2 5.1G5.1D SD1(ms)
n.s. n.s. 20.8G9.8 22.8G16.1 34.5G17.1SD2(ms) n.s. n.s. 63.3G16.4
56.3G12.3 61.0G20.4A.Vossetal.
286Phil.Trans.R.Soc.A(2009)complexity. Applyingmultifractal
analysis, Shiogai (2007) demonstratedthatthe neurogenic control of
the HRbecomes more signicant compared
withmyogeniccontrolwithageing.(ii)
SexeffectsUnderhealthyconditions,
sexdifferencesinHRVhavebeenobservedacrossall ages (Ryanet al. 1994;
Beckers et al. 2006). It has beensuggestedthat abenecial autonomic
control of the heart infemales agedless than45yearsmay contribute
to the lower risk of coronary heart disease and
seriousarrhythmiasinfemales.
Wedemonstratethissexdifferencebycomparingage-matchedhealthyfemaleswiththegroupof
olderhealthymalesinvestigatedinthe previous paragraph. Table 2
shows the numerical results for the sexcomparison (group test T2),
and gure 2(iii) provides an example tachogram andword distribution
of 64 different three-letter word types of a representative
olderfemale. In particular, indices of SDyn indicate a higher
occurrence of word typesandahigher complexityof HRdynamics
inwomenthaninmen(gure2(ii)versus(iii)).(b )
HRVunderpathophysiological conditionsAreductioninHRcomplexitywas
reportedinpatients withCHF(fractalscaling properties; Peng et al.
1995), myocardial infarction (MI, SDyn; Voss et al.1996, 1998) and
other cardiovascular diseases. Clinical research, where
measuresfromNLD have been applied, focuses mainly on cardiology and
internalmedicine. Inparticular, thosetechniques havebeenusedfor
HRVanalysis inpatients with ischaemic heart diseases (IHDs) and CHF
and those threatened bysevere arrhythmias. Todemonstrate the power
of some HRVindices
derivedfromNLDforseparatingthehealthyfrompathologicalHRV,wecomparethreedifferent
groups of patients (dilatedcardiomyopathy(DCM), ischaemic
heart850115085011508501150 (a) (i) (i)(ii) (ii)(iii)
(iii)(b)0.20.10.20.10.20.10 30 0 021 102 123words time (min)210 231
312 333Figure2. Examples of (a) tachograms (30 min; BBI,
beat-to-beat intervals (ms)) and(b)
worddistributionsof64differentthree-letterwordtypes(probability;SDyn)froma(i)younghealthymale,(ii)olderhealthymaleand(iii)olderhealthyfemale.Thedashedarrowsindicatethemostsignicant
word distribution probability pw110 differentiating between the
young and olderhealthy
males.Thesolidarrowsindicatethemostsignicantworddistributionprobabilitypw021differentiatingbetweentheolderhealthymalesandhealthyfemales(table2).287
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)failure
(IHF) and myocardial infarction) to that of sex- and age-matched
healthysubjects (REF). The short-termindices fromNLD(SDyn and
compressionentropy) aresignicantlydifferent inall threepatient
groups whencomparedwithhealthysubjects(table3).
DFAshowsonlyatrendforgroupdifferences.AlthoughPoincare
mapanalysisisoriginallyanonlinearmethod,
thetypicallyextractedindicesSD1andSD2(see2)aremoreorlessinsensitivetononlinearcharacteristics.Surprisingly,theindex
SD1wasable todifferentiatethehealthysubjectsfromall patients,
partlyincontrasttothetime-domainindexrmssd,which is known to be
highly correlated with SD1. Figure 3 shows the tachogram,SDyn word
distribution and Poincare plot for a representative patient from
eachinvestigatedgroupand ahealthy
subject.Bycomparingshort-versuslong-termrecordings inrisk
stratication, we coulddemonstrate that inpatients withischaemic
heart failure (lowrisk: survivors, nZ179 and high risk:
patientswhodiedduetoacardiaceventduringafollow-upperiodof2years,nZ29)a1(DFA)differentiatesbothgroupssignicantly(lowrisk30
min: a1Z1.2G0.22,low risk 24 hours: a1Z1.2G0.19, high risk 24
hours: a1Z1.07G0.25;signicances: lowrisk 30 min versus 24 hours,
n.s.; high risk and lowrisk:30 minand24hours, bothp!0.01). There
were nosignicant differences inshort-termversuslong-terma1.Several
large clinical studies have demonstratedthe potential of
measuresbasedonNLDandfractal analysis. Someof
themarediscussedbrieyinthefollowingexamples. Bigger et al. (1996)
was therst toreport theabilityofTable 3.
Resultsofthegroupcomparisonspatientswithischaemicheartfailure(IHF),
dilatedcardiomyopathy(DCM)andaftermyocardial
infarction(MI)versusREF(10subjectsineverygroup). (Tests: T1, IHF
versus REF; T2, DCM versus REF; T3, MI versus REF; signicance
value:n.s., not signicant;p!0.05,p!0.01,p!0.001; TD,
time-domainindices; FD, frequency-domainindices; A, fractal
measures; B, entropymeasure; C, symbolic dynamics measures; D,
Poincare plot representation; CE, compression entropy; for
parameter denitions and units, see table 2.)grouptests
patients(meanvalueGs.d.)parameter T1 T2 T3 IHF DCM MIage n.s. n.s.
n.s. 54.20G5.25 52.60G9.14 58.25G4.98TD meanNN n.s. n.s. n.s.
920.0G97.9 925.4G103.9 952.7G103.1sdNN n.s. n.s. n.s. 43.3G14.0
41.4G18.3 38.2G10.5rmssd n.s. n.s.20.0G7.1 25.8G15.1 17.1G4.3FD
LF/HF n.s. n.s. n.s. 4.52G2.29 2.75G2.45 3.89G1.62LFn n.s. n.s.
n.s. 0.79G0.09 0.64G0.18 0.77G0.08HFn n.s. n.s. n.s. 0.21G0.09
0.36G0.18 0.23G0.08A a1 n.s. n.s. n.s. 1.29G0.15 1.17G0.22
1.20G0.16a2 n.s. n.s. n.s. 1.10G0.10 1.16G0.17 1.01G0.15B CE
0.53G0.07 0.53G0.11 0.54G0.05C Shannon n.s. n.s.2.63G0.41 2.58G0.64
2.49G0.30Forbword n.s. n.s.33.90G6.21 31.20G14.65
38.63G4.17Renyi025 n.s. n.s.3.27G0.27 3.25G0.60 3.10G0.19pW321
0.1G0.2 0.5G0.7 0.1G0.1D SD1 14.1G5.0 18.3G10.7 12.0G3.2SD2 n.s.
n.s. n.s. 59.5G19.4 55.3G24.4 52.5G14.9A.Vossetal.
288Phil.Trans.R.Soc.A(2009)powerspectrumbasedonlong-termscalingindicestopredict
deathafterMI.Theystudied715patientswithrecentMI,274healthysubjectsand19patientswithhearttransplants.Theslopeofthepower-lawrelationshipwasfoundtobesomewhatsteeper(morenegative)inMIandmuchsteeperforhearttransplant00.10.20.10.20.10.20.10.27500
30time (min)1150750115075011507501150 (a) (i)
(i)(ii)(iii)(iv)(ii)(iii)(iv)(i)(ii)(iii)(iv)(c)(b)8001000800100080010008001000021
102 123 210 231words312 333 600
800SD2SD1SD2SD1SD2SD2SD1SD11000Figure 3. Examples of (a) tachograms
(BBI (ms)), (b) SDyn word distributions of 64 different
three-letter word types (probability) and (c) Poincare plots
(BBI(nC1) (ms)) for a representative
patientfromeachinvestigatedgroupandahealthysubject((i)REF,
healthysubject;(ii)DCM, dilatedcardiomyopathy; (iii) IHF, ischaemic
heart failure; (iv) MI, myocardial infarction); BBI,
beat-to-beatintervals; SD1, standarddeviationshort-termvariability;
SD2, standarddeviationlong-termvariability.289
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)patients.
They demonstrated that a power-law regression coefcient belowK1.372
is signicantly associated with total cardiac and arrhythmic
mortality. Amultivariate approach (Voss et al. 1998), using all
domains and especially SDyn,revealed the best prediction for
all-cause mortality as well as for suddenarrhythmic death. Inthis
study, 572 survivors of acute myocardial infarctionwereenrolled.
Within the follow-up period, 43 patientsdied
(all-causemortality),of whom13diedfromventricular
tachycardia/ventricular brillation, 14fromsuddenarrhythmicdeath,
22fromsuddendeathand34fromcardiacdeath.A combination of four HRV
parameters from all domains (time and frequencydomain,
NLD)inthismultivariateapproachimprovedthediagnosticprecisionmorethantwofold.
Ma kikallioet al. (1999)examinedtraditional
HRVindicesalongwithshort-termfractal-likecorrelationproperties(DFA)andpower-lawscalingin159post-MIpatientswithejectionfraction(EF)lessthan35percentwith4-yearfollow-up.
Amongall of theanalysedvariables, reduceda1(DFA)was the strongest
univariate predictor of mortality in patients with depressed
leftventricularfunction(EFlessthan35%)afteracuteMI.IntheDIAMONDstudy,acohortof446survivorsofacuteMIwithEFlessthan35per
cent was investigatedanda reductioninthe short-termfractalexponent
a1 (DFA) was the most powerful predictor of all-cause
mortality(Huikuri et al. 2000). The exponent predicted both
arrhythmic and non-arrhythmiccardiacdeath.Tapanainen et al. (2002)
showed that several other HRV indices were also ableto predict
mortality in the univariate analysis, but in a multivariate model,
afteradjustingforclinicalvariablesandleftventricularEF,
a1(DFA)wasthemostsignicantHRV-basedcontributor(comprisingasetoflinearindicesandpowerspectrum1/f
power-lawslope)topredictsubsequentmortality. InastudybyStein et al.
(2005), abnormal nonlinear HRV (short-term fractal scalingexponent,
power-lawslopeandSD12(Poincare
plotrepresentation))hasbeenassociated with mortality post-MI. The
results suggest that decreasedlong-termHRVandincreasedrandomness of
HRare eachindependent riskfactors formortality post-MI. However, as
with traditional HRV parameters,
thisrelationshipmightbeblurredbycoronaryarterybypassgraftsurgerypost-MIorbydiabeticautonomicneuropathy.Guzzetti
et al. (2000) found signicantly lower normalized LF power and
lower1/f slope in chronic heart failure patients compared with
controls. Moreover, thepatients who diedduringthe
follow-upperiodpresentedfurther reducedLFpower and steeper 1/f
slope than the survivors. They concluded that spectral
andnonlinearanalysesof HRVbothhaveprognosticrelevanceindependentof
thetime-domainmeasuresofHRVinpatientswithCHF.Another study
investigating 499 patients with CHF has also shown
thepredictivevalueofalteredshort-andlong-termscalingpropertiesformortality(Ma
kikallio et al. 2001). A short-term fractal scaling exponent of
a1!0.9 was
thestrongestpredictorofmortality(univariateandmultivariate).Aninterestingndingof
that studywasthattheHRVindiceswerestrongpredictors of mortality in
patients with mild/moderate CHF, but failed
toprovideindependentprognosticinformationforseverecasesofCHF.Recently,
multiscale indices of HRV have been included in clinical trials.
MSEstratied441patients fromthe intensive care unit
bymortalityandwas
anindependentpredictorofdeathoccurringdayslater(Norrisetal.2008).A.Vossetal.
290Phil.Trans.R.Soc.A(2009)Thepaperof Huetal.
(2008)providesanexampleof howconceptsof NLDand fractal analysis may
enhance our knowledge regarding physiologicaland pathophysiological
regulation of HR. The authors demonstrated
thatscale-invariantcardiaccontrol
occursacrosstimescalesvaryingfromminutestoapproximately24hours.
Lesioningof themammaliancircadianpacemaker(suprachiasmatic nucleus,
SCN) completely abolishes the scale-invariant patternat time scales
greater thanapproximately4 hours. At time scales less
thanapproximately 4 hours, the scale invariance was persistent
after SCN lesions, butwithadifferentpattern.
Thisstudyrevealedtheinuenceof
theSCNonHRuctuationsovermultipletimescales,
whichpreviouslycouldnotbeexplainedbysimplepacemakermodelsof24hoursrhythmicity.ItwasconcludedthattheSCNservesasamajornodeinthecardiaccontrol
networkandimpartsscale-invariantcardiaccontrol acrossawiderangeof
timescaleswiththestrongesteffectsbetweenapproximately4and24hours.4.
SummaryMethods of NLD and fractal analysis have opened up new ways
to analyse HRV.Although time- and frequency-domain methods enable
the quantication
ofHRVondifferenttimescales,nonlinearmethodsprovideadditionalinformationregardingthedynamicsandstructureofbeat-to-beattimeseries.Insummary,wecanstatethefollowing.There
are several indices derived fromNLDproven to be powerful
riskstratiers and contribute towards enhanced diagnostics of
cardiovasculardiseases,e.g.DFA,MSEandSDyn.ThereisavarietyofotherpotentialindicesfromNLDthatseempromising,buthaveyettobevalidatedinfurtherclinicaltrials(Maestrietal.2007).NLDprovideadditional
andindependentinformationaboutphysiological
aswellaspathophysiologicalcardiovascularregulation.Temporal
changesinHRVandHRdynamicsoftendependonthebaselinecharacteristicsofthepatient.Altered
HRV and HR dynamics have prognostic signicance for theprogressionof
adisease (e.g. coronaryarterydisease) andfor mortality(e.g.after
acute MI). Conversely, the HRV indices are limited in scope
fordifferentiating betweenpathophysiological states or patients.
However,
whenappliedtotheindividualpatientoveratimeperiod,theseindicesmayprovetobeclinicallyuseful,differentiatingtheprogressionofdisease.Furthermore,theymight
provide a valuable addition to current patient monitoring
systems.Therefore, cardiovascular variability based risk
stratication might be
morepowerfulinlongitudinalstudies.Afurtherimprovementinthediagnosticsofcardiovasculardiseasesandriskstraticationfor
arrhythmic fatal events is expected, andpartly proven, bycombining
NLD methods of HRV analysis with additional cardiovascular
signals,i.e. bloodpressureandrespiration. Theanalysisof
interactions, couplingsandsynchronizations is also a powerful tool
for research in cardiovascular regulation.291
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)Finally,
coupling models accounting for a large range of time scales
andintervals are central to describing complexsystems andtherefore
to
biology(Coveney&Fowler2005).TherearesomeimportantpointsthatonehastoconsiderwhenapplyingthemethodsfromNLD.(i)
Oneparameteralone(independentfromthedomain)cannotsufcientlydescribecomplexphysiological
systems, suchasHRcontrol. Therefore,multivariate approaches should
be considered. NLD parameters incombination with standard linear
parameters usually improve theperformanceofHRVanalysis.(ii) The
type of underlying disease often determines the applicability of
HRVindicesfordiagnosticsorriskstratication.(iii)
Thereareanumberoffactorsthatmightaffecttheresultsobtainedbynonlinear
methods and consequently have to be considered, e.g.
recordingduration, degree of stationarity, superimposed noise and
signal pre-processing(ltering).(iv) NLDindices are often introduced
with special xed presettings (e.g.windowlength, number of bins)
that havebeenprovedandoptimizedin various studies and have to be
considered in new comparativeinvestigations.In conclusion, methods
derived from NLD have provided new insights into theHRVchanges
under various physiological andpathophysiological conditions.They
provide additional prognostic information and complement
traditionaltime-andfrequency-domainanalysesofHRV.Today,thequestionisnolongerabout
whether or not methods fromNLDshouldbe applied; however, it
isrelevanttoaskwhichofthemethodsshouldbeselectedandunderwhichbasicandstandardizedconditionsshouldtheybeapplied.ReferencesAmaral,
L. A., Goldberger, A. L., Ivanov, P. C. &Stanley, H. E. 1999
Modeling heart ratevariabilitybystochasticfeedback. Comput. Phys.
Commun. 121122, 126128.
(doi:10.1016/S0010-4655(99)00295-7)Babloyantz, A. &Destexhe, A.
1988Isthenormal heartaperiodicoscillator?Biol. Cybern.
58,203211.(doi:10.1007/BF00364139)Bar, K. J., Boettger, M. K.,
Koschke, M., Schulz, S., Chokka, P., Yeragani, V. K. & Voss, A.
2007Non-linear complexity measures of heart rate variability in
acute schizophrenia.
Clin.Neurophysiol.118,20092015.(doi:10.1016/j.clinph.2007.06.012)Baselli,
G., Porta, A. & Pagani, M. 2006 Coupling arterial windkessel
with peripheral vasomotion:modeling the effects on low-frequency
oscillations. IEEETrans. Biomed. Eng. 53,
5364.(doi:10.1109/TBME.2005.859787)Baumert, M., Walther, T., Hopfe,
J., Stepan, H., Faber, R. &Voss, A. 2002Joint symbolicdynamic
analysis of beat-to-beat interactions of heart rate and systolic
blood pressure in
normalpregnancy.Med.Biol.Eng.Comput.40,241245.(doi:10.1007/BF02348131)Baumert,
M., Baier, V., Haueisen, J., Wessel, N., Meyerfeldt, U.,
Schirdewan, A. & Voss, A. 2004Forecasting of life threatening
arrhythmias using the compression entropy of heart
rate.MethodsInf.Med.43,202206.A.Vossetal.
292Phil.Trans.R.Soc.A(2009)Baumert, M., Baier,V.& Voss,A.2005
Estimatingthecomplexityof
heartrateuctuationsanapproachbasedoncompressionentropy. NoiseLett.
4, L557L563. (doi:10.1142/S0219477505003026)Beckers, F., Verheyden,
B. &Aubert, A. E. 2006Agingandnonlinear heart ratecontrol
inahealthy population. Am. J. Physiol. Heart Circ. Physiol. 290,
H2560H2570. (doi:10.1152/ajpheart.00903.2005)Bigger Jr, J. T.,
Steinman, R. C., Rolnitzky, L. M., Fleiss, J. L., Albrecht, P.
& Cohen, R. J. 1996Power lawbehavior of RR-interval
variabilityinhealthymiddle-agedpersons, patients withrecentacute
myocardial infarction, and patients with heart transplants.
Circulation 93, 21422151.Brennan, M., Palaniswami, M. & Kamen,
P. 2002 Poincare plot interpretation using aphysiological model of
HRVbasedonanetworkof oscillators. Am. J. Physiol. Heart
CircPhysiol.283,H1873H1886.Chafn, D. G., Goldberg, C. C. &
Reed, K. L. 1991 The dimension of chaos in the fetal heart
rate.Am.J.Obstet.Gynecol.165(Pt1),14251429.Christini, D. J.,
Bennett, F. M., Lutchen, K. R., Ahmed, H. M., Hausdorff, J. M.
& Oriol, N. 1995Application of linear and nonlinear time series
modeling to heart rate dynamics analysis.
IEEETrans.Biomed.Eng.42,411415.(doi:10.1109/10.376135)Costa, M.,
Goldberger, A. L. &Peng, C. K. 2002 Multiscale entropy analysis
of complexphysiologictimeseries.Phys.Rev.Lett.89,068
102.(doi:10.1103/PhysRevLett.89.068102)Costa, M., Goldberger,A. L.
& Peng, C.-K. 2005 Multiscaleentropy analysis of biological
signals.Phys.Rev.E71(Pt1),021
906.(doi:10.1103/PhysRevE.71.021906)Coveney, P. V. &Fowler, P.
W. 2005 Modelling biological complexity: a physical
scientistsperspective.J.R.Soc.Interface2,267280.(doi:10.1098/rsif.2005.0045)Eckmann,
J. P. & Ruelle, D. 1985 Ergodic theory of chaos and strange
attractors. Rev. Mod.
Phys.57,617656.(doi:10.1103/RevModPhys.57.617)Garnkel,A.1983Amathematicsforphysiology.Am.J.Physiol.245,R455R466.Glass,
L. & Mackey, M. C. 1988 From clocks to chaos: the rhythms of
life. Princeton, NJ: PrincetonUniversityPress.Goldberger, A. L.
1991Is thenormal heartbeat chaoticor homeostatic?News Physiol. Sci.
6,8791.Goldberger, A. L. &West, B. J. 1987Applicationsof
nonlineardynamicstoclinical
cardiology.Ann.NYAcad.Sci.504,195213.(doi:10.1111/j.1749-6632.1987.tb48733.x)Goldberger,
A. L., Rigney, D. R., Mietus, J., Antman, E. M. &Greenwald, S.
1988Nonlineardynamics in sudden cardiac death syndrome: heart rate
oscillations and
bifurcations.Experientia44,983987.(doi:10.1007/BF01939894)Goldberger,A.L.,Amaral,L.
A.,Hausdorff,
J.M.,Ivanov,P.C.,Peng,C.K.&Stanley,H.E.2002Fractaldynamicsinphysiology:alterationswithdiseaseandaging.Proc.NatlAcad.Sci.USA19(Suppl.1),24662472.(doi:10.1073/pnas.012579499)Gomes,
M. E., Souza, A. V., Guimaraes, H. N. & Aguirre, L. A. 2000
Investigation of
determinisminheartratevariability.Chaos10,398410.(doi:10.1063/1.166507)Grassberger,P.&Procaccia,I.1983aMeasuringthestrangenessofstrangeattractors.PhysicaD9,189208.(doi:10.1016/0167-2789(83)90298-1)Grassberger,
P. & Procaccia, I. 1983b Estimation of the Kolmogorov entropy
from a chaotic
signal.Phys.Rev.A28,25912593.(doi:10.1103/PhysRevA.28.2591)Guevara,
M. R., Glass, L. &Shrier, A. 1981Phase locking,
period-doublingbifurcations, andirregular dynamics
inperiodicallystimulatedcardiac cells. Science 214, 13501353.
(doi:10.1126/science.7313693)Guzzetti, S., Mezzetti, S., Magatelli,
R., Porta, A., De Angelis, G., Rovelli, G. & Malliani, A.
2000Linearandnon-linear24
hheartratevariabilityinchronicheartfailure.Auton.Neurosci.86,114119.(doi:10.1016/S1566-0702(00)00239-3)Guzzetti,S.etal.2005Symbolicdynamicsofheartratevariability:aprobetoinvestigatecardiacautonomic
modulation. Circulation 112, 465470.
(doi:10.1161/CIRCULATIONAHA.104.518449)293
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)Hadamard,J.1898Lessurfacesacourburesoppose
esetleurslignesgeode
siques.J.Math.PuresAppl.4,2773.Hejjel,L.&Gal,I.2001Heartratevariabilityanalysis.ActaPhysiol.Hung.88,219230.(doi:10.1556/APhysiol.88.2001.3-4.4)Ho,
K. K., Moody, G. B., Peng, C. K., Mietus, J. E., Larson, M. G.,
Levy, D. & Goldberger, A. L.1997Predictingsurvival
inheartfailurecaseandcontrol subjectsbyuseoffullyautomatedmethods
for deriving nonlinear and conventional indices of heart rate
dynamics. Circulation96,842848.Hu, K., Scheer, F. A., Buijs, R. M.
&Shea, S. A. 2008Theendogenous circadianpacemakerimparts a
scale-invariant pattern of heart rate uctuations across time scales
spanning
minutesto24hours.J.Biol.Rhythms23,265273.(doi:10.1177/0748730408316166)Huikuri,H.V.,Makikallio,T.H.,Peng,C.K.,Goldberger,A.L.,Hintze,U.&Mller,
M.2000Fractal correlation properties of RRinterval dynamics and
mortality in patients withdepressed left ventricular function after
an acute myocardial infarction. Circulation101, 4753.Ivanov, P. C.,
Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. G.
&Struzik,Z. H. E. 1999Multifractalityinhumanheartbeat dynamics.
Nature (Lond.) 399, 461465.(doi:10.1038/20924)Javorka, M.,
Trunkvalterova, Z., Tonhajzerova, I., Javorkova, J., Javorka, K.
& Baumert, M. 2008Short-termheart
ratecomplexityisreducedinpatientswithtype1diabetesmellitus.
Clin.Neurophysiol.119,10711081.(doi:10.1016/j.clinph.2007.12.017)Kamen,P.W.&Tonkin,A.M.1995ApplicationofthePoincare
plottoheartratevariability:anewmeasureof
functionalstatusinheartfailure.Aust.NZJ.Med.25,1826.Kamen, P. W.,
Krum, H. &Tonkin, A. M. 1996Poincare plot of heart
ratevariabilityallowsquantitative displayof parasympathetic nervous
activityinhumans. Clin. Sci. (Lond.)
91,201208.Kaplan,D.&Glass,L.1995Understandingnonlineardynamics.NewYork,NY:Springer.Kaplan,
D. T., Furman, M. I., Pincus, S. M., Ryan, S. M., Lipsitz, L. A.
& Goldberger, A. L.
1991Agingandthecomplexityofcardiovasculardynamics.Biophys.J.59,945949.Khoo,
M. C. 2008 Modeling of autonomic control in sleep-disordered
breathing. Cardiovasc. Eng.
8,3041.(doi:10.1007/s10558-007-9041-9)Kleiger, R. E., Miller, J.
P., Bigger, J. T. & Moss, A. R. 1987 Multicenter
post-infarction
researchgroup.Decreasedheartratevariabilityanditsassociationwithincreasedmortalityafteracutemyocardialinfarction.Am.J.Cardiol.59,256262.(doi:10.1016/0002-9149(87)90795-8)Kobayashi,M.&Musha,T.19821/f
uctuationofheartbeatperiod.IEEETrans.Biomed.Eng.29,456457.(doi:10.1109/TBME.1982.324972)Kurths,
J., Voss, A., Saparin, P., Witt, A., Kleiner, H. J. & Wessel,
N. 1995 Quantitative
analysisofheartratevariability.Chaos5,8894.(doi:10.1063/1.166090)Laitio,T.T.etal.2002Relationofheartratedynamicstotheoccurrenceofmyocardialischemiaaftercoronaryarterybypassgrafting.Am.J.Cardiol.89,11761181.(doi:10.1016/S0002-9149(02)02300-7)Lake,
D. E., Richman, J. S., Grifn, M. P. &Moorman, J. R.
2002Sampleentropyanalysisofneonatal heart rate variability. Am. J.
Physiol. Regul. Integr. Comp. Physiol.283,
R789R797.Li,M.&Vitnyi,P.A.1997AnintroductiontoKolmogorovcomplexityanditsapplications.NewYork,NY:Springer.Lin,
D. C. & Hughson, R. L. 2001 Modeling heart rate variability in
healthy humans: a
turbulenceanalogy.Phys.Rev.Lett.86,16501653.(doi:10.1103/PhysRevLett.86.1650)Lombardi,
F. 2000 Chaos theory, heart rate variability, and arrhythmic
mortality. Circulation 101,810.Lombardi, F., Porta, A., Marzegalli,
M., Favale, S., Santini, M., Vincenti, A., DeRosa, A.
&participating investigators of ICD-HRVItalian Study Group.
2000 Heart rate variabilitypatterns before ventricular tachycardia
onset inpatients withanimplantable
cardioverterdebrillator.Am.J.Cardiol.86,959963.(doi:10.1016/S0002-9149(00)01130-9)A.Vossetal.
294Phil.Trans.R.Soc.A(2009)Maestri,R., Pinna, G. D., Balocchi,R.,
DAddio, G.,Ferrario, M., Porta,A., Sassi,R., Signorini,M. G. &
La Rovere, M. T. 2006 Clinical correlates of non-linear indices of
heart rate
variabilityinchronicheartfailurepatients.Biomed.Tech.(Berl.)51,220223.Maestri,R.etal.2007Nonlinearindicesofheartratevariabilityinchronicheartfailurepatients:redundancy
and comparative clinical value. J. Cardiovasc. Electrophysiol.18,
425433. (doi:10.1111/j.1540-8167.2007.00728.x)Makikallio, T. 1998
Analysis of heart rate dynamics by methods derived
fromnonlinearmathematicsclinical applicability and prognostic
signicance, ch. 2. PhD thesis, OuluUniversityLibrary,Oulu.Ma
kikallio, T. H., Seppanen, T., Niemela, M., Airaksinen, K. E.,
Tulppo, M. & Huikuri, H. V. 1996Abnormalities inbeat tobeat
complexityof heart ratedynamics inpatients withapreviousmyocardial
infarction. J. Am. Coll. Cardiol. 28, 10051011.
(doi:10.1016/S0735-1097(96)00243-4)Makikallio, T. H., Hoiber, S.,
Kober, L., Torp-Pedersen, C., Peng, C. K., Goldberger, A.
L.,Huikuri, H. V. &TRACEinvestigators. 1999Fractal analysis of
heart rate dynamics as apredictor of mortality in patients with
depressed left ventricular function after acute
myocardialinfarction.Am.J.Cardiol.83,836839.(doi:10.1016/S0002-9149(98)01076-5)Makikallio,
T. H., Huikuri, H. V., Hintze, U., Videbaek, J., Mitrani, R. D.,
Castellanos, A.,Myerburg, R. J., Moller, M.
&forDIAMONDStudyGroup. 2001Fractal
analysisandtime-andfrequency-domainmeasuresofheartratevariabilityaspredictorsofmortalityinpatientswithheartfailure.Am.J.Cardiol.87,178182.(doi:10.1016/S0002-9149(00)01312-6)Norris,
P. R., Anderson, S. M., Jenkins, J. M., Williams, A. E. &
Morris Jr, J. A. 2008 Heart
ratemultiscaleentropyatthreehourspredictshospital
mortalityin3,154traumapatients. Shock30,1722.Novak, V., Novak, P.,
de Champlain, J., Le Blanc, A. R., Martin, R. & Nadeau, R. 1993
Inuenceofrespirationonheartrateandbloodpressureuctuations.J.Appl.Physiol.74,617626.Oxenham,H.&Sharpe,N.2003Cardiovascularagingandheartfailure.Eur.J.HeartFailure5,427434.(doi:10.1016/S1388-9842(03)00011-4)Parati,
G., Di Rienzo, M., Bertinieri, G., Pomidossi, G., Casadei, R.,
Groppelli, A., Pedotti, A.,Zanchetti, A. &Mancia, G.
1988Evaluationof
thebaroreceptor-heartratereexby24-hourintra-arterialbloodpressuremonitoringinhumans.Hypertension12,214222.Peng,
C. K., Havlin, S., Stanley, H. E. &Goldberger, A. L. 1995
Quantication of
scalingexponentsandcrossoverphenomenainnonstationaryheartbeattimeseries.
Chaos5, 8287.(doi:10.1063/1.166141)Pincus, S. M. 1991 Approximate
entropy as a measure of system complexity. Proc. Natl Acad.
Sci.USA88,22972301.(doi:10.1073/pnas.88.6.2297)Pompe,B.,Blidh,P.,Hoyer,D.&Eiselt,M.1998Usingmutualinformationtomeasurecouplingin
the cardiorespiratory system. IEEE Eng. Med. Biol. Mag.17, 3239.
(doi:10.1109/51.731318)Porta, A., Guzzetti, S., Montano, N.,
Furlan, R., Pagani, M., Malliani, A. &Cerutti, S. 2001Entropy,
entropyrateandpatternclassicationastoolstotypifycomplexityinshortheartperiodvariabilityseries.IEEETrans.Biomed.Eng.48,12821291.(doi:10.1109/10.959324)Porta,A.etal.2007Anintegratedapproachbasedonuniformquantizationfortheevaluationofcomplexity
of short-termheart periodvariability: applicationto 24 hHolter
recordings inhealthyandheartfailurehumans.Chaos17,015
117.(doi:10.1063/1.2404630)Richman, J. S. &Moorman, J. R. 2000
Physiological time-series analysis using
approximateentropyandsampleentropy.Am.J.Physiol.HeartCirc.Physiol.278,H2039H2049.Ritzenberg,
A. L., Adam, D. R. &Cohen, R. J.
1984Periodmultuplingevidencefornonlinearbehaviourofthecanineheart.Nature307,159161.(doi:10.1038/307159a0)Ryan,S.M.,Goldberger,A.L.,Pincus,S.M.,Mietus,J.&Lipsitz,L.A.1994Gender-andage-relateddifferencesinheartratedynamics:
arewomenmorecomplexthanmen?J. Am. Coll.Cardiol. 24,17001707.Saul,
J. P., Albrecht, P., Berger, R. D. &Cohen, R. J. 1987Analysis
of longtermheart
ratevariability:methods,1/fscalingandimplications.Comput.Cardiol.14,419422.295
Review.NonlineardynamicsforanalysingHRVPhil.Trans.R.Soc.A(2009)Schumacher,A.2004Linearand
nonlinearapproachestotheanalysisofRR
intervalvariability.Biol.Res.Nurs.5,211221.(doi:10.1177/1099800403260619)Schwab,
K., Eiselt, M., Putsche, P., Helbig, M. &Witte, H. 2006
Time-variant parametricestimationof
transientquadraticphasecouplingsbetweenheartratecomponentsinhealthyneonates.Med.Biol.Eng.Comput.44,10771083.(doi:10.1007/s11517-006-0120-7)Shiogai,Y.2007Nonlineardynamicsofcardiovascularaging.PhDthesis,DepartmentofPhysics,LancasterUniversity,UK.Stein,
P. K., Domitrovich, P. P., Huikuri, H. V. &Kleiger, R. E. 2005
Cast investigators.Traditional and nonlinear heart rate variability
are each independently associated withmortalityaftermyocardial
infarction. J. Cardiovasc. Electrophysiol. 16, 1320.
(doi:10.1046/j.1540-8167.2005.04358.x)Takens,F. 1981 Detecting
strangeattractors in turbulence.In Dynamical systemsand
turbulence,vol.898(edsD.A.Rand&L.-S.Young).LectureNotesinMathematics,pp.366381.Tapanainen,
J. M., Thomsen,P. E., Kober, L., Torp-Pedersen,C., Makikallio, T.
H., Still, A.
M.,Lindgren,K.S.&Huikuri,H.V.2002Fractalanalysisofheartratevariabilityandmortalityafter
an acute myocardial infarction. Am. J. Cardiol. 90, 347352.
(doi:10.1016/S0002-9149(02)02488-8)TaskForceofthe EuropeanSocietyof
Cardiologyand theNorthAmericanSocietyof PacingandElectrophysiology
1996 Heart rate variabilitystandards of measurement,
physiologicalinterpretationandclinicaluse.Circulation93,10431065.Truebner,
S., Cygankiewicz, I., Schroeder, R., Baumert, M., Vallverdu, M.,
Caminal, P., Vazquez,R., Baye s de Luna, A. & Voss, A. 2006
Compression entropy contributes to risk stratication
inpatientswithcardiomyopathy.Biomed.Tech.(Berl.)51,7782.Tulppo, M.
P., Kiviniemi, A. M., Hautala, A. J., Kallio, M., Seppanen, T.,
Makikallio, T. H. &Huikuri, H. V. 2005 Physiological background
of the loss of fractal heart rate
dynamics.Circulation112,314319.(doi:10.1161/CIRCULATIONAHA.104.523712)Tuzcu,
V., Nas, S., Borklu, T. & Ugur, A. 2006 Decrease in the heart
rate complexity prior to
theonsetofatrialbrillation.Europace8,398402.(doi:10.1093/europace/eul031)Vinet,
A., Chialvo, D. R. &Jalife, J. 1990 Irregular dynamics of
excitation in biologic
andmathematicalmodelsofcardiaccells.Ann.NYAcad.Sci.601,281298.(doi:10.1111/j.1749-6632.1990.tb37307.x)Voss,
A., Dietz, R., Fiehring, H., Kleiner, H. J., Kurths, J., Saparin,
P., Vossing, H. J. & Witt, A.1993HighresolutionECG,
heartratevariabilityandnonlineardynamics:
toolsforhighriskstratication. InComputers incardiology, pp. 261264.
Los Alamitos, CA: IEEEComputerSocietyPress.Voss, A., Kurths, J.,
Kleiner, H. J., Witt, A., Wessel, N., Saparin, P., Osterziel, K.
J., Schurath, R. &Dietz, R. 1996 The application of methods of
non-linear dynamics for the improved and predictiverecognition of
patients threatened by sudden cardiac death. Cardiovasc. Res.31,
419433.Voss, A., Hnatkova, K., Wessel, N., Kurths, J., Sander, A.,
Schirdewan, A., Camm, A. J. & Malik,M.
1998Multiparametricanalysisof
heartratevariabilityusedforriskstraticationamongsurvivors of acute
myocardial infarction. Pacing Clin. Electrophysiol. 21(Pt 2),
186192. (doi:10.1111/j.1540-8159.1998.tb01086.x)Weiss, J. N.,
Garnkel, A., Spano, M. L. & Ditto, W. L. 1994 Chaos and chaos
control in
biology.J.Clin.Invest.93,13551360.(doi:10.1172/JCI117111)Wolf, M.
M., Varigos, G. A., Hunt, D. & Sloman, J. G. 1978 Sinus
arrhythmia in acute
myocardialinfarction.Med.J.Aust.2,5253.Ziv,J.&Lempel,A.1977Universalalgorithmforsequentialdatacompression.IEEETrans.Inf.Ther.23,337343.(doi:10.1109/TIT.1977.1055714)A.Vossetal.
296Phil.Trans.R.Soc.A(2009)
