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Delignières,D.,Almurad,Z.M.H.,Roume,C.&Marmelat,V.(inpress).Multifractalsignaturesofcomplexitymatching.ExperimentalBrainResearch.DOI:10.1007/s00221-016-4679-4

Multifractalsignaturesofcomplexitymatching

DidierDelignières1,ZainyM.H.Almurad1,2,ClémentRoume1&VivienMarmelat1,3

1.EA2991Euromov,UniversityofMontpellier,France2.FacultyofPhysicalEducation,UniversityofMossul,Irak

3.CenterforResearchinHumanMovementVariability,UniversityofNebraskaatOmaha,USA

Abstract:

The complexity matching effect supposes that synchronization between complexsystemscouldemerge frommultiple interactionsacrossmultiple scales, andhasbeenhypothesized to underlie a number of daily-life situations. Complexity matchingsuggests that coupled systems tend to share similar scaling properties, and thisphenomenon is revealed by a statisticalmatching between the scaling exponents thatcharacterize the respective behaviors of both systems. However, some recent paperssuggested that this statistical matching could originate from local adjustments orcorrections, rather than fromagenuine complexitymatchingbetween systems. In thepresent paper we propose an analysis method based on correlation betweenmultifractal spectra, considering different ranges of time scales. We analyze severaldatasets collected in various situations (bimanual coordination, interpersonalcoordination,walking in synchronywith a fractalmetronome). Our results show thatthismethod is able to distinguish between situations underlain by genuine statisticalmatching,andsituationswherestatisticalmatchingresultsfromlocaladjustments.

Key-words:Synchronization,coordination,complexitymatching,multifractals

Correspondingauthor:ProfDidierDelignièresEA2991,Euromov,UniversiyofMontpellier700avenueduPicSaintLoup34090Montpellier,Francedidier.delignieres@umontpellier.frTel:+33411759005Fax:+33411759024

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Introduction

The concept of complexity matching (West et al. 2008) states that the exchange ofinformationbetweentwocomplexnetworksismaximizedwhentheircomplexitiesaresimilar. This particular property requires both networks to generate 1/f fluctuations,andhasbeeninterpretedasakindof“1/fresonance”betweennetworks(Aquinoetal.2011).Insuchasituation,acomplexnetworkrespondstoastimulationbyanotherasafunctionof thematchingof theirmeasuresofcomplexity, i.e. thematchingof their1/ffluctuations. Incontrast, theresponseofacomplexnetworktoaharmonicstimulus isveryweakascomparedwiththatobtainedwithanothernetworkofsimilarcomplexity(Aquinoetal.2010;Mafahimetal.2015).A direct conjecture exploiting the complexity matching effect is when two complexsystems become coupled, they should attune their complexities in order to optimizeinformationexchange.ThisconjecturehasbeeninitiallytestedbyStephenetal.(2008),inataskwhereparticipantshadtosynchronizefingertapswithachaoticmetronome.Resultsshowedthatdespitetheunpredictablenatureofthestimuliparticipantswhereroughly able to synchronizewith the chaoticmetronome, with amix of reaction andproaction. As expected the authors observed a close matching between the scalingproperties of the inter-beat interval series of chaotic signals and those of thecorrespondinginter-tapintervalseriesofparticipants.

Marmelat and Delignières (2012) evidenced similar results in an inter-personalcoordination task where participants oscillated pendulums in synchrony. Resultsrevealedlowlocalcorrelationsbetweentheseriesofoscillationperiodsproducedbythetwoparticipantsofeachdyad.Theauthorsanalyzedthescalingpropertiesoftheseriesof periods produced by participants, and evidenced a very close correlation betweenfractal exponents. Similar results were evidenced by Abney, Paxton, Dale, and Kello(2014),intheanalysisofspeechsignalsduringdyadicconversations.However, the idea that the matching of scaling exponents could be considered anunambiguous signature of complexitymatching remains questionable, and it could benecessarytodistinguishbetweenthestatisticalmatching(i.e.,theconvergenceofscalingexponents) and the genuine complexity matching effect (i.e., the attunement ofcomplexities). For example Fine et al. (2015), in an experiment on rhythmicinterpersonalcoordination,observedas inpreviousexperimentsa typicalmatchingofscaling exponents, but suggested that this statistical matching could just result fromlocalphaseadjustments,andnotfromaglobalattunementofcomplexities.Delignièresand Marmelat (2014) analyzed series of stride durations produced by participantsattemptingtowalkinsynchronywithafractalmetronome.Theytriedtosimulatetheirempiricalresultsbymeansofamodelbasedon localcorrectionsofasynchronies,andshowed that this model was able to adequately reproduce the statistical matchingobservedinexperimentalseries.Theauthorsconcludedthatwalkinginsynchronywithafractalmetronomecouldessentiallyinvolveshort-termcorrectionprocesses,andthatthe close correlation observed between scaling exponents could in such a case justrepresent the consequence of local correction processes. Torre et al. (2013) alsosupported this hypothesis in a tapping task where participants synchronized withdifferent non-isochronous auditory metronomes. They evidenced that inter-tapintervalscouldbemodeledbasedonthepreviousinter-beatintervalofthemetronomeandacorrectionofpreviousasynchroniestothemetronome,independentlyofthelevelof1/ffluctuationsofthemetronome(i.e.whitenoiseorpinknoise).

