NAZE, Sébastien, et al . Robustness of connectome harmonics to local gray matter and long-range white matter connectivity changes. NeuroImage , 2020, vol. 224, p. 117364

Article

Reference

Robustness of connectome harmonics to local gray matter and long-range white matter connectivity changes

NAZE, Sébastien, et al.

NAZE, Sébastien, et al . Robustness of connectome harmonics to local gray matter and long-range white matter connectivity changes. NeuroImage , 2020, vol. 224, p. 117364

DOI : 10.1016/j.neuroimage.2020.117364 PMID : 32947015

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http://archive-ouverte.unige.ch/unige:144757

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ContentslistsavailableatScienceDirect

NeuroImage

journalhomepage:www.elsevier.com/locate/neuroimage

Robustness of connectome harmonics to local gray matter and long-range white matter connectivity changes

Sébastien Naze

^a,b,∗

, Timothée Proix

^c

, Selen Atasoy

^d

, James R. Kozloski

^a

aIBM T.J. Watson Research Center, Yorktown Heights, New York, USA

bIBM Research Australia, Melbourne, Victoria, Australia

cDepartment of Basic Neurosciences, Faculty of Medicine, University of Geneva, Geneva, Switzerland

dDepartment of Psychiatry, University of Oxford, UK

a b s t r a c t

Recently,ithasbeenproposedthattheharmonicpatternsemergingfromthebrain’sstructuralconnectivityunderlietherestingstatenetworksofthehumanbrain.

Theseharmonicpatterns,termedconnectomeharmonics,areestimatedastheLaplaceeigenfunctionsofthecombinedgrayandwhitemattersconnectivitymatricesand yieldaconnectome-specificextensionofthewell-knownFourierbasis.However,itremainsunclearhowtopologicalpropertiesofthecombinedconnectomesconstrain thepreciseshapeoftheconnectomeharmonicsandtheirrelationshipstotherestingstatenetworks.Here,wesystematicallystudyhowalterationsofthelocaland long-rangeconnectivitymatricesaffectthespatialpatternsofconnectomeharmonics.Specifically,theproportionoflocalgraymatterhomogeneousconnectivity versuslong-rangewhite-matterheterogeneousconnectivityisvariedbymeansofweight-basedmatrixthresholding,distance-basedmatrixtrimming,andseveraltypes ofmatrixrandomizations.Wedemonstratethattheproportionoflocalgraymatterconnectionsplaysacrucialrolefortheemergenceofwide-spread,functionally meaningful,andoriginallypublishedconnectomeharmonicpatterns.Thisfindingisrobustforseveraldifferentcorticalsurfacetemplates,meshresolutions,orwidths ofthelocaldiffusionkernel.Finally,usingtheconnectomeharmonicframework,wealsoprovideaproof-of-conceptforhowtargetedstructuralchangessuchasthe atrophyofinter-hemisphericcallosalfibersandgraymatteralterationsmaypredictfunctionaldeficitsassociatedwithneurodegenerativeconditions.

1. Introduction

Understanding the structure-function relationships in large-scale brain networks is an active research topic in neuroscience (Sporns etal.,2004; Honey etal.,2010; Miietal.,2016).Inclinical neuro- science,theserelationshipscanelucidatetheroleofstructuralchanges inneurologicaldiseasesandtheirrespectivesymptoms.Graphtheoret- icalanalysisofbrainconnectivityhasledtonewinsightsaboutcor- ticalwiringpatterns(JirsaandMcIntosh,2007;BullmoreandSporns, 2009)suchassmall-worldtopology,presenceofhubs,hierarchicalprop- erties,andenabledthedevelopmentofquantitativemeasuresofnet- workresilience (Rubinov andSporns, 2010).These metricsarenow commonlyusedtounderstandtheorganizationofbrainfunctionand dysfunction(Fornitoetal.,2015),includingAlzheimer’sdisease(Stam etal.,2006,2008),dementia(Agostaetal.,2013;Vecchioetal.,2015), schizophrenia(Alexander-Blochetal.,2013;Golloetal.,2018;vanden Heuveletal.,2010,2013;Lynalletal.,2010)andHuntington’sdisease (Harringtonetal.,2015;McColganetal.,2015).

SeveralstudieshaveappliedthegraphLaplacian,adiscreteversion ofthecontinuousLaplaceoperator,tobrainconnectivitymatricesand inferredprogressionofneurodegenerativediseases(Rajetal.,2012), brain malformation (Wang et al., 2017), time of attention switch- ingin a cognitivetask (Huang etal., 2018; Medaglia etal., 2018), macroscalecouplinggradientbetweenbrainregions(PretiandVanDe

∗Correspondingauthor.

E-mailaddress:[email protected](S.Naze).

Ville, 2019),andaltereddynamic connectivityin patientswith con- cussion(Sihagetal..2020).Thesestudiesfocusonlongrangewhite- matterconnectivityandusebrainconnectivitymatricesderivedfrom parcellationschemesthatrangefromafewdozenstoseveralhundred regions-of-interests(ROI)(Desikanetal.,2006;Destrieuxetal.,2010).

Recently,anotherframeworkhasbeenproposedfortheapplicationof graphLaplaciantothehumanconnectome,whereinthelocalconnec- tivityofthegraymattercorticalstructureestimatedfromthemagnetic resonanceimaging(MRI) datais combinedwiththelong-rangecon- nectivityofthewhite-matterthalamo-corticalfibersestimatedfromthe diffusionMRI (dMRI)datawithoutincorporatingany parcellationof thecorticalsurface(Atasoyetal.,2016,2017a).Thisdenselysampled connectomemodelhastheadvantageofofferingtheminimumamount ofdiscretizationpossibleinthegivenresolutionoftheMRIanddMRI dataandyieldingtheclosestapproximationofthecontinuousLaplace operator,asthegraphLaplacianconvergestoitscontinuouscounter- partwhenthenumberofuniformlysampleddatapointstakenfromthe underlyingmanifoldincreases(BelkinandNiyogi,2008).Incontrastto otherparcellation-basedapproaches,theinputstructureofthisframe- work consistsofa graph,whereeachnoderepresents avertexfrom thecorticalsurfacemeshwithoutapplyinganyparcellation.Hence,this approachprovidesanincreasein thenumberofnodesbytwoorders ofmagnitudecomparedtoothermethodsutilisingbrainparcellations.

TheeigenvectorsofthisdenseconnectomeLaplacian(called”connec- tomeharmonics”)yieldasetoffrequency-orderedharmonicpatterns emergingonthecortexandprovide aconnectome-specificextension ofthewell-knownFourierbasistothehumanbrain.Connectomehar-

https://doi.org/10.1016/j.neuroimage.2020.117364

Received28January2020;Receivedinrevisedform14August2020;Accepted7September2020 Availableonline16September2020

1053-8119/© 2020TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

Fig.1. Overviewoftheworkflowfortheconstructionoftheconnectomeharmonics.Localconnectivityfromcorticalsurfacemesh(bottomleft)andlong-range connectionsfromtractography(topleft)arecombinedinahigh-resolutionstructuralconnectome(middle),fromwhichagraphLaplacianLiscomputedbasedon theadjacency(A)andthedegree(D)matricesofthecombinedconnectivities.Connectomeharmonics(right)aretheeigenvectorsofthegraphLaplacian.

monicsproducedbythisframeworksuggestarelationshipbetweenlow frequencyharmonicsandbrain’srestingstatenetworkssuchasthede- faultmodenetwork(DMN)andrevealunique andfrequency-specific changesinbrainactivitymeasuredbyfunctionalmagneticresonance imaging(fMRI)data(Atasoyetal.,2017b,2018).

However,connectomereconstructionmethodsgenerallysufferfrom manycaveatsthathindertheaccurateandreproduciblereconstruction ofconnectomes(JbabdiandJohansen-Berg,2011;Jonesetal.,2013).

Here,weaimtoprovideasystematicassessmentofconnectomehar- monicpatternsbyquantifyingtheirsensitivitytovariationsofstructural connectivity,andapplytheresultingframeworktocharacterizebrain changesduringneurodegeneration.Wedemonstratetheeffectsofdif- ferentframeworkparametersontheemergenceofreliableconnectome harmonicsusingwidelyacknowledgedandvalidatedtemplatedatasets.

Bysystematicallyalteringpropertiesofthelocalconnectivityofvari- ouscorticalsurfaces,characteristicsoflong-rangeconnections,andthe proportionoflocalandlong-rangeconnectivities,weevaluatethesensi- tivityoftheframeworktoproducereliableconnectomeharmonics.Our resultsrevealthatbothlocalgraymatterconnectivityandlong-distance whitematterfibertractsdeterminetheexactshapeoftheconnectome harmonicpatterns,yetthelocalgraymatterconnectivityplaysacrucial roleintheemergenceoffunctionallymeaningfulspatialpatternsonthe corticalsurface.Oursensitivityanalysisfurtherextendspreviousappli- cationsofconnectomeharmonicsbyintroducingstructuralwhitematter andgraymatteralterationstoobservetheirconsequencesonfunctional patternsemergingonthecorticalsurface.Finally,wealsodiscussim- plicationsofthepresentedconnectomealterationspossiblyrelatingfor existingdiseaseconditionssuchasHuntington’sandotherneurodegen- erativedisorders.

2. Methods

2.1. Multi-modalimagingframeworkanddataset

Weimplemented the systematic constructionof connectomehar- monics(Fig.1),thatweintegratedintoanexistingprocessingpipeline (SCRIPTS, (Proix et al., 2016)). Briefly,SCRIPTS is an open-source pipelinethatprocessesMRIanddMRItobuildsubject-specificsurface meshes,parcellationsandcorrespondingconnectivitymatrices.Weex- tendedthispipelinewithnewfeaturesallowingforcomputingsubject- specificandsubject-averagedhigh-resolutionsurface-basedconnectiv- itymatrices,andconnectomeharmonics.Theframeworkcombineslocal andlong-rangeconnectivitymatricestoformahigh-resolutionstruc- turalconnectome,onwhichthegraphLaplacianisapplied,andfrom whichtheconnectomeharmonicsarecomputed.Onlyparametersrel- evanttothehigh-resolutionconnectomeandtheconstructionofthose harmonicsareexploredinthiswork.

Toalleviatetheknownshortcomingsassociatedwithsubject-specific imagingmethods(Bürgeletal., 2006;Willatsetal., 2014)from our study,weusedvalidatedtemplatedatasetsfromopen-sourcestudiesto generate ourresults.Forthesurfacemesh,weusedsubject-averaged FreeSurfertemplatescvs_avg35_in_MNI152(default),of20,000vertices, aswellasfsaverage5(20,484vertices)andfsaverage4(5,124vertices).

AllsurfaceswereregisteredintheMNIspacesotheimpactofdiffer- entmeshresolutionscouldbe assessedwithoutmanual intervention.

For thewhite-matterstreamlines, weused the Gibbsdataset, which contains20,712,081streamlinescomputedbyprobabilistictractogra- phyfrom169subjects(HornandBlankenburg,2016).Allstreamlines areregisteredin MNIcoordinatesandnormalizedtominimizeinter- subject differences of brain sizesand shapes. The datasethas been crossvalidatedacrosscortico-corticalandcortico-thalamo-corticalat- lases (Behrens et al., 2003; Bürgel et al., 2006; Hornand Blanken- burg,2016).SeeSupplementaryText1forfurtherdetailsabouttheac- quisitionandprocessingofimagesfromtheGibbsconnectomedatabase, wherebyindividualsubjectswereregisteredinMNIspaceandnormal- izedusingtheDARTELprocedure(Ashburner,2007)inordertocorrect forinter-subjectdifferencesinbrainvolumeandestimateagroupcon- nectome.

2.2. Localandlong-rangeconnectivity

Gray matterintracorticalconnectivityincludesbothbranchedax- onalramificationswithinthegraymatter(typicallyatlengthsof0−2 mm (BraitenbergandSchuz,2013),andtheintrinsichorizontalcon- nectivitythatcanreachupto8mmwithinthegraymatteralongun- branchedaxonsthatrunparalleltothecorticalsurface(Gonzlez-Burgos etal.,2000;Melchitzkyetal.,2001;Vogesetal.,2010a,b;Braitenberg andSchuz,2013).Thehorizontalgraymatterconnectionscanterminate inpatchesacrossbrainregions,resultinginanisotropicpropagationof brainactivitythroughout thegraymatterandthus playingarolein firstorderprocessingamongsensoryareas(VogesandPerrinet,2012).

WhitematterconnectivityisusuallyderivedfromdMRIandtractogra- phyanalysisandincludesbothstreamlinesoflengthslessthan40mm (Bajadaetal.,2019),includingUfibers,knowntobeunreliablycounted due tolimitationsindMRIspatialresolution(Gigandetetal., 2008), andmedium and long-rangestreamlinesthat are80−160mm long, believedtobeassociatedwithareasinvolvedinfunctionalintegration (Bajadaetal.,2019).NotethattoshowU-fibersasshortas3mmre- quiresaspecifictractographyalgorithmparameterofstepsizelessthan 1mmandahighresolutionMRI,whichdiffersfromthestandardproto- col(Songetal.,2014).Inourstudy,weusethetermlocalconnectivityto refertointrinsichorizontalgraymatterconnectivityoflength1−6mm, butnotincludingthesinglecorticalcolumnscaleoflessthat1mm.This choiceisconsistentwiththecorticalsurfacesusedherein,wherebythe

Table1

Summarytableofparameters.

Parameter Default value; [range] Unit Interpretation

f i 0; [0; 8; 21; 89] ∅ Smoothing coefficient of the cortical surface mesh

z C 1; [0–10] standard deviation of weight distribution Adjacency weight threshold used for binarization of long-range connectivity matrix

Λ_s 2; [1; 2] node distance (number of graph edges away) Width of the local connectivity kernel

𝜂 0; [0–100] % Percentage of long-range connection trimmed based on

averaged white matter track length

𝜅 0 [0–100] % Percentage of interhemispheric long-range connections

randomly removed ( callosectomy )

𝜌 0 [0–100] % Percentage of local connections randomly removed

( anisotropy )

averageedgelengthsare2.9mm,3.1mmand5.6mmforfsaverage5, cvs_avg35andfsaverage4,respectively(seealsoSuppl.Figure1C).We usethetermlong-rangeconnectivitytorefertoanywhitematterconnec- tionfiberswithaminimumtracklengthof10mm.Thislowerboundof 10mmforatractographystepsizeof2mmisconsistentwithprevious seminalworkwhereintracksshorterthan5mmwerediscardedwhena tractographystepsizeof1mmwasused(Zaleskyetal.,2010)inorder toavoidtheinclusionofspurioustracks(seeSuppl.Figure1B).The shortestofthewhitematterconnectionsarereferredtoasshort-range whitematterconnections,butshould notbe confusedwiththelocal graymatterconnectivitydescribedabove.

2.3. Computationofhigh-resolutionconnectome

Wecomputed the vertex-basedhigh-resolutionconnectomeusing streamlinesreconstructedbytractographyandverticesfromthecortical surfacemeshes.Wemanuallycheckedthattracksandmeshwereprop- erlyaligned(seeSuppl.Figure1A),fordistancebetweentrackbounds andnearestcorticalsurfacemesh(seeSuppl.Figure1B).Intersections betweenthestreamlineswithFreeSurfer’scorticalsurfacemeshtem- plateswereassessedusingthecoordinatesoftwopointsofthetracksat eachbound(seeSuppl.Figure2).Ifanintersectionwasfoundforeach boundsiandj,theconnectionweightC_i,jbetweentheverticesnearest totheintersectionpointsisincrementedby1.Otherwise,ifforoneor bothboundsthetrackwastooshorttoreachthesurfacemesh(nointer- sectionpoint),alinearextensionof3mmusingthedirectiondefinedby linearlyinterpolatingtheboundandthethirdlastcoordinatesfromthe boundwasaddedtothetrack,andtheintersectionwasassessedagain.

Ifnointersectionisfoundinoneorbothboundsevenafterextension, thetrackisdiscarded.Lengthoftrackboundandsizeoftheextension areadjustableparameters,andseveralcorticalmeshesareavailableus- ingwhite-mattergray-matterboundary(WMGM)orgraymatter(pial) surfaces.

2.4. Computationofconnectomeharmonics

Asin(Atasoyetal.,2016),weusethegraphLaplacianLasthedis- cretecounterpartoftheLaplaceoperatorappliedtothebrainconnectiv- itymatrix.ThegraphLaplaciancanbeinterpretedasaparticularcase ofthediscreteapproximationofthecontinuousLaplace-Beltramioper- ator,ageneralizationoftheLaplaceoperatortoRiemannianmanifolds.

WeusethegraphLaplacian(Levy,2006)definedas:

𝐿=1 2

((𝐷−𝐴)+(𝐷−𝐴)^𝑇)

, (1)

whereAistheadjacencymatrixofthecombinedconnectomeconsisting ofwhite-matterandgray-matterconnectivity,andDitsdegreematrix.

Notethat otherformulaeexistfor computingthediscrete Laplacian.

Ourchoiceofthis Laplacianformulaismainlydrivenbyitsnumeri- calstabilityasdiscussedin(Levy,2006).Thecombinedconnectomeis theadjacencymatrixAobtainedbycombiningthelocalconnectivity adjacencymatrixA_𝓁 derivedfrom thecorticalsurfacemeshandthe

long-rangeconnectivityadjacencymatrixA_cconstructedbytractogra- phymethods:

𝐴=𝐴_𝓁∪𝐴_𝑐, (2)

wherebybothA_𝓁andA_carem×mmatriceswithmbeingthenumber ofverticesofthecorticalsurfacemesh.Thelong-rangeconnectivityad- jacencymatrixA_cis derivedbyremovingtheweakestweightsofthe z-scored long-rangeconnectivitymatrixC^z (referredinthisarticleas thresholding):

𝐴_{𝑐𝑖,𝑗}= {

1 if𝐶_𝑖,𝑗^𝑧 >𝑧_𝐶 ,

0 otherwise (3)

forall𝑖,𝑗=1,…,𝑚,with𝑧_𝐶beingtheadjacencyweightthreshold.Note that weapplied z-scorenormalization tothelong-rangeconnectivity matrixCinordertohaveintegervaluesof𝑧_𝐶 relatingtothestandard deviationofconnectionweights:

𝐶^𝑧=( 𝐶−𝜇_𝐶)

∕𝜎_𝐶, (4)

where𝜇_𝐶 isthemeanconnectivitystrengthofthelong-rangeconnec- tomeC,and𝜎_𝐶itsstandarddeviation.A_𝓁representsthelocalgraymat- terconnectivitymatrix.Asin(Atasoyetal.,2016),twonodesarelocally connectedwhentheyareconnectedthroughthemeshasdirectneigh- bors.

WethendecomposethegraphLaplacianLintoafinitenumberof eigenvalues𝜆kandeigenvectors,orconnectomeharmonics,𝜓k: 𝐿𝜓_𝑘=𝜆_𝑘𝜓_𝑘.

2.5. Connectomealterations

Localandlong-rangeconnectivitiescanbealteredbychangingpa- rameters at several stagesof theframework andaresummarized in Table1.Atearlystages,smoothingofthecorticalsurfacemesh(f_i)and thenumberofstreamlinestoretainfromtractographyareimportant parameterstoconsiderinbuildingconnectivitymatrices(weusedby defaultallavailablestreamlines).Atlaterstages,differentparameterse- lectionsarepossibleforcreatingadjacencymatricesfromweightedcon- nectivitymatricesbythresholding(z_C),fortrimming(𝜂,𝜅)ofthelong- rangewhitematterconnections,andforthediffusionkernelwidth(Λ_s), meshresolution(n_v),andanisotropy(𝜌)ofthelocalgraymattercon- nections.Theseparametersinfluencetheoutcomeoftheframeworkby modifyingthegraphstructure,theproportionoflocalorlong-rangecon- nectionsinthecombinedconnectomeusedforcomputingharmonics,or selectivelyremovingspecificconnectionsorconnectivitypatterns.

Defaultparametervalueswerechosenbasedonapreliminarypa- rameterspaceexplorationsothatharmonicscouldbegeneratedmost reliablywhilevaryingasubsetofotherparameters.MappingoftheDe- faultModeNetwork(DMN)totheDesikan-Killianyatlasisprovidedin Suppl.Table1.

2.5.1. Localconnectivityalterations

Cortical surfacemeshsmoothing. Corticalsurfacemeshes areoften averagedorsmoothedinstudieslookingataveragedbrainproperties (Fischl et al., 1999). Here, smoothing was performed to soften the strongcurvatureofthesurfacepriortocomputingtheconnectomeby thesurface-tractsintersectionroutine(seeSuppl.Figure2).Because linearlyextendedtractscanterminateobliquelyornotcrossthemesh, this routine can result in a reassignment of the track bound to a differentcorticalsurfacenode,orthetracknotbeingassignedatall,for computingthebrainwideconnectivitymatrix.Herewhenindicated, corticalsurfacemeshesfromFreesurfer’stemplatesweresmoothedus- ingthe

smoothPatch

functionfromanopensourceMATLABtoolbox (https://www.mathworks.com/matlabcentral/fileexchange/26710- smooth-triangulated-mesh)usingtheLaplaciansmoothingwithinverse vertice-distancebasedweightsandbyvaryingthesmoothingcoefficient f_i.ThenumbersofsmoothingiterationsweretakenfromtheFibonacci sequence,upto89iterations,andvisuallyinspected.

Local diffusion kernel width and mesh resolution. We applied two widthsΛ_sofdiffusionkernelsoverthecorticalsurfacetocomputethelo- calgraymatterconnectivitymatrix(Atasoyetal.,2016).WidthsΛ_scom- prisedeitheroneortwonearestneighbors,eachresultinginadifferent proportionoflocaltolong-rangeconnectionwhenthenumberoflong- rangewhitematterconnectionsarefixed.Ifinsteadtheproportionoflo- caltolong-rangeconnectionswasheldconstant,alargerlocaldiffusion kernelwidthallowstoincreasethenumberoflong-rangeconnections incorporatedtothecombinedconnectome.Forexample,alocal:long- rangeproportionof1:1withalocalkernelofonlyimmediateneigh- borsresulted inacombinedconnectomecomprisingaround100,000 long-rangeconnections.Thesame1:1proportionwithalocalkernel spanningimmediateneighborsandtheirneighborsresultedinacom- binedconnectomecomprisingaround400,000long-rangeconnections.

Thedefaultlocalkernelwidthchosenthroughoutthestudywastwo,but theresultsdonotchangewithonlydirectneighborconnections(Fig.8).

Tovarythemeshresolution,weusedameshwith20,484vertices(fsav- erage5)andacoarserversionof5,124vertices(fsaverage4).Changesin meshresolutionaffectthepositionofverticesandcanresultinadiffer- entvertexattributionoftrackboundsorindiscardedtracks.

Anisotropy. Weintroducetheremovalofcorticalsurfacemeshedges inaprocesstermedanisotropy,wherebymeshedgesareremovedeither randomly(withprobability 𝜌),byascending order,orbydescending orderofedgelengths.Avisualizationoftheresultinggraphsstructure forgradualchangesto𝜌isprovidedinSuppl.Fig.10forillustrative purpose.

2.5.2. Long-rangeconnectivityalterations

Weperformedtwodistinctoperationstoalterthelong-rangecon- nectivity:thresholding,whichisbasedonthenumberoftracksbetween vertices(weight-based),andtrimming,whichisbasedontheaverage tracklengthbetweenvertices(distance-based).

Long-rangeconnectivitythresholding. Forthresholding,thez-scored weightedlongrangeconnectivitymatrixC^z (Eq.4)wasbinarizedac- cordingtoanadjacencyweightthresholdvalue𝑧_𝐶.Theratioroflocal versuslong-rangeconnectionsisdefinedas𝑟=tr(𝐴_𝓁)∕(tr(𝐴_𝓁)+tr(𝐴_𝑐)), whichisthenumberoflocalconnectionsdividedbythesummednum- bersoflocalandlong-rangeconnections.Becausethelocalgraymatter connectivitymatrixisdeterminedbythecorticalsurfacemeshandthe diffusionkernel,wekeptitconstant(unlessotherwisementioned)and variedthewhitematterconnectivitythreshold𝑧_𝐶,whichtherebycon- trolstheproportionroflocalconnections.Asweightsrepresentthenum- berofstreamlinesconnectingdistantnodes,when𝑧_𝐶increases,onlythe mostprominenttracksofthewhitematterremain.

Long-range connectivity trimming. Trimming was simulatedby re- movingsomepercentage𝜂ofthelongrangeconnectivityentriesbased ontheiraveragetracklength.Differentscenarioswereimplemented:1) removingthelongesttracksfirst;2)removingtheshortesttracksfirst;

and3)removingtracksinrandomorder.Notethatweappliedtrimming

tothelong-rangewhitematterconnectionsafterapplyingthethreshold 𝑧_𝐶=1,therebysettingtheproportionoflocalconnectionstor≃0.7.

Thus,thereportedpercentageoftrimmingaffectedonlytheremaining long-rangewhitematterconnectionsafterthresholding.

Callosectomy. Finally,weintroducetheremovalofinter-hemispheric connections in a process termed callosectomy, whereby inter- hemisphericconnectionsareremovedeitherrandomly(withprobability 𝜅) orbydescendingorderoftracklengths.Avisualizationofthere- sultinggraphsstructureforgradualvaluesof𝜅 isprovidedinSuppl.

Figure10forillustrativepurpose.

2.6. Comparisonmetrics

Toassessthesensitivityofagivenparameteroftheframework,we compared connectomeharmonicsby computing Pearsoncorrelations between theharmonicpatternsemergingfromunaltered andaltered connectomes.WealsocomparedeachharmonictotheDMNusingmu- tualinformationasin(Atasoyetal.,2016).Thesemetricsservediffer- entpurposes:Pearsoncorrelationpermitsquantificationof similarity betweenharmonics,whilemutualinformationindirectlymeasuresthe structure-functionrelationshipsbetweeneachharmonicandtheDMN.

2.6.1. Correlation

Eachconnectomeharmonicisrepresentedbyavectorofthesizeof thenumberofverticesinacorticalsurfacemesh.Whensmoothingthe corticalsurfacemeshorchangingitsresolution,theorderingofvertices isaffected.Toovercomesuchdisorganization,theharmonicvectorsare projectedontothecoarserspaceoftheDesikan-Killianyatlas,whichis conservedbyregionmapping.Werefertothiscoarseratlasspaceas thelowerresolutionalternativetothehigherresolutionmeshspace.The similaritybetweenharmonicsoftwodifferentmeshesisthenassessed in thelowerresolutionatlasspacebycomputingthecorrelationPbe- tweenthetwocoarseharmonicvectors.Astwosetsofconnectomehar- monicsbecomeidentical,averyhighcorrelationvalueresults(P→1) forpairsofharmonicswiththesameindex,resultsinadiagonalhar- monicscorrelationmatrix(Fig.6,7and8).Divergencefromthediag- onalmatrixreflectsadecreaseincorrelationbetweenharmonicsofthe sameindexacrossthetwosets,reflectingadisorganizationoftheeigen- decompositionoutputintheformofeigenvalue/eigenvectorpairs.

2.6.2. Mutualinformation

TheMutualInformation(MI)betweenaconnectomeharmonic𝜓k

andtheDMNvectorv_DMNwascomputedasfollows:

𝑀𝐼(𝜓_𝑘,𝑣DMN)=

∑𝑁 𝑛=1

∑𝑀 𝑚=1

𝑝_{𝜓𝑘,𝑣DMN}(𝑛,𝑚)log

(𝑝_{𝜓𝑘,𝑣DMN}(𝑛,𝑚) 𝑝_𝜓𝑘(𝑛)𝑝DMN(𝑚)

)

(5)

wherep(n)istheprobabilitydistributionofn,andp(n,m)isthejoint probability distributionof nandm, withv_DMN beingthevectorrep- resentingtheDefaultModeNetworkinatlasspace.Thevaluesofthe harmonic vectorwerediscretized intoN=16bins, whiletheDMN is representedbyabinaryvector(M=2bins),whetherthecorticalmesh vertexispartoftheDMNornot(Atasoyetal.,2016).Mappingofthe DefaultModeNetwork(DMN)totheDesikan-Killianyatlasisprovided inSuppl.Table1.

2.7. Surrogateandstatistics

Different surrogates of the connectomeharmonics wereobtained byusingdifferentmethodologicaltypesofrandomizationofthehigh- resolutionconnectome.WeusedtheMonte-Carlomethodwiththenull hypothesis thatharmonicsestimatedfromthestructuralconnectivity matrixafterrandomizingthelong-rangeconnectivity(randomizedhar- monics𝜓̃_𝑘)havethesamesimilarity,asmeasuredbyMI,comparedto theonesgeneratedfromtheoriginalconnectome.Eachtypeofrandom- izationwasrepeated100times,involvedshufflinglong-rangebutnot

Fig. 2. Weight-based thresholding of long- rangewhitematterconnectionsincreasesMI betweenconnectomeharmonicsandthede- faultmode network(DMN). (A)Proportionr of local gray matter connections in the adja- cencymatrixAofthegraphLaplacian,fordif- ferent threshold valuez_C applied tothe long- rangewhitematterconnectivity.(B)Distribution ofconnectomeweights(numberofwhitematter streamlinesWbetweenvertices)inlog-logscale, describedasprobabilitygiventhattheweightis positive(P(W|W>0)).Verticallinescorrespond to𝑧𝐶valueswithsamecolorcodeasA).(C-D) Mutualinformation(MI)betweenthefirst100 connectomeharmonicsandtheDMNforarange ofz_Cresultinginarangeofproportionroflocal connections.Colorcodeisconsistentacrosspan- els.(SeeSuppl.Figure3forMIwithotherRSNs).

localconnectivity,andwasperformedusingthebrainconnectivitytool- box(RubinovandSporns,2010)whereinaselectedratioofconnections areshuffledwhilekeepingthedegreeofeachnodeunchanged(i.e.the totalnumberofconnectionspernoderemainedthesameforeachnode).

Thisconstraintmaintainsthehubnessofthegraph,andassuchisnotas destructivetographfeaturesasisacompletelyunconstrainedshuffling method.Allowingforhubnessdeteriorationwasnotwithinthecurrent scopeofworkbutmaybeinvestigatedinthefuture.Wetermtheran- domizationtypeinvolvingshufflingonlyinter-hemisphericconnections ofthelong-rangeconnectivitymatrixinter,andrandomizationinvolving shufflingonlyintra-hemisphericconnectionsofthelong-rangeconnec- tivitymatrixintra.Forinter+intrarandomization,bothinter-andintra- hemisphericconnectionswereshuffledbyisolatingeachquadrantofthe matrixandshufflingthemindependently.Forglobalrandomization,all long-rangeconnectionsareshuffledinasinglebatch.

Wecomputedasurrogatesummarystatisticbycalculatingtheproba- bility,𝑝_𝑀𝐼

surr,thatthe𝑀𝐼surrbeinggreaterthantheoriginal𝑀𝐼origfor 𝑁surrsurrogates ofasubsetofn_k consecutiveconnectomeharmonics 𝜓𝑘∈𝐾={𝑘0,…,𝑘0+𝑛_𝑘},startingatrankk₀.Inthemostcommoncasethrough thismanuscript,𝑘0=7and𝑛_𝑘=5asweareinterestedinharmonics7 to11.

𝑝_𝑀𝐼

surr = 1+

𝑁∑surr 𝑛=1

𝑛∑_𝑘−1 𝑚=0

{1 if𝑀𝐼orig_𝑘

0+𝑚≤𝑀𝐼surr_𝑛,𝑘 0+𝑚

0 otherwise 1+𝑁surr×𝑛_𝑘

where𝑁surr=100isthenumberofsurrogatesforeachrandomization type.Statisticswerecomputedforallvaluesof𝑧_𝐶butonlyreportedfor 𝑧_𝐶=1sinceitisthecriticalvalueforwhichlargescalepatternsemerge onthecorticalsurface.Wecorrectedp-valuesformultiplecomparisons usingtheBenjamini-Hochbergprocedurewithfalse-discoveryrate0.1 (McDonald,2009).

3. Results

To understand the effect of the two different types of struc- tural connectivitiescomposingthe humanconnectomeasdefined in

(Atasoyetal.,2016),namelythelocalgraymatterconnectivityandthe long-rangewhitematterconnectivity,wealteredseveralpartsof the frameworkinfluencingeitherthelong-rangewhiteconnectivityonly, localgraymatterconnectivityonly,orbothtogether.

3.1. Effectsofwhitematterconnectionchangesonconnectomeharmonics, affectingonlylong-rangeconnectivity

Weight-basedthresholding

Wefirstinvestigatedtheeffectofapplyingdifferentthresholdval- ues z_C totheweightsofthelong-rangewhite-matterconnectionsin- cludedinthehumanconnectome(seeMethods).Asexpected,theratio 𝑟=tr(𝐴𝓁2)∕(

tr(𝐴𝓁2)+tr(𝐴𝑐2))

ofthenumberoflocalovertotalnumber ofconnectionsincreasesasafunctionoftheadjacencyweightthresh- olds𝑧_𝐶(Fig.2A)appliedtothewhitematterconnectivity.Thisleadsto theincrementalremovaloflong-rangeconnectionswithlowestweights (Fig.2B).Wecomputedthemutualinformation(MI)betweeneachcon- nectomeharmonic patternandtheDMN,projected ontothecortical surfacemesh(seeMethods).Weobservedthatforlowfrequencyconnec- tomeharmonics,theMIgenerallyincreaseswithincreasing𝑧_𝐶(Fig.2C).

Thissuggeststhatlocalconnectionsplayakeyroleintheemergenceof functionallyrelevantconnectomeharmonicpatterns.Forhighpropor- tionsoflocaloverlong-rangeconnectivity(e.g.,resultingfromz_C>0.6), connectomeharmonics𝜓_𝑘∈𝐾={2,3,4,7,8,9,10,11}yieldedthehighestMIindi- catingthestrongestoverlapoftheseharmonicswiththeDMN.These MIvalues(MI ≃ 0.05)areconsistentwiththeonespreviouslyreported in(Atasoyetal.,2016).Toprovetherobustnessoftheframework,we alsoreproducedthosefindingsinasingleHCPsubject(seeSuppl.Fig- ure5A-D).

OurworkfocusesontheDMN,buttheMImeasuredbetweenhar- monicsandotherrestingstatenetworks(RSN)followsasimilarpattern (Suppl.Figure3),wherebyhigherz_CleadstohigherMI.Interestingly,a firstpeakiscommonlyobservedacrossseveralRSNsaroundthe3^rdhar- monic,andabroadersecondpeakisobservedbeforeoraroundthe9^th, whichseemsmorespecifictoeachRSN.Athirdpeakbetweenthe30^th and40^thharmonicsispresentfortheDMN.Inthefollowing,ournet- workofinterestremainstheDMN,andtheMIissystematicallyreported

Fig.3. MutualInformation(MI)betweenconnectomeharmonicsandtheDMNforseveralrandomizedversionsoflong-rangewhitematterconnectivity.

Meanandstandarddeviationofconnectomeharmonics’𝜓_𝑘∈𝐾={7,8,9,10,11}MIsfordifferentproportionsoflocaltolong-rangeconnections(parameterizedbythe adjacencyweightthreshold𝑧_𝐶asinFigure2).Differenttypesofconnectomesurrogatesareshown:original(norandomization),inter(interhemisphericonly),intra (intrahemisphericonly),inter+intra(interandintrahemisphericseparately),andglobal(interandintrahemispherecombined),seeMethods).⋆indicates𝑝<0.05of Monte-Carlostatisticaltestof𝜓_𝑘∈𝐾={7,8,9,10,11}MIvalueswiththeDMNfororiginalharmonicsvs.surrogatedataMIdistributionsfor𝑧_𝐶=1ofthedifferenttypesof randomizations.MIwithotherRSNsforthoserandomizationsareshowninSuppl.Figure9.

forharmonics𝜓_𝑘∈𝐾={7,8,9,10,11}asmotivated bytherangereportedto matchtheDMNpatternin(Atasoyetal.,2016)andFig.2.

Randomizations

Wethenassessedwhetherthespecificorganizationofthelong-range connectionssignificantlyaffectstheconnectomeharmonicpatternsand theirrelationtotheDMN.

Wegeneratedsurrogatedatawithseveraltypesofrandomizations ofthelong-rangeconnections(i.e.inter,intra,inter+intra, andglobal, see Methods) for each valueof 𝑧_𝐶 inorder to studytheireffects on MIbetweenconnectomeharmonicsandtheDMN.Localconnectivity waskept fixedandMIs between randomizedconnectomeharmonics

̃

𝜓𝑘∈𝐾={7,8,9,10,11}andtheDMNwerecomputed (Fig.3).Aspreviously, long-rangeconnectionswereeliminatedbyincreasingthethresholdz_C (andthereforetheratior)whichisentirelydependentonthenumberof streamlines(weight)ofeachlong-rangewhitematterconnection,with weakerconnectionsremovedfirst.

The difference of MI between the original connectome harmon- ics𝜓_𝑘∈𝐾={7,8,9,10,11} and theDMN compareto thosesame harmonics wheninter-hemispheric connectionsarerandomized, withoutsignifi- cantdifference(∀𝑧_𝐶,𝑝>0.1;Fig.3originalvsinter).However,assoon asintra-hemisphericconnectionsarerandomized,theMIwiththeDMN decreases significantly (𝑝<0.05 for original vs. intra, original vs. in- ter+intra, andoriginalvs. global, FDRcorrected,reported for 𝑧_𝐶=1 forclarity,correspondingtoaproportionr≃0.7oflocalconnections).

Suppl.Figure7showstheresultingfirst10harmonicsprojectedonto thecorticalsurfacemeshforvisualizationoftheobtainedspatialpat- ternsforthosedifferenttypesofrandomizations.

Regardlessoftherandomizationmethod,MIincreaseswithhigher proportionsoflocaltolong-rangeconnections(asinFig.2).Suppl.Fig- ure8showsthattheseresultsarealsoobservedwhentheMIiscom- putedonthe3^rdharmonicaloneorawiderrangeofharmonics(e.g.

from2to20,𝜓̃_{𝑘∈𝐾={2…20}}).Together,theseobservationssuggestthat functionallyrelevant features of the low-frequencyconnectomehar- monicpatternsobserved forproportionsof localconnectionsr >0.4 (correspondingtoz_C>0.3)disappearwithintrahemisphericandglobal randomization.ThisisfurthersupportedbySuppl.Figure9,showing

thattheseobservationsarevalidnotonlyfortheDMNbutalsoforother RSNs.WemainlyreporttherelationtotheDMNforconsistencywith theoriginalstudy(Atasoyetal.,2016).

Distance-basetrimming

We investigated the roleof shorter versus longer streamlines in thewhitematterconnectivitybyremovingthelong-rangeconnections basedontheirstreamlinesaveragelengths,aprocessthatwecalltrim- ming.

Whenthelongestwhite-matterfibersweretrimmedfirst(Fig.4A), theMIbetweenconnectomeharmonics𝜓𝑘∈𝐾={7,8,9,10,11}andtheDMN showedarapidincrease(i.e.for0–40%whitematterconnectionscut), whereaswhenshortestconnectionsweretrimmedfirst(Fig.4B)weob- servedlowMIvaluesuntilacertainthresholdwascrossedcorrespond- ingto ~ 60%ofwhitematterconnectionsbeingremoved.Thisresult indicatesthatthepresenceoflong-rangestreamlinesisamaincontribut- ingfactorfortheobservedlowMIbetweenharmonicsandtheDMN.

Trimminglongrangeconnectionsinrandomorderalsoshowedlower MIvaluesfor0–40%cut(Fig.4C),andfurtherincreasesofMIvalues above0.05fortrimmingabove60%.Thisisnotsurprising,sincethedis- tributionoffiberlengthsisskewedtowardstheshortfibers(seeSuppl.

Figure1B,middlehistogram),leadingtoahigherprobabilityofremov- ingshorterstreamlinesfirst,andthusconfirmingourfindingsreported above.

Callosectomy

Next,weassessedthecontributionofinter-hemisphericconnections tothefunctionalrelationofconnectomeharmonicstotheDMN(Fig.5).

Inter-hemisphericconnectionsweretrimmedbyascendinganddescend- ingfiberlengthsorrandomlyinaprocessreferredtoascallosectomy (as in Wang etal., 2017). Note that thealteration based on length is deterministic,while the randomremoval is not,andthus can be runmultipletimestoestablishstatistics.Fig.5showsthatharmonics 𝜓_𝑘∈𝐾={7,8,9,10,11}arenotstronglyaffectedbythecallosectomyuntilat least 50%of theinter-hemisphericconnectionsareremoved.There- lationof harmonics𝜓_𝑘∈𝐾={7,8,9,10,11} totheDMN is moresensitiveto

Fig.4. Effectofdifferentwhitematterfiberlengthsontheemergenceoffunctionalharmonicspatterns.Mutualinformation(MI)betweentheDMNand connectomeharmonics’𝜓_{𝑘∈𝐾={7,}8,9,10,11}fordifferentpercentagesoftrimmingofthewhitematterconnectivity.Trimmingwasperformedbyeliminatinglongesttracks first(A),shortesttracksfirst(B),andinrandomorder(C).Baselineconnectomeusedfor𝑧𝐶=1correspondingtor≃0.7beforetrimming.

Fig.5. MutualInformationbetweenconnectomeharmonicsandtheDMNforgradualremovalofinter-hemisphericwhitematterconnectivity.Mutual information(MI)betweenDMNandharmonics𝜓_{𝑘∈𝐾={7,}8,9,10,11}forgradualtrimmingofinter-hemisphericconnections(termedcallosectomy)bydescending(A), ascending(B)ordersoftracklength,andrandomly(C).Adjacencyweightthresholdissetto𝑧_𝐶=1,correspondingtotheproportionr≃0.7oflocalconnections beforealteration.⋆indicates𝑝<0.05ofMonte-CarlostatisticaltestbetweendistributionsofMIsusing100samples.Notethatweusedamaximumcallosectomyof 99%inordertoavoidtotallydisconnectedhemispheres.

shortthanlonginter-hemisphericfiberswhenthosecorrespondingcon- nectionsareremovefirst(Fig.5A-B).Whenrandomlytrimmed,MIbe- tweentheDMNandharmonics𝜓𝑘∈𝐾={7,8,9,10,11}isnotsignificantlydif- ferent(𝑝>0.1using100samples,afterBonferronicorrectionformulti- plecomparisons).Suppl.Figure11showstheresulting10firsthar- monics projected onto thecortical surfacemesh forvisualization of theobtainedspatialpatternsforgradualremovalofinter-hemispheric connections.

Aspecialcaseoccurswhenallinter-hemisphericconnectionsarere- moved(i.e.at100%callosectomy)suchthatleftandrighthemispheres areseparatedandpatternsappearconsecutivelyononehemisphereand theother(Suppl.Figure12A,middle).Insuchacase,eigenvaluesare organizedby pairsi.e.(𝜆1,𝜆2),(𝜆3,𝜆4),(𝜆5,𝜆6),…asshownin Suppl.

Figure12B,withtwozero-valuedeigenvalues𝜆1and𝜆2indicatingthe twoisolatedsub-graphs (Chung,1996),oneforeachhemisphere.As such,wealsore-constructedtheharmonicsonthewholebrainbycom- biningeigenvectorscomputedoneachhemisphereseparately(Suppl.

Figure12A,right).WeassessedtheMIbetweenthesenewlygenerated harmonicsandtheDMNfordifferentvaluesofwhitematterconnectiv- itythresholdz_C(Suppl.Figure12C)andobservethesamepatternof higherMIasz_Cincreases.

3.2. Effectsofcorticalsurfacemeshchangesonconnectomeharmonics, affectingbothlocalgraymatterandlong-rangewhitematterconnections

Weassessedhowchangesinthecorticalsurfacemeshgeometryand resolutionaffectthepatternsofconnectomeharmonics.

Smoothing

Firstly,weinvestigatedthechangesinharmonicpatternswhenusing agraymattersurfacecorticalmesh(Freesurfer’spialsurface)versusa white-mattergray-matterboundarysurface(WMGM,Freesurfer’swhite surface(Fig.6andSuppl.Figure2).Inordertoinvestigatetheeffect oftheprecisegeometryofthecortexontheparticularshapeofthecon- nectomeharmonic patterns,we appliedmeshsmoothingtodifferent meshesofthecorticalsurfaceanditerativelyincreasedtheamountof smoothing.Wemeasuredthecorrelationbetweenharmonicsintheat- lasspacetocompareharmonicpatternsemergingondifferentcortical surfacemeshes(seeMethods).Notethatsmoothingofthecorticalsur- facemeshalsoaffectsthewhite-matterconnectome,asthelong-range connectivitymatrixisgeneratedbasedonthetrack-meshintersections.

WefoundthatwhenusingtheWMGMsurfacemesh,smoothingdid nothaveasignificantinfluenceonthelowestfrequencyharmonicsup tothe15^th,indicatedbyhighdiagonalvaluesinthecorrelationmatrices inFig.6,constructedbycorrelatingtheoriginalcorticalsurface(rows) andthesmoothedsurfaces(columns).Higherfrequencyharmonicswere moderatelyaffectedbythesechanges,asseenbyhighvaluesofcorrela- tionshiftingfrombeingexactlyonthediagonaltobeingafewelements off diagonal,indicatingare-orderingofeigenvalue-eigenvectorpairsby onlyafewranks.Whenusingthegraymatterpialsurface,weobserved adegradationofthecorrelationbetweenharmonicpatterns(lowerval- uesinthediagonalofthecorrelationmatrix,Fig.6,right).Inthiscase, correlationvaluesforlowfrequencyharmonics𝜓_𝑘∈𝐾={1,2,3,4,5}remained unaffected,butdecreasedforlowfrequencyharmonics𝜓_{𝑘∈𝐾={6,…,}20}. Here,lowfrequencyharmonicscorrespondingtothelowest20eigen- valuesoftheeigenspectrumrepresentonly0.1%oftheharmonicspec-

Fig.6. Lowfrequencyharmonicsarerobusttocorticalsurfacechanges.High-resolutionconnectomeandconnectomeharmonicsarerecomputedfordifferent smoothinglevelsofWMGMcorticalsurface(left),orpialsurface(right)andcomparedtooriginalWMGMsurfacemeshusingPearsoncorrelationinatlasspacefor thefirst100harmonics.Adjacencyweightthresholdissetto𝑧_𝐶=1,correspondingtoaproportionr≃0.7oflocalconnections.

Fig.7. Lowfrequencyharmonicsarerobusttomeshresolutionchanges.Region-wisecorrelationofconnectomeharmonics𝜓_{𝑘∈𝐾={1,…,}100}usingfsaverage4 (5,124vertices)andfsaverage5(20,484vertices)corticalsurfacemeshtemplatesfromFreeSurfer.Proportionroflocalconnectionsisindicatedbydifferentdegrees ofconnectomeadjacencyweightthreshold𝑧_𝐶.Insetsshowamagnifiedversionofthecorrelationmatrixforconnectomeharmonics𝜓_{𝑘∈𝐾={1,…,}20}.

trum.Changeswereevenmoreprominentinhigherfrequencymodes wherethediagonallinedisappear.Wealsoreproducedthosefindingsin asingleHCPsubject(Suppl.Figure5E),showingthattheon-diagonal correlationvaluesareverystrongincludingforhigh-frequencyharmon- ics,whichsupportsthattheframeworkisrobusttoperturbationsofthe WMGMsurfacesuchassmoothing.

Meshresolution

Wealsoexaminedtheimpactofusingdifferentmeshsizestorep- resentthecorticalsurface(Fig.7),whilevaryingz_Candthereforethe proportionoflocalconnectionsr.

We foundthat whentheproportion oflocal connectivity is high comparedtothelong-rangeconnections(i.e.r>0.7),thepatternsof the emerginglow frequency harmonics 𝜓k<15 were robust tothese changesinlocalgraymatterconnectivity,asseenbythediagonalon the correlation matrices for z_C > 1. Note that sincedifferent mesh resolution implies thattheordering ofvertices fromone meshreso- lution to theother doesnot pair, the correlation matricesarecom- putedinatlasspaceratherthaninvertexspace,resultinginmorenoisy values.

Fig.8. Lowfrequencyharmonicsarerobusttolocaldiffusionkernelwidthchanges.Vertex-wisecorrelationofconnectomeharmonics𝜓_{𝑘∈𝐾={1,…,}100}using cvs_avg35templatedecimatedto20,000vertices,usingΛ_𝑠=1(Λ₁)vs.Λ_𝑠=2(Λ₂)neighboringverticesaslocalconnectivitykernel.Proportionroflocalconnections foreachkernelsizes(separatedbyadashΛ₁−Λ₂)isindicatedbydifferentdegreesofconnectomeadjacencyweightthreshold𝑧_𝐶.Insetsshowamagnifiedversion ofthecorrelationmatrixforconnectomeharmonics𝜓_{𝑘∈𝐾={1,…,}20}.

Fig.9.MutualInformationbetweenconnectomeharmonicsandtheDMNforgradualdisruptionsoflocalgraymatterconnectivity.Mutualinformation (MI)betweenDMNandharmonics𝜓𝑘∈𝐾={7,8,9,10,11}forgradualremovaloflocalgraymatterconnections(termedanisotropy)bydescending(A),ascending(B)order ofedgelengthandinrandomorder(C).Adjacencyweightthresholdissetto𝑧_𝐶=1correspondingtoaproportionr≃0.7oflocalconnectionsbeforealteration.⋆ indicates𝑝<0.05ofMonte-CarlostatisticaltestbetweendistributionsofMIsusing100samples.

3.3. Effectsofgraymatterconnectionchangesonconnectomeharmonics, affectingonlylocalconnectivity

Localdiffusionkernelwidth

Wethenfocused onalterationsaffectingonlythelocalgray mat- ter connectivity, first by changing the spatial extent Λ_s of the lo- cal connectivity(Fig.8) while varying z_C andtherefore thepropor- tionoflocal tolong-rangeconnectionsr.Aspreviouslyreported, we found thatwhen the local connectivity was sufficiently strong with respect to thelong-range connections(i.e. r > 0.7), the patterns of the emerging low frequency harmonics 𝜓k<15 were robust to the widening or narrowing of the local gray matter connectivity kernel width, as seen by a clear diagonal on the correlation matrices for z_C>1.

Note thatthe proportionr of local gray matterconnections also varyaccordingtothelocalconnectivitykernel widthΛ_s sincewider kernelscreatemorelocalconnections.Wealsoreproducedthosefind- ingsinasingleHCPsubject(Suppl.Figure5F),showingthattheon- diagonalcorrelationvaluesareverystrongincludingforhigh-frequency harmonics, which supports that the framework is robust to such changes.

Anisotropy

Thelastmodificationofthelocalgraymatterconnectivityconsisted ofremovingedgesfromthemeshformingthelocalconnectivitymatrix, aprocesstermedanisotropy.Asforthecallosectomy,weperformedsuch trimmingeitherbyascendingorbydescendingorderofedgelengths,or randomly(Fig.9).Noteagainthatthealterationbasedonedgelength isdeterministic,whiletherandomremovalisnotandthuscanberun multipletimestoestablishstatistics.Itcanbeobservedthatwhenedges areremovedrandomlyorwhenlongestedgesareremovedfirst(Fig.9A and9C),theMIbetweenharmonics𝜓_𝑘∈𝐾={7,8,9,10,11}andtheDMNde- creasesforanisotropy𝜌above10%(𝑝<0.05using100samples,after Bonferronicorrection).Intheothercase,whenshortestedgeswerere- movedfirst(Fig.9B),theMIbetweenharmonics𝜓_𝑘∈𝐾={7,8,9,10,11}and theDMNdecreasesfaster,for𝜌of5%.Suppl.Figure7showsthere- sulting10firstharmonicsprojectedontothecorticalsurfacemeshfor visualizationoftheobtainedspatialpatterns.

4. Discussion

Inthiswork,wehavepresentedaquantitativeevaluationof how changesinthestructuralconnectomeaffecttheemergenceofconnec-

tomeharmonic patterns.Ononehand,wedemonstratethatthepro- portionoflocalgraymattertolong-rangewhitematterconnectionsis acriticaldeterminantfortheemergenceofspatialharmonicpatterns onthecorticalsurfaceconsistentwithfunctionalbrainnetworks.On theotherhand,changestothecorticalsurfacemeshonlyweaklyaf- fectthelowestfrequencyconnectomeharmonics.Therelationshipbe- tweenthestructuralconnectomeandtheRSNsisacoveredtopic(Honey etal.,2007;Decoetal.,2011),andmanyfactorshavebeenshownto influencethis relationship,includingthetopologyof theconnectome (Goñietal.,2014;Messé etal.,2015).Theimportantdifferentiatorof theconnectomeharmonicsframeworkisthatitnotonlyuseswhitemat- terconnectivity(asotherstudiesdo)butinadditionincorporateslocal graymatterconnectivityof1-6mm,whichweshowiscriticaltoobtain ameaningfuldecompositionofstructureintofunctionalpatternsonthe corticalsurfacethatpossiblysupportRSNs.Notably,ourfindingsthat suchlocalgraymatterconnectivitymustaccountfor30–80%ofallcorti- calconnectionsisconsistentwithneuroanatomicalobservationsincats androdents(Stepanyantsetal.,2009;Boucseinetal.,2011).

Interpretationoflocalgraymatterconnectivityconstructedfrom thecorticalsurfacemesh

Interconnectingcorticalsurfacemeshedgestomodelgraymatter connectivityisrelativelycommon(SpieglerandJirsa,2013;Eckeretal., 2013;Loetal.,2015;Spiegleretal.,2020,2016).Thespatialextentof thelocalconnectivitykernelwidthinourstudyisconsistentwithgray mattertracingstudies(Gonzalez-Burgosetal.,2000;Melchitzkyetal., 2001;BraitenbergandSchuz,2013).Otherapproachesforconstruct- inglocalgraymatterconnectivityfromthecorticalsurfacemeshhave beenproposed.Forexample,Ecker etal.(2013)usedtheradiusand perimeterofacorticalpatch,whosesurfaceisgivenbyafixedratioof thetotalcorticalsurface,asaproxyforgraymatterintrinsiccortico- corticalwiringcost.This cost,theydetermined,is alteredinautism.

Othertechniquescouldalsobeapplied,wherebylocalconnectivitybe- tweenadjacentregionsofanatlas-basedparcellationisestimatedbythe boundarylengthbetween regions.Theanalysisofsuchaframework, throughwhichgraphspectralpropertiesandtheirrelationshipstocon- nectomeharmonicscanbeobserved,isanexcitingdirectionforfuture studies.

Roleoflocalgraymatterversuslong-rangewhitematter connections

Wefound thatthe local gray matterconnectionsencoded in the connectivityofthecorticalsurfacemeshispreponderantforobserving highmutualinformationbetweenlowfrequencyconnectomeharmon- ics𝜓𝑘∈𝐾={7,8,9,10,11}(asinAtasoyetal.,2016)andtheDMN.However, withoutthepresenceoflong-rangeinterhemisphericconnections,spa- tialmapsarerestrictedtoseparatehemispheres,andthuswholebrain mapsarenotobserved.Yet,ourresultsdemonstratethatlong-rangecon- nectionsalonearenotsufficientfortheemergenceofwholebrainmaps whenusingahigh-resolutionconnectomeofseveralthousandsnodes, astheyhavethesideeffectofcausingbrainnetworkstodivideinto many isolated componentswithout theformation of large-scale pat- ternsonthecorticalsurface.This“isolatedcomponents” issueoccurs becauseunderthiscondition,thegraphisdisconnected.Connectedness meansmathematicallythatthereexists(atleast)apathfromonever- textoanyothervertexthroughthegraph.Withinthehigh-resolution connectome,thewhitematterconnectivitymatrixoftendoesnotfulfill thiscondition;thelocalconnectionsthroughthecorticalsurfacemesh alleviatesthisproblem.Hence, ourfindings emphasizethatacareful blendof localandlong-rangeconnectivityisnecessaryfortheemer- genceoffunctionallymeaningfulharmonicpatternsspanningthewhole corticalsurface.Ourresultsconfirmthatconnectomeharmonicsnatu- rallyprovidearepresentation,wherelowfrequenciesexhibitmorero- bust,stable,andgenericpatternsacrosssubjects,whilehighfrequency harmonicsprovidemorevariable,sensitive,andpossiblymoresubject- specificpatterns.Ourfindingthatlocalconnectivityispreponderantfor

observinghighMIbetweenconnectomeharmonicsandtheDMNdoes notinvalidatepreviousstudiesthatretrievedadegreeofcorrelationbe- tweenthewhite-matterconnectivityandtheDMN(Hornetal.,2014), thefunctionalconnectivitycomputedfromