Dynamic Functional Connectivity as a complex random walk: Definitions and the dFCwalk toolbox

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Dynamic Functional Connectivity as a complex random walk: Definitions and the dFCwalk toolbox

Lucas Arbabyazd, Diego Lombardo, Olivier Blin, Mira Didic, Demian Battaglia, Viktor Jirsa

To cite this version:

Lucas Arbabyazd, Diego Lombardo, Olivier Blin, Mira Didic, Demian Battaglia, et al.. Dynamic

Functional Connectivity as a complex random walk: Definitions and the dFCwalk toolbox. MethodsX,

Elsevier, 2020, 7, pp.101168. �10.1016/j.mex.2020.101168�. �hal-03092856�

ContentslistsavailableatScienceDirect

MethodsX

journal homepage:www.elsevier.com/locate/mex

MethodArticle

Dynamic Functional Connectivity as a complex random walk: Definitions and the dFCwalk toolbox

Lucas M.Arbabyazd^a, Diego Lombardo^a, OlivierBlin^a,b,Mira Didic^a,c, Demian Battaglia^a,∗,Viktor Jirsa^a,∗

aUniversité Aix-Marseille, INSERM UMR 1106, Institut de Neurosciences des Systèmes, F-13005 Marseille, France

bAP-HM, Timone, Service de Pharmacologie Clinique et Pharmacovigilance, F-13005 Marseille, France

cAP-HM, Timone, Service de Neurologie et Neuropsychologie, F-13005 Marseille, France

abstract

Functional Connectivity, describing the interaction between brain regions beyond their anatomical interconnection, is highly dynamic even when no task is performed (“resting state”) and it remains a methodological challenge to properly describe its changes in time without strong assumptions. We have developedaframeworktodescribethedynamicsofFunctionalConnectivity(dFC)estimatedfrombrainactivity time-series as a as a smooth reconfiguration process, combining “liquid” and “coordinated” aspects. Our frameworkconsidersdFCasacomplexrandomwalkinthespaceofpossiblefunctionalnetworks.Unlikeother previous approaches, our method does not require the explicit extraction of discreteconnectivity statesbut trackschangesinacontinuoustimefashion.

• WeintroducedseveraldFCrandomwalkmetrics.First,dFCspeedanalysesextractthedistributionofthetime- resolvedrateofreconfigurationofFCalongtime.Thesedistributionshaveaclearpeak(typicaldFCspeed,that canalreadyserveasabiomarker)andfattails(denotingdeviationsfromGaussianitythatcanbedetectedby suitablescalinganalysesofFCnetworkstreams).

• Second, meta-connectivity (MC) analyses identify groups of functional links whose fluctuations co-vary in timeandthatdefineveritabledFCmodulesorganizedalongspecificdFCmeta-hubcontrollers(differingfrom conventional FCmodules and hubs). The decompositionof whole-braindFC by MCallows performing dFC speedanalysesseparatelyforeachofthedetecteddFCmodules.

• Wepresent hereblocksand pipelinesfordFC randomwalkanalysesthataremadeeasily availablethrough adedicatedMATLAB^R toolbox(dFCwalk),openlydownloadable.Althoughweappliedsuchanalysesmostlyto fMRIrestingstatedata,inprincipleourmethodscan beextendedtoanytypeofneuralactivity(fromLocal FieldPotentialstoEEG,MEG,fNIRS,etc.)orevennon-neuraltime-series.

© 2020TheAuthors.PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

DOI of original article: 10.1016/j.neuroimage.2020.117156

∗ Corresponding authors.

E-mail addresses: [email protected] (D. Battaglia), [email protected] (V. Jirsa).

https://doi.org/10.1016/j.mex.2020.101168

2215-0161/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

article info

Method name: dFC random walk analyses

Keywords: Neuroimaging, fMRI, Functional Connectivity, Chronnectome

Article history: Received 28 July 2020; Accepted 27 November 2020; Available online 1 December 2020

Specificationstable

Subject Area: Neuroscience

More specific subject area: Neuroimaging; functional connectomics

Method name: dFC random walk analyses

Name and reference of original

method: ^• Battaglia, D., Boudou, T., Hansen, E.C.A., Lombardo, D., Chettouf, S., Daffertshof er, A., Mcintosh, A.R., Zimmermann, J., Ritter, P., Jirsa, V., 2020. Dynamic Functional Connectivity between order and randomness and its evolution across the human adult lifespan. NeuroImage 222, 117,156. doi:10.1016/j.neuroimage.2020.117156 and

• Lombardo, D., Cassé-Perrot, C., Ranjeva, J.-P., Le Troter, A., Guye, M., Wirsich, J., Payoux, P., Bartrés-Faz, D., Bordet, R., Richardson, J.C., Félician, O., Jirsa, V., Blin, O., Didic, M., Battaglia, D., 2020. Modular slowing of resting-state dynamic Functional Connectivity as a marker of cognitive dysfunction induced by sleep deprivation. NeuroImage 222, 117,155. doi:10.1016/j.neuroimage.2020.117155

• Hansen, E.C.A., Battaglia, D., Spiegler, A., Deco, G., Jirsa, V., 2015. Functional connectivity dynamics: modeling the switching behavior of the resting state.

NeuroImage 105, 525–535. doi:10.1016/j.neuroimage.2014.11.001.

Resource availability: The dFCWalk toolbox can be downloaded at the link:

https://github.com/FunDyn/dFCwalk.git under the Creative Commons Zero v1.0 Universal license

Methoddetails

While structural connectivity refers to the existence of an anatomical connection between two neuronal regions –with the compilation of structural links between all pairs of anatomically interconnected regions known as the connectome [44]–, the notion of functional connectivity (FC) intendstocapturetheexistenceofcoordinatedactivitybetweentwobrainnetworknodes.Therefore, functional connectivity is necessarilyway moreflexible than the underlyingstructural connectome [37], being potentially the manifestation of collective emergent dynamics [9,30]. Notably, during the resting state (rs; [21]), the flexible sampling of a rich repertoire of possible dynamical states compatiblewitha givenconnectome[18,23] isexpectedtolead toa complexreconfigurationofFC alongtime,thecharacterizationofwhichistheendeavorofthenascentfieldof“chronnectomics”[12]. Indeed,thebrainis restlessevenatrestandrs FCcontinually fluctuatesina waywhich displaysa nonrandomspatiotemporalorganization[22].ThereisgrowingevidencethatsuchdynamicFunctional Connectivity (dFC), in both rest and task conditions, can be used as a sensitive biomarker of the efficiency and flexibility of cognitive processing [7,14,43] as well as of pathological alterations of restingstatedynamics[8,15,28].

To the growth in the number of dFC studies, has corresponded an analogous inflation in the numberofpossiblemethods toextractandquantitatively parameterizedFC fromfMRI dataorother time-seriesofneuralactivity(suchasEEG,MEG, fNIRS…).Weinvitethe readertorefertoe.g. Preti et al. [40] for a non-exhaustive review.Many approaches have tried to characterize “states of FC”

that wouldbe visitedatdifferent timesby theevolution ofFC. Stateextraction hasbeenachieved, e.g.,bydirectclusteringofinstantaneousFCnetworksobservedindifferenttemporalwindows[2],by temporalnetworkapproaches[45] orbyfittingofstatisticalgenerativemodelsofconnectivitystates andstatetransitions [3,48].However, statisticalevidenceinfavour fortheactual non-stationarityof rs FC [25,51] orthe actual existence of well-distinct FC states [42] is not completely conclusive. A problemwithstate-extraction-baseddFCmethodsisindeedthat,byassumingimplicitlytheexistence ofFCstates,theywillfindthemevenwhenadescriptionofdFCintermsofdiscretestatesisnotfully pertinent.

In Battaglia et al. [5] and Lombardo et al. [34] we have introduced a novel methodological framework that circumvents the problem of detecting FC states or assessing stationarity or non- stationarity of dFC. In this new framework –which we here refer to as dFCwalk paradigm–, we conceptualize dFC as a smooth reconfiguration flow across continuallymorphing FC configurations.

In thedFCwalkvision,an instantaneously observedFC network isseenasa “point” inthespaceof possible FC network realisations. By evolvingin time, this point performs a stochastic exploration of the high-dimensional space of possible FC, describing a veritable random walk. In the dFCwalk paradigm, dFC properties are quantified as descriptors of the dFC random walks implemented by collective brain dynamics during a specific recording or imaging session. Each session is indeed mapped toadifferentpath inFCspace.RelativelysmallvariationsofFCfromone observationtime to thenext willresultinshortflightlengthsandmoreextensivenetworkreconfigurations inlarger flightlengths.BykeepingtheintervalbetweentwoconsecutiveFCnetworktime-resolvedestimations constant,shorterandlongerflightlengthscanalsonaturallybereinterpretedasassociatedtoaslower orfasterspeedofdFCreconfiguration.AfirstaimofthedFCwalkanalysisframeworkwillbetherefore tosuitablyquantifyanddescribedistributionsofdFCspeed(notably,theirmode,givingatypicaldFC speed,andtheirspread).

Beyond the speed at which dFC travels along a walk, it will be also important, as a second aim, to describe the general geometry of the dFC paths. Random walks giving rise to stochastic paths, a naturalwaytocharacterize their“shape” will beto measuretheir fractalscaling properties.

Indeed the fractal geometries of stochastic paths will be different, depending on the degree of correlationbetweentheflightlengthstravelledatconsecutivetimes[36],varyingfrom“memoryless”

(asinBrownianmotion)topositiveornegativesequentialautocorrelations(resultinginnon-Gaussian stochasticprocesses,suchasLevywalks).Atoolsuitableforthescalinganalysisofrandomwalkpaths isDetrendedFluctuationAnalysis(DFA;[39,46,50]),whichwehaveadaptedtodFCwalksinBattaglia et al.[5].Such analysis provides an estimation ofthe fractalgeometry ofdFC paths, in termsof a DFAexponent,αDFA=0.5formemorylesswalks,andαDFA >0.5(αDFA<0.5)forwalkswithpositive (negative)sequentialcorrelationsinflightlengths.

Exactlyastypicalstatic(time-averaged)FCanalysisignoretime,thepreviouslymentioneddFCwalk analyses introduced in Battaglia et al. [5] ignore space. However, FC reconfiguration may occur at differentspeedsfordifferentsetsoflinks.Furthermore,thefluctuationsofcertainFClinksmaycovary withthefluctuationofotherFClinks–formingadFCmodule– butberelativelyindependentfromthe fluctuation of other sets of links. Therefore, it may make sense to compute a different dFC speed distributionfordifferentdFCmodules(modulardFCspeeds),sinceeachofthesemodulesperformsits own specificrandom walk.Athird aimofthedFCwalkframework willthusbe theidentificationof theseeventualdFCmodulesfluctuatinginparallelaccordingtoindependentrandomwalks.Todoso, wehaveintroducedinLombardoetal.[34]andedge-covariance-basedmethodwhichwecalledMeta- Connectivity(MC)analysis,andwhichisstronglyrelated(althoughnotidentical)tootheredge-centric functionalconnectivityapproachesintroducedelsewhere[4,10,20].Inthisapproach,dFCmodulesare extractedapplyingconventional modulardecompositionalgorithms–as theLouvain method– tothe MC matrix. Indeed, ifwe consider FC linksas nodes of a new network, dual (see e.g. [6]for the notion)totheoriginalFCnetwork,theMCmatrixentriescanbeinterpretedas“linksbetweenlinks”, or, as we say, meta-connections. In this vision, MC is still a graph, although over a broader set of nodesthantheoriginalFCgraphandanyusualgraph-theoreticalmethodvalidforstandardnetworks can alsobeappliedto MC.Asdiscussedby Lombardoetal.[34],linkswhicharestronglycovarying tendtobeincidenttoacommonregion.Therefore,dFCmodulestendtoincludestarsubgraphswith well-definedcenterregions–meta-hubs– and divergent links whichtend to be all strong or all weak simultaneously dependingon time. Thesemeta-hubs act as“puppet masters” moving the“threads”

in specific dFC modules andreaching to regions whichcan be far away andpotentially distributed brain-wide.Therefore,meta-hubsarelocalizedcontrollersofnon-localizedfunctionalsystemsandare inthissenseverydifferentfromconventionalFChubs(see[34]foradetaileddiscussion).

Altogether,inthetwomanuscriptsbyBattagliaetal.[5]andLombardoetal.[34]wehaveforged a rich toolboxofinterdependent analyses andmetrics that haveprovedto beable totrackchanges ofdFC propertiesacrossagingorinrelationwithvariationsofcognitive performance.Wehavenow made this toolbox available inthe form of a collection of MATLAB^R functions(dFCwalk toolbox,

downloadlink: https://github.com/FunDyn/dFCwalk.git,underCreativeCommons Zerov1.0Universal license(CC0))thatallowcodingcomplexanalysisdesigns suchastheonesofBattagliaetal.[5]and Lombardoetal.[34] with justafew lines ofhigh-level code.We describehere inthefollowing in a step-by-step wayhow to implementthe various dFC random walk analyses we have mentioned, providingspecificexampleapplicationsforillustration.

TheinputneededfordFCwalkanalyses

The set ofmethods which we willreview here canin principle be applied towhatever type of multivariate time-series. The examples here includedare restingstate BOLD time-series fromfMRI experiments,howeverpreciselythesameanalyses(withparameters,such aswindowsizes,adjusted to the specificities of the signals of interest) may be applied to EEG or MEG or whatever other neuronalactivitytime-series.As amatteroffact,inClawsonetal.[13]andPedreschietal.[38]we havepresentedveryrelatedanalysesperformedonspikingactivityrasterplotsormultichannellocal field potentials. Evenmore generally,such analyses may be used fornon-neural multivariate time- series,as,e.g.EMGs[47]orfunctionalconnectivityanalysesingeosciences[49].

Wedefine thereforeasinputa setoftime-seriesTS_i(t)withi =1…N and0≤t ≤TwhereN is thenumberofnetwork nodes(brainregions, voxelsorchannelsforneuroscience datasets)andT is thenumberoftimesamplesinthetime-series.

FunctionalconnectivityanddFCstream

Oncegivenasetoftime-seriesTS(t),itispossibletocomputethetime-averaged(static)Functional Connectivity(FC)matrix,withentries:

FC_{i j}=Corr

TS_i(^t),TS_j(^t)

,with0≤t≤T

ThismatrixgivesthenormalizedpairwisePearsoncorrelationbetweentime-series,averagedover the totalsessiontime T (Fig. 1A).Note thatall entries areretained inthematrix, independentlyof whetherthecorrelation valuesaresignificantornotorwithoutfixinganythreshold.Thisisbecause weconsiderFCentries(andotherfeaturesingeneralinthefollowing)as“features” thatcanbeuseful fortracking trends orseparating cohorts,i.e. we compute them fortheir potential predictive value morethanfortheirassessmentitself.

InthedFCwalktoolboxthefunctiontocomputeastaticFCmatrixoutoftime-seriesTSisTS2FC, takingasinput a generic TS matrixvariable (differentcolumns aredifferent regions, differentrows aredifferenttimes)andproducingasoutputthestaticFCentries(ineither‘matrix’or‘vector’format, seelater).

Togeneratenotasingletime-averagedmatrixofFCbutanentirestreamofFCmatricesevaluated at differenttimes along the session–i.e. a temporal network [27,32], calledthe “dFC stream”–, the dFCwalktoolboxusesaslidingwindowapproach(Fig.1B,top).IndividualtemporalframesofthedFC streamareevaluatedusingthesameformulaasabove,restrictedtotheconsideredtime-window:

FC_{i j}(^tk)=Corr

TS_i(^t),TS_j(^t)

,witht_k≤t≤t_k+W

wheret_k isthe starttime ofthe k-thtemporal frame ofthenetwork andWis afixed observation windowlength.Framestarttimesareseparatedbyaslidingstep࢞τ^,sothattk=t_k-1 +࢞τ =k࢞τ^.

Thisslidingincrementisoftenfixedtobe equaltothewindowsizeWsothat consecutivenetwork frames are evaluated on non-overlappingtime-series segments, howeverit can be fixed to a value of choice. Typically, we used ࢞τ = Wfor dFC speed distribution analyses, but set it to values as smallas࢞τ =1time-stepinthetime-series(e.g.oneTRforfMRItime-series)foranalysesrequiring more continuousdFC streams such asDFAor MC analyses. Inthe dFCwalktoolbox the function to computean entiredFC streamout oftime-seriesTSisTS2dFCstream,takingasinputagenericTS matrixvariable,awindow sizeWand,optionally,a slidingincrement࢞τ ^{(which isotherwisefixed}

asdefaultat࢞τ =W).Theoutputisthetemporalnetworkitself.

Fig. 1. From Functional Connectivity to Functional Connectivity Dynamics. (A) Traditionally, correlations between neural activity time-series TS i(t) of N different brain region nodes i and j (left) are averaged over long times and compiled into the entries FC ijof a ‘static’ N- times- N Functional Connectivity (FC) matrix (right). (B) Sliding windows of a shorter temporal duration, it is possible to estimate a stream of time-resolved FC (t) networks, which we call the dFC stream (top). The variation between a FC frame at a time t 0and the next non-overlapping frame at a time t 0+ W is measured by the dFC speed V dFC,W(t 0) where W is the chosen window size. The degree of similarity (inter-matrix correlation) between FC (t) networks observed at different times is then represented into a F -times -F recurrence matrix, or dynamic Functional Connectivity (dFC) matrix, where F T is the total number of probed windows (i.e. frames in the temporal network given by the dFC stream), depending on window size and overlap (bottom). (C) Alternatively, one can consider each individual FC link as a dynamic variable FC ij(t) attached to the graph edge between two regions i and j (top). Generalizing the construction of the FC matrix in panel (A), we can thus extract a N(N-1)- times- N(N-1) matrix of covariance between the time-courses of different FC ij(t) links. We re-baptized this inter-link covariance matrix as Meta-Connectivity (MC) matrix.

BothTS2FCandTS2dFCstreamcanproduceoutputs intwo differentformats. TheFCmatrices are symmetricand therefore,despitetheir size is N ×N,they justhave L= N(N-1)/2 independent entries, where L is the number oflinks, corresponding to the lower triangularpart of the matrix.

Whenusingthe‘matrix’format(or‘2D’,defaultforTS2FC)theoutputFCmatrix(oreachofthedFC streamframes)areformattedasactualN×Nmatrices.Therefore,theoutputofTS2dFCstream,in matrixformat,willbea3DtensorofsizeN×N×F,whereFisthenumberofframesproducedgiven theactual choicesofWand࢞τ ^{(frameswithwindows withtrimmedlength thatmayoccur atthe}

terminal part of thetime-series are dropped). In ‘vector’ format (or ‘1D’ forTS2FC,and therefore

‘2D’ defaultfor TS2dFCstream^{), on the contrary, only the} independent entries of FC or of each FC frame will be provided as output, so that the output of TS2FC will be a vector of size L × 1 andthe outputof TS2dFCstreama 2D matrixofsize L× F(each oftheF framesbeinggivenin vectorformat).Thevectorformathastheadvantageofproducingsmalleroutputvariablesoccupying less spacein memory, since dFC stream variables can be quite large forvery long time-series and largenumbersofnodes.IthasalsotheadvantagethatFCsin‘vector’formatarenaturallyformatted as ‘points’ in a vector space which makes more straightforward the generation of dimensionally reduced representations of the dFC stream random walk via projection methods, such as e.g. t- stochasticneighbourhoodembedding[26],accessibleinthestandardMATLAB^R Statisticaltoolboxvia thefunctiontsne.

Forthesake ofcomputingspeed, allcomputationsare alwaysperformedinthecompact ‘vector’

formatevenwheninputsareprovidedin‘matrix’format(anautomaticconversionto‘vector’format isperformedbythedFCwalktoolboxfunctions).ThedFCwalktoolboxfunctionsMatrix2Veccanbe calledtoperformconversionsfrom2DFC matricesinN× Nmatrixformatto1D vectorFCsofsize L×1orfrom3DdFCstreamsinN×N×Ftensorformatto2DdFCstreamsofsizeL×F.

RecurrenceanalysisviathedFCmatrix

After extracting a dFC stream, itis possible to studyits recurrence structure, verifying whether theFC(t)networksobservedatcertaintimesaretransientlystabilizedandpossiblyreturnindifferent epochsorwhethertherearespecialtransientsofanomalouslyfastnetworkreconfiguration.Todoso, onecanstudythesimilaritybetweenFCframesobservedatdifferenttimes,evaluatingtheso-called dynamicFunctionalConnectivity(dFC)matrixofadFCstream.Weintroduced thismatrixforthefirst timeinHansenetal.[23],butitconstitutesinrealityjustanexampleofrecurrenceanalysis,common intime-seriesanalyses [29],hereadapted totemporalnetworks. Tomeasurethesimilaritybetween two networksFC(t1)andFC(t2)we useplain correlation betweentheupper triangularpartsof the twomatrices.TheentriesofthedFCmatrixarethengivenby:

dFC(^t1,t₂)=Corr[U pperTri(^FC(^t1)),U pperTri(^FC(^t2))^]

One couldhoweverredefinethe analysistouseanyother moresophisticated metricofnetwork similarity(e.g.JaccardsimilaritycoefficientinPedreschietal.[38]).NotethattheobtaineddFCmatrix depends on the window-sizeW and the sliding step ࢞τ ^{adopted to evaluate the dFC stream. As}

shownintheexampleofFig.1B(bottom),dFCmatricesforrestingstatefMRIsessionsusuallydisplay characteristicpatterns composedout ofsquare-shaped red-huedblocks,corresponding to epochs of transiently increased similarity betweenconsecutiveFC(^t) ^{network frames. Such epochs ofrelative} FCstabilityincrease–or‘dFCknots’– areintertwined withtransientsofrelativeinstability,appearing as light blue stripes in the dFC matrix, denoting strong dissimilarity from previously visited FC(t) networks. Duringsuch transients–or‘dFCleaps’–FC(^t) ^{quicklymorphsbefore} stabilizingagaininto thenextdFCknot.AsdiscussedinBattagliaetal.[5],thisalternationbetweenknotsandleaps(cf.also Fig.2)can beseenastheexistenceofa“liquidclustering” in dFCstreams,wheresharplyseparated FC statesdo notexist, butepochof transientstabilizationintoalternative FC configurations canbe neverthelessbe detected.Longerandmorepersistent“knots” areobservedforincreasing age[5]or afterSleepDeprivation[34].InClawsonetal.[13]orPedreschietal.[38] wherewe haveconducted similaranalysis(usingJaccard similaritycoefficientasnetworksimilaritymeasure)onneuralactivity atthe micro-scalewithin hippocampalformationlocal circuits,dFC analyses ledon thecontraryto