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Our goal in the present paper is to seek for statistical signatures that couldunambiguously distinguish between genuine global complexity matching and localcorrections or adjustments. To date, most papers that tried to evidence complexitymatchingeffectsworkedonthebasisofmonofractalanalysis.Hereweproposetoadoptmultifractalanalysesbecausetheyallowforamoredetailedpictureofthecomplexityoftimeseries,andthetailoringoffluctuationsthatistypicalofcomplexitymatchingcouldbe considered as the product ofmultifractality (Stephen andDixon 2011).One of themost appealing hypotheses about the origin of fractal fluctuations in the behavior ofcomplexsystemsreferstotheideathattheinteractionsbetweensystem’snetworksaregoverned by multiplicative cascade dynamics (Ihlen and Vereijken 2010). Suchdynamicsissupposedtogeneratemultifractal,ratherthanmonofractalfluctuations,andindeed Ihlen and Vereijken (2010) showed that it was the case in most previousanalyzed series in the literature. Stephen and Dixon (2011) consider that complexitymatchingshouldbeconceivedasaproductofmultiplicativecascadedynamics,entailingacoordinationoffluctuationsamongmultipletimescales.Whilemonofractal processes are characterizedby long-term correlations and a singlescaling exponent, inmultifractal time series subsetswith small and large fluctuationsscale differently, and their description requires a hierarchy of scaling exponents(Podobnik and Stanley 2008).We propose to assess statistical matching through thepoint-by-point correlation function between the sets of scaling exponents thatcharacterizethecoordinatedseries.InthepresentpaperweusedtheMultifractalDetrendedFluctuationanalysis(MF-DFA)introduced by Kantelhardt et al. (2002), which is an extension of the DetrendedFluctuationAnalysis (DFA,Pengetal.1993). JustasDFA,MF-DFAallows toselect therange of intervals over which exponents are estimated. Usually authors considersintervals from8or10datapoints, inordertoallowaproperassessmentofstatisticalmoments,uptoN/4orN/2(Nrepresentingthelengthoftheanalyzedseries),inorderto get at least four or two estimates of these moments. Quite often, however, seriespresentdifferentscalingregimesovertheshortandthelongterm,andauthorsperformseparateestimatesoverdifferentrangesof intervals(DelignièresandMarmelat2014).Herewepropose to estimate the set ofmultifractal exponents in first over the entirerangeofavailable intervals(i.e., from8toN/2),andthenovermorerestrictedranges,progressivelyexcludingtheshortestintervals(i.e.,from16toN/2,from32toN/2,andthen from 64 to N/2). We expect to find in both cases (local corrections or globalmatching), a strong correlation pattern between exponents when considering longlengthintervals(i.e,64toN/2).Ifsynchronizationisjustbasedonlocalcorrections,weconsiderthatthisclosestatisticalmatchinginlongintervals is justtheconsequenceofthe short-term, local couplingbetween the two systems.As local correctionsbetweenunpredictable systems remains approximate,we hypothesize that correlations shoulddramaticallydecreasewhenintervalsofshorterdurationsaretakenintoconsideration.In contrast, in the case of genuine complexitymatching, the synchronization betweensystems is supposed to emerge from interactions across multiple scales. We thenhypothesize to find in this case close correlations, even when considering the entirerangeofintervals,fromtheshortesttothelongest.

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Methods

Inthepresentpaperwere-analyzeasetofexperimentalserieswhichwerepreviouslyused by Delignières and Marmelat (2014), in a first attempt to derive statisticalsignature of complexity matching from monofractal analyses. All studies have beenapproved by the local ethics committee and have therefore been performed inaccordancewiththeethicalstandardslaiddowninthe1964DeclarationofHelsinki.Allparticipantsgave their informedconsentprior to their inclusion in thestudy.We firstbrieflypresentthethreesetsofseriessubmittedtoanalysis.Bimanualcoordination.

The first set of series was collected in an experiment where twelve participantsperformed bimanual oscillations (Torre and Delignières 2008a; Torre andWagenmakers 2009). This kind of bimanual coordination task has been extensivelystudied in the dynamical systems approach to coordination, in order to evidence theemergentpropertiesthatunderliethemacroscopicbehaviorofcomplexsystems(Hakenetal.1985;Schöneretal.1986).Thetwolimbsareconsideredasasystemofcoupledoscillators,andbimanualcoordinationrepresentsaniceexampleofclosecoordinationbetween complex (sub)systems, embedded to form a global functional system. In thepresentcontext,thisfirstsetofseriesrepresentsalimitcase,wherecoordinationshouldbeachievedbycomplexitymatchingprocesses.

Participants were instructed to perform smooth and regular forearm oscillationsholding two joysticks, synchronizing the reversalpointsof themotionof the joysticks(in-phase coordination). This experiment used the synchronization-continuationparadigm:during30secparticipantssynchronizedtheirmovementswithavideomodel,inducinganinitialfrequencyof1.5Hzinafirstcondition,and2.0Hzinasecond.Thenthemodel was removed and participants had to continue following the initial tempoduring600cycles.Foreachhand,wecomputedtheseriesofperiods,definedasthetimeintervalsbetweentwosuccessivereversalpointsinmaximalpronation.Themeanperiodofoscillationsofthe effectorswas 665.24ms (+/- 63.01) in the 1.5 Hz condition, and 524.05ms (+/-66.52) in the 2.0 Hz. The standard deviation of asynchronies (i.e., the time intervalsbetweentherespectivepronationreversalpointsofthetwohands)was18.65ms(+/-5.26) in the 1.5 Hz condition and 14.45 ms (+/- 2.62) in the 2.0 Hz condition,corresponding to a mean relative phase was-6.33° (+/- 3.76), and -5.27° (+/- 6.01),respectively.Interpersonalsynchronization

Thesecondsetofserieswerecollectedinanexperimentoninterpersonalcoordination(Marmelat andDelignieres2012). In contrastwith theprevious example, these seriesrepresent coordination between two physically independent systems that interact forachieving a commongoal. Twenty-twoparticipantswere randomlypaired into elevendyads.Participants ineachdyadwere instructed toperformsynchronizedoscillationswith pendulums, in the sagittal plane, following an in-phase pattern of coordination.Theywereinstructedtooscillateatthepreferredfrequencyofthedyad,asregularlyaspossible.Thetaskwasperformedinthreeconditions,characterizedbyincreasinglevelsofcouplingbetweenparticipants. Intheweakcouplingcondition,auditionwaslimitedwithearplugs,andparticipantswereinstructedtovisuallyfixatargetinfrontofthemonthewall.Inthenormalcouplingcondition,visualandauditoryfeedbackswerefully

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available,andparticipantswere invitedtovisually fixtheirpartner’spendulum.Inthestrongcouplingcondition,participantswereinstructedtocrosstheirfreearms(arm-in-arm),inordertoaddhapticinformationtovisualandauditoryfeedbacks.Seriesof512oscillations were collected in each condition. For each participant, we computed theseriesofperiods,definedasthetimeintervalsbetweentwosuccessivereversalpointsinmaximalextension.

Resultsshowedthatdyadswereabletoperformadequatelythiscoordinationtask,withameanrelativephaseof-2.15°(+/-8.64)inthelowcouplingcondition,-1.67°(+/-7.50)inthenormalcouplingcondition,and-2.36°(+/-7.15)inthestrongcouplingcondition.

Themeanperiodofoscillationsoftheeffectorswas1035.06ms(+/-130.59),1018.47ms (+/- 126.70), and 989.35 ms (+/- 51.95), respectively. The standard deviation ofasynchronieswas 46.51ms (+/- 7.50), 34.78ms (+/- 9.10) and 37.71ms (+/- 6.06),respectively.Walkinginsynchronywithafractalmetronome

In this experiment participants had to walk in synchrony with a fractal metronome.Elevenparticipantswereinvolvedinthisexperiment.Theywalkedonatreadmill,andhadtosynchronizetherightheelstrikeswithmetronomesignalsadministeredthroughanearphone.Metronomesignalspresentedfractalfluctuationswithameanαexponentof about0.9, ameanvalueof1135ms, and their standarddeviationwasadjusted forobtaining a coefficient of variation of 2%. Series of 512 strides, defined as the timeintervalsbetweentwosuccessiverightheelstrikes,werecollected.Resultsshowedthatparticipants were able to maintain synchrony with the metronome, with a meanasynchronyofabout-52.8ms(+/-46.9).Themeanstridedurationwas1135.53ms(+/-51.95),andthestandarddeviationofasynchronieswas46.94ms(+/-11.68).

MultifractalDetrendedFluctuationanalysis(MF-DFA)

InthepresentpaperweusedtheMF-DFAmethod,initiallyintroducedbyKantelhardtetal(2002).Considertheseriesx(i),i=1,2,…,N.Inafirststeptheseriesiscenteredandintegrated:

X k( ) = x i( )− 1N

x i( )i=1

N

∑⎡⎣⎢

⎤⎦⎥i=1

k

∑ (1)

Next,theintegratedseriesX(k) isdividedintoNnnon-overlappingsegmentsoflengthnandineachsegments=1,...,NnthelocaltrendisestimatedandsubtractedfromX(k).

Thevarianceiscalculatedforeachdetrendedsegment:

F2 n, s( ) = 1n

X k( )− Xn,s k( )⎡⎣ ⎤⎦k=(s−1)n+1

sn

∑2

(2)

andthenaveragedoverallsegmentstoobtainqthorderfluctuationfunction

Fq (n) =1Nn

F2 n, s( )⎡⎣ ⎤⎦s=1

Nn

∑q/2⎧

⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

1/q

(3)

whereqcantakeanyrealvalueexceptzero.Inthepresentworkweusedintegervaluesforq,from-15to+15.NotethatEq.(3)cannotholdforq=0,becauseofthedivergingexponent.Alogarithmicaveragingprocedureisusedforthisspecialcase:

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F0 (n) = exp12Nn

ln F2 n, s( )⎡⎣ ⎤⎦s=1

Nn

∑⎛⎝⎜

⎞⎠⎟

(4)

Repeatingthiscalculationforalllengthsnprovidestherelationshipbetweenfluctuationfunction Fq(n) and segment length n. If long-term correlations are present, Fq(n)increaseswithnaccordingtoapowerlaw:

Fq n( )∝ nh(q) (5)

Thescalingexponenth(q) isobtainedastheslopeof the linearregressionof logFq(n)versus logn. Note that for stationary time series,h(2) is identical to thewell-knownHurstexponentH,andthereforeh(q)iscalledthegeneralizedHurstexponent.

For positive values of q the generalized Hurst exponent h(q) describes the scalingbehavioroflargefluctuations,whilefornegativevaluesofq,h(q)describesthescalingbehavior of small fluctuations. Formonofractal time seriesh(q) is independent ofq ,whileformultifractaltimeseriessmallandlargefluctuationsscaledifferentlyandh(q)isadecreasingfunctionofq.

The results of theMF-DFA can then be converted into themore classicalmultifractalformalismbysimpletransformations(Kantelhardtetal.2002):first,generalizedHurstexponentsh(q)arerelatedtotheRenyiexponentsτ(q)definedbythestandardpartitionfunction-basedmultifractalformalism:

τ (q) = qh(q)−1 (6)

Formonofractal signalsτ(q) is linear functionofq, and formultifractal signalsτ(q) isnonlinear function of q . Another way to characterize multifractal process is thesingularityspectrumf(α)whichisrelatedtoτ(q)throughtheLegendretransform:

α (q) = dτ (q)dq

(7)

f (α ) = qα −τ (q) (8)

where f(α) is the fractaldimensionof thesupportofsingularities in themeasurewithLipschitz-Hölder exponent α. The singularity spectrum of monofractal signal isrepresented by a single point in the f(α) plane,whereasmultifractal process yields asinglehumpedfunction.

Thefocus-basedapproachtomultifractalanalysis

The classical MF-DFA algorithm was shown to perform as well as other multifractalmethods (Oswiecimka et al. 2006; Schumann and Kantelhardt 2011). However,especiallywhenappliedonempiricalseries it isknowntooftenproduce theso-called“inversed”spectra,exhibitingazig-zagshapesratherthantheexpectedparabolicshape(Makowiecetal.2011;Muklietal.2015).Mukli et al. (2015) have recently introduced a focus-based approachwhich allows toavoid this pitfall. The standard method assesses the scaling exponents h(q)independently for each q value. This procedure can be considered adequate if anassumptiononhomogeneous structuringholds for the scaling function.This propertyhowevermaynotalwaysbepresentespeciallyinempiricalsignals.

Theoretically, the moment-wise scaling functions, for all q values, should converge

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towardacommonlimitvalueatthecoarsestscale.Indeed,substitutingsignallength(N)tointervallength(n)inEq.(3)yields:

Fq (N ) =1NN

F2 N , s( )⎡⎣ ⎤⎦s=1

NN

∑q/2⎧

⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

1/q

= F2 N , s( )q/2{ }1/q = F N , s( ) (9)

F N , s( ) canthenbeconsideredthetheoreticalfocusofthescalingfunctions,andMuklietal.(2015)proposedtousethisfocusasaguidingreferencewhenregressingforh(q).Inessence,onecaniterateonh(q)astheidealmultifractalwithitsgivenfocusandsetofassociated slopes best fitting to the observeddata of the scaling function.Mukli et al.(2015) showed that this method allowed to successfully avoid the occurrence ofinversedspectra.

Asexplainedintheintroduction,wefirstappliedMF-DFAconsideringthewidestrangeof intervals, from 8 to 256 (N/2). We then replicated this analysis by progressivelyfocusingonlongerintervals:16to256,32to256,and64to256.

WefinallycomputedforeachqvaluethecorrelationbetweentheindividualLipschitz-Hölder exponents characterizing the two coordinated systems, α1(q) and α2(q),respectively,yieldingacorrelation functionr(q).Aspreviouslyexplained,weexpectedtofindinallcasesacorrelationfunctioncloseto1,forallqvalues,whenonlythelargestintervals are considered. Increasing the range of considered intervals should have anegligibleimpactonr(q)whencoordinationisbasedonacomplexitymatchingeffect.Incontrast, if coordination is based on local corrections, a decrease in r(q) should beobserved,asshorterandshorterintervalsareconsidered.

Results

Bimanualcoordination.

Wepresent inFigure1 (upperpanels) theaveragedmultifractal spectraof theperiodseries,consideringintervalsfrom8to256points,inthe1.5Hzcondition(a)andthe2.0condition(b).Thespectraoftherightandthelefthandarecloselysuperimposedinbothconditions.Thecorrelationfunctionsbetweenthemultifractalspectraarepresentedinbottompanels.Correlationcoefficientsareplottedagainsttheircorrespondingqvalues.Four correlation functions are displayed, according to the shortest interval lengthconsideredduringtheanalysis(8,16,32,or64).Inallcasesthecorrelationsfunctionsdisplayed very high values, close to 1.0 (Figure 1, c and d). Especially in the 1.5 Hzcondition the correlations between spectra appeared maximal, over all q values andwhatevertheconsideredintervalsrange.Correlationsappearedslightlylower(around0.90), in the 2.0 Hz condition, for negative values of q, andwhen the entire range ofintervalswereconsidered.

Interpersonalsynchronization.

Wepresent inFigure2 (upperpanels) theaveragedmultifractal spectraof theperiodseriesinthelowcoupling(a),normalcoupling(b)andstrongcoupling(c)conditions.Asforthepreviousexperiment,weobservedaclosesuperimpositionofthetwoaveragedspectra, and the superimposition appeared closer as coupling strength increased. Thecorrelation functionsbetween themultifractal spectraarepresented inbottompanels(d,eandf).Inallcasesthecorrelationfunctiondisplayedveryhighvalues,closeto1.0,whenanalysisfocusedtolongintervals(i.e.32toN/2or64toN/2).

8

1

1.5Hz 2.0Hz

a b

c d

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

Figure1:Bimanualcoordination.Upperpanels:Multifractalspectrafortheright(blackcircles) and the left (white circle) effectors, for the 1.5 Hz (a) and the 2.0 Hz (b)conditions.Bottompanels: Correlation functions r(q), for the four rangesof intervalsconsidered(8toN/2,16toN/2,32toN/2,and64toN/2),forthe1.5Hz(c)andthe2.0Hz(d)conditions.qrepresentsthesetofordersoverwhichtheMF-DFAalgorithmwasapplied.

Walkinginsynchronywithafractalmetronome

WepresentinFigure3(panela)theaveragedmultifractalspectraofthestridedurationseries(blackcircles)andthemetronomeseries(whitecircles).Inthisexperimentashiftwasobservedbetweenthetwoaveragedspectra,indicatingalowerlevelofcorrelationin participants series, with respect to the corresponding metronomes series. Thecorrelationfunctionsbetweenthemultifractalspectraarepresentedintherightpanel(b).Incontrastwiththepreviousresults,theconsideredrangeofintervalshadastrongimpactonthecorrelationfunction.Whenthesmallestrangewasconsidered(64toN/2),the correlation function remained close to 1.0. When the range of interval wasprogressivelyenlarged,thecorrelationvaluesdecreased,especiallyfornegativeqvaluescorresponding to low variance epochs in the series. When all available intervals areconsidered(i.e.8toN/2),r(q)presentsnonsignificantvalues.

9 2

LowCoupling NormalCoupling StrongCoupling

a b c

d e f

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%Co

rrela1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

Figure 2: Interpersonal synchronization. Upper panels: Multifractal spectra forparticipantA(blackcircles)andparticipantB(whitecircle),forthelow(a),normal(b)and strong (c) coupling conditions.Bottompanels: Correlation functions r(q), for thefourrangesofintervalsconsidered(8toN/2,16toN/2,32toN/2,and64toN/2),forthelow(d),normal(e)andstrong(f)couplingconditions.

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a b

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

Figure3:Walkinginsynchronywithafractalmetronome.Panela:Multifractalspectrafor the participant (black circles) and the metronome (white circle). Panel b:Correlationfunctionsr(q), forthefourrangesof intervalsconsidered(8toN/2,16toN/2,32toN/2,and64toN/2).

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Inorder toprovideaclearerpictureof theevolutionofcorrelationswith therangeofconsidered intervals in the three experiments,we present in Figure 4 a set of scatterplots representing the relationships between theα(2) samples characterizing the twocoordinatedsystems.Thesegraphsshowaglobalnarrowingofexponent’ssamples,asthe considered range of intervals increases. However, the decrease of correlation,especiallyinthethirdexperiment,clearlyarisesfromaweakeningoftherelationshipsbetweenexponents.

4

64-N/2 32-N/2 16-N/2 8-N/2

a

b

c

0.6

0.8

1

1.2

1.4

1.6

1.8

0.6 0.8 1 1.2 1.4 1.6 1.8

α(2)Ler)Ha

nd

α(2)RightHand

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

α(2)Ler)Ha

nd

α(2)RightHand

0.55

0.65

0.75

0.85

0.95

1.05

1.15

0.6 0.7 0.8 0.9 1 1.1 1.2

α(2)Le(

Han

d

α(2)RightHand

0.5

0.55

0.6

0.65

0.7

0.75

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1

0.55 0.65 0.75 0.85 0.95

α(2)Le(

Han

d

α(2)RightHand

0.4

0.6

0.8

1

1.2

1.4

1.6

0.4 0.6 0.8 1 1.2 1.4 1.6

α(2)Par)cipan

tB

α(2)Par)cipantA

0.5

0.6

0.7

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1

1.1

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1.3

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tB

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0.6

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0.65

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t

α(2)Metronome

0.55

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t

α(2)Metronome

0.7

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t

α(2)Metronome

0.65

0.7

0.75

0.8

0.85

0.9

0.75 0.85 0.95 1.05

α(2)Par)cipan

t

α(2)Metronome

Figure 4: Scatter plots of the samples of Hölder exponents α(2) characterizing thecoupledseries,inthethreeexperiments,forthefourrangesofintervalsconsidered(64toN/2,32toN/2,16toN/2,and8toN/2).Upperrow(a):bimanualcoordination,F1;Middlerow(b): interpersonalcoordination,strongcoupling;Bottomrow(c):walkinginsynchronywithafractalmetronome.

DiscussionThe present results are based on the analysis of behavioral series of relatively shortlength. 512 data points could be considered insufficient for deriving reliable results,especiallywithmultifractal analyses.However, the application of time series analysessupposesthatthesystemunderstudyremainsinstablestateduringthewholewindowof observation, and in behavioral experiments the lengthening of the task could raiseproblems of fatigue or lack of concentration (Delignières et al. 2005; Marmelat andDelignières2011).Ontheotherhand,anumberofimprovementshavebeenintroducedin fractalanalyses,whichcouldallowtoconsiderwithacertainconfidencetheresultsobtainedfromrelativelyshortseries(Delignièresetal.2006;AlmuradandDelignières2016). Such series lengths are generally considered as an acceptable compromisebetween the requirements of time series analyses and the limitations of hebavioralhumanexperiments(Gilden1997;Chenetal.1997;Gilden2001;Chenetal.2001).

11

Weareawarethatthepresentresultsshouldbeconsideredwithcaution,andhavetobeconfirmedbyfurtheranalyses.However,thedifferencesweobservedbetweenthethreeexperimentsareconsistentwithourinitialhypothesesandseemsufficientlylargetoberegardedwithsomeconfidence.

Complexitymatchingvsdiscretelocalcoupling

Themainresultofthispaperisthecleardistinctionbetweenthetwofirstexperiments(bimanual coordination and interpersonal coordination) and the third one (walkingwithafractalmetronome).Inthetwofirstcasesthecorrelationfunctionrevealedaclearstatistical matching between multifractal spectra, whatever the range of intervalsconsidered in the analysis. In contrast, in the third set of data the close statisticalmatchingappearedonlywhentherangewasrestrictedtothe lengthiest intervals(i.e.,from64to256),andwasprogressivelyalteredwhenwiderrangeswereconsidered.

Theseresultssuggestthatinthetwofirstexperimentssynchronizationwasgovernedbyaglobal,multiscalecoordinationbetweenthetwointeractingsystems,andinthethirdexperiment synchronization was the result of local corrective processes. As in thisexperimentparticipantshadtosynchronizewithaseriesofdiscretestimuli,onecouldhypothesize that these correction processeswork on a discrete, step-to-step basis, assuggested by Marmelat and Delignières (2012). One could argue, however, that thishypothesisshouldbeconsideredwithcaution,asthetaskusedinthethirdexperimentstronglydiffersfromthoseusedintheothers,andespeciallythewalkingtaskinvolvesverylargemassescomparedtotheothertasks.Note,however,thatTorreetal.(2013)proposedasimilarhypothesisinanexperimentwhereparticipantshadtosynchronizefingertapswithfluctuatingmetronomes.Itcouldbeinteresting,forreinforcingthishypothesis,tocheckwhetheramodelbasedonsuchdiscrete, localcorrectionprocesses,couldgenerateseriesyieldingcomparableresults.Wethen tried tosimulateseries thatcouldresult froma localcouplingwithafractalmetronome. In thismodeling studywe considered that the organismproducesintrinsically long-range correlated series. Indeed, a number of previous studies haveshown that organismsproduced long-range correlated series in self-paced conditions,and that, during synchronizationwith a regularmetronome, this sourceof long-rangecorrelation was still at work and had to be considered for properly modeling thesynchronisation process (Torre and Delignières 2008b; Torre and Delignières 2009;Delignières and Marmelat 2014). We then proposed that the organism corrects theinterval it intended to intrinsically produceon thebasis of theprevious asynchronies(DelignièresandMarmelat2014).Weworkedwithatwo-orderauto-regressivemodel:

x(i) = y(i)− a ASYN(i −1)[ ]− b ASYN(i − 2)[ ]+ cε(i) , (10)

ASYN(i) = ASYN(0)+ x(k)− z(k)k=0

i

∑k=0

i

∑ (11)

wherex(i)representstheseriesofperiodseffectivelyproducedbytheorganism,y(i)theseries of virtual periods produced by the organism, and z(i) the fractal metronome.ASYN(i)istheseriesofasynchroniesbetweentheeventsproducedbytheorganismandthe signals of themetronome. y(i)and z(i)were bothmodeled as fractional GaussiannoiseswithH=0.9(mean=1000andSD=20).Finallyε(i)isawhitenoiseprocesswithzeromeanandunitvariance.

12

This model suggests that periods are corrected on the basis of the two previousasynchronies, ahypothesis consistentwith the cross-correlation functionsobtained inthisexperiment(DelignièresandMarmelat2014).

Forthepresentsimulationsweuseda=0.2,b=0.4,c=12,andwegenerated12seriesof512datapoints.WepresentinFigure5(panela)theaveragedmultifractalspectraforthe ‘participant’ (black circles) and the ‘metronome’ (white circles). Note that thesesimulated results are characterized by a shift of the first spectrum, similar to thatobserved in experimental data (see Figure 3). Panel b represents the correlationfunctionsr(q),forthefourrangesofintervalsconsidered.Ascanbeseenweobtainedapatternofresultssimilartothatobtainedwithexperimentalresults:whenfocusingonlong-term intervals the correlation function remained close to one, and progressivelyextinguishedasmoreandmoreshorter-termintervalsweretakenintoconsideration.

5

a b

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.8 0.9 1 1.1

F(α)

α

Par1cipant

Metronome

-0.2

0

0.2

0.4

0.6

0.8

1

-15 -10 -5 0 5 10 15

Correla1

onα1(q)/α2(q)

q

6432168

Lowvariance Highvariance

p=.05

Figure 5: Simulation of the synchronization to a fractal signal by local corrections ofasynchronies.Panela:Multifractal spectra for the ‘participant’ (blackcircles)and the‘metronome’ (white circle). Panel b: Correlation functions r(q), for the four rangesofintervalsconsidered(8toN/2,16toN/2,32toN/2,and64toN/2).

Anotherdifferencebetweenour experiments lies in the couplingbetween systems: inbimanual coordination and interpersonal synchronization the systems mutuallyinteracted but synchronization with a fractal metronome is characterized by thepresenceofa‘master’(themetronome)anda‘slave’(theparticipant).Interestingly,ourresults revealed an asymmetry in the alterationof correlationswhen larger ranges ofintervalswereconsidered:weespeciallyobservedadramaticdecreaseofcorrelationsfor negativeq-values, corresponding to low-variance epochs in the signals (Figure 3).This result was particularly obvious in the third experiment, and suggests that highvariance epochs in themetronome signals allow a better synchronization, reinforcingthehypothesisofadiscrete,perceptualbasisofsynchronization.

Theseresultsquestionanumberof recentexperimentsdealingwith theuseof fractalmetronomeswiththeperspectiveofrehabilitationpurposes(Stephenetal.2008;Hoveetal.2012;Kaipustetal.2013;Torreetal.2013;Rheaetal.2014;Marmelatetal.2014).In their seminal paper, Stephen et al. (2008) suggested that synchronization with achaotic metronome was not based on local adjustments but rather on a global,multiscale coordination with the metronome. The present results cast doubt on thisconclusion (see also Delignières & Marmelat, 2014; Torre et al., 2013). Fractalmetronomes have recently sparked a great interest, suggesting that they couldmimic

13

natural variability, and be used for conceiving artificial devices for training andrehabilitation, based on the complexitymatching effect.Mimicking natural variability,especially with discrete signal series, seems not sufficient to generate the global andmultiscalecoordinationhypothesizedincomplexitymatching.

Complexitymatchingvscontinuouslocalcoupling

Our results, however, do not prove that the strong statistical matching observed inbimanualcoordinationandinterpersonalcoordinationisduetocomplexitymatching,asdefinedintheintroduction.Recently,Fineetal(2015)questionedtheglobalcomplexitymatching hypothesis, and suggested that a local and continuous coupling betweensystemscouldunderlainthestatisticalmatchingobservedinsuchsituations.Thedynamicalsystemsapproachtocoordinationpromotedaphenomenologicalmodelbasedona continuous couplingbetweenoscillators (Hakenet al. 1985; Schöneret al.1986). This so-called HKB model accounts for coordination by non-linear couplingbetweentwohybridlimit-cycleoscillators,basedonthetwooscillators’statevariables(positionandvelocity):

!!x1 +δ !x1 + λ !x13 + γ x1

2 !x1 +ω2x1 = ( !x1 − !x2 )[a + b(x1 − x2 )

2 ]

!!x2 +δ !x2 + λ !x23 + γ x2

2 !x2 +ω2x2 = ( !x2 − !x1)[a + b(x2 − x1)

2 ] (12)

wherexiis thepositionofoscillator i, and thedotnotationrepresentsderivationwithrespecttotime.Theleftsideoftheequationsrepresentsthelimitcycledynamicsofeachoscillatordeterminedbyalinearstiffnessparameter(ω)anddampingparameters(δ,λ,andγ),andtherightsiderepresentsthecouplingfunctiondeterminedbyparametersaandb.Thismodelhasbeenproventoadequatelyaccountformostempiricalfeaturesinbimanualcoordinationtasks,suchasthedifferentialstabilityofin-phaseandanti-phasecoordinationmodes, and the transition fromanti-phase to in-phase coordinationwiththeincreaseofoscillationfrequency(Hakenetal.1985;Schöneretal.1986).Onecanindeedsupposethatthiskindofcontinuouscouplingcouldbeattheoriginofthestrongstatisticalmatchingobservedincoordinationexperiments(Fineetal.2015),but the multifractal signature proposed in this paper could be unable to distinguishbetween genuine complexity matching and such local and continuous coupling. Asolutionfordisentanglingthesetwohypothesesistoanalyzetheseriesproducedbythismodel,andtocomparethemtothoseempiricallyobserved.

However, inordertoaccountforempiricalfeatures,theoriginalHKBmodelshouldbeslightlymodified.Especially,ithasbeenproventhatinbimanualcoordination,theseriesof periods produced by each effector and the series of relative phase contained long-range correlations, a property that the original HKB model was unable to generate(Torre andDelignières 2008a). The authors proposed to account for this behavior byreplacingthefixedlinearstiffnessparameterωinequations(12)byatwoindependentdiscrete series ω1,i and ω2,i, exhibiting long-range correlation properties, andrepresentingtheinnerfrequenciesofoscillators1and2,respectively,atcyclei.

!!x1 +δ !x1 + λ !x13 + γ x1

2 !x1 +ω1,i2x1 + Qε1,t = ( !x1 − !x2 )[a + b(x1 − x2 )

2 ]

!!x2 +δ !x2 + λ !x23 + γ x2

2 !x2 +ω 2,i2x2 + Qε2,t = ( !x2 − !x1)[a + b(x2 − x1)

2 ] (13)

14

Note that they also introduced white noise terms of strength Q in the limit cycleequations,assuggestedbySchöneretal.(1986).TorreandDelignières(2008a)showedthat a relevant set of parameters allowed to simulate a stable in-phase coordinationbetween the two oscillators, despite the intrinsic long-range correlated fluctuationsinjected in each oscillator. Their results replicatedmost empirical results, in terms ofmean and standard deviation of relative phase, and also concerning the presence oflong-rangecorrelationsintheseriesofrelativephase.

Weusedthismodelforgeneratingpairsofseriesofcoordinatedperiods.Weattemptedto generate series reproducing the main features of the inter-personal coordinationexperiment,inthestrongcouplingcondition.WeusedthesamesetofparametersthanTorreandDelignières(2008a):δ=0.5,λ=0.02,γ=1.0,Q=0.4.Weusedthe“hoppingmodel” (Delignièresetal.2008;TorreandDelignières2008a) forsimulating the long-rangecorrelatedseriesω1,iandω2,iaroundameanvalueof4π.Forstabilizingin-phasecoordinationbetweenthetwosystems,thecouplingparametersweresettoa=12andb=6.Thesevaluesweremuchstrongerthanthosecommonlyreportedintheliterature(Fink et al. 2000; Assisi et al. 2005; Leise and Cohen 2007), but were necessary forobtainingastablecoordination(Delignièresetal.2008;TorreandDelignières2008a).Wesimulated12setsofseriesof512datapoints.Simulatedseriesbroadlyreproducedexperimentalresults,withameanrelativephaseof0.74° (+/- 6.93). The mean period of oscillations was 1001.14 ms (+/- 50.07). Thestandarddeviationofasynchronieswas19.34ms(+/-3.38).WepresentinFigure6(aandb)theresultofthemultifractalanalysisofthesecoordinatedseries.Ascanbeseen,the correlation function r(q) exhibited very high values, even when considering thewidest range of intervals. This result appears similar to that obtained with theexperimental series (Figure2), suggesting that this kindof local, continuous coupling,couldindeedunderlaytheobservedstatisticalmatchingbetweenseries.However, a closer look to the coupled period series casts some doubts about thisconclusion.WepresentinFigure6twoexamplessubsetsofcoupledseries(50points),thefirstgraph(c)correspondingtorepresentativeexperimentalseries,andthesecond(d)tosimulatedseries.Thesegraphssuggestthatwhileprovidingcomparablestatisticalresults, interpersonalcoordinationandthecoupledoscillatormodelworksdifferently.The simulated series present very close dynamics, resulting from the continuouscouplingofpositionsandvelocitiesinthemodel.Incontrast,experimentalseriesappearmore independent, at least on this local scale. In order to quantify these localdependences, we computed the average local cross-correlation coefficient, using asliding window of 15 points (Marmelat and Delignieres 2012). The average cross-correlationcoefficientwas0.87forsimulatedseries,butclosetozeroforexperimentalseries.

Onecouldargue,however,thatalthoughthestiffnessseriesω1,iandω2,iincludedinthemodelareindependent,theystillhavethesameaveragevalues,andthatmorerealisticresults could be obtained by introducing an asymmetry, i.e. a difference between thenatural frequencies of the two systems. In order to check this point, we performedadditionalsimulations introducinga10%detuningbetweenthe twooscillators (meanω1,i=4π,andmeanω2,i=4.4π).Allothersparameterswereunchanged.Thisnewsetofsimulations gave essentially similar results, suggesting that the symmetry of the firstmodel cannotper se explain the strong local convergence of the simulated series.Wepresent in Figure 6 (panel e) an example subset of series simulated with these newsettings.

15

6

a b

c

d

e

!0.2%

0%

0.2%

0.4%

0.6%

0.8%

1%

!15% !10% !5% 0% 5% 10% 15%

Correla1

on%α1(q)/α2(q)%

q"

64%32%16%8%

Low%variance% High%variance%

p"=%.05%

950$

1000$

1050$

1100$

1150$

1200$

0$ 5$ 10$ 15$ 20$ 25$ 30$ 35$ 40$ 45$ 50$ 55$ 60$

Perio

d$(m

s)$

Cycle$#$

Par:cipant$A$

Par:cipant$B$

950$

970$

990$

1010$

1030$

1050$

1070$

1090$

1110$

1130$

0$ 5$ 10$ 15$ 20$ 25$ 30$ 35$ 40$ 45$ 50$ 55$ 60$

Perio

d$(m

s)$

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Oscillator$1$

Oscillator$2$

8608809009209409609801000102010401060

0 5 10 15 20 25 30 35 40 45 50 55 60

Perio

d(m

s)

Cycle#

Oscillator1

Oscillator2

Figure6:Panela:Averagemultifractalspectraforthetwosimulatedoscillators.Panelb:Correlationfunctionsr(q),forthefourrangesofintervalsconsidered(8toN/2,16toN/2, 32 to N/2, and 64 to N/2). Panel c: Representative subsets of coupledexperimentalseries(50points).Paneld:Representativesubsetsofcoupledsimulatedseries(symmetricmodel).Paneld:Representativesubsetsofcoupledsimulatedseries,witha10%detuning.

Theseresultssuggest thata localcontinuouscoupling,asproposed in theHKBmodel,can indeed mimic the statistical matching supposed to emerge from complexitymatching. However, this kind of coupling generates a very close correspondencebetween the trajectories of oscillators, which seems unrealistic in view of theexperimentalobservations.

16

Conclusion

Complexitymatching is a very innovative hypothesis,which has recentlymotivated anumber of theoretical and experimental contributions. In this paper we introduce amethod,basedonmultifractalanalysis, fordistinguishingbetweengenuinecomplexitymatchingand localdiscretecoupling.Ourresultsshowthatsomesituations thatwereconsidered prototypic of complexitymatching,where participants had to synchronizewith a fractal metronome, seem controlled through local adjustments. In contrast,genuinecomplexitymatchingseemsoccurringwhentwocomplexsystemsaremutuallycoupled.

Complexitymatchingandlocaldiscretecoupling,however,arenotnecessarilyexclusive.Onecouldconceive,forexample,thatinsometaskssynchronizationcouldbeachievedthrough a mix of the two processes. Moreover, the respective contribution of eachprocesses could differ among participants (see, for example, Delignieres and Torre2011). Further investigations, focusing on individual series and based on cross-correlationsshouldprovidesomeinsightsaboutthishypothesis.Conflictofinterest

Theauthorsdeclarethattheyhavenoconflictofinterest.

AcknowledgmentsWe thank Prof. Andras Eke who kindly provided us with the Matlab code for themultifractalfocus-basedmethod.

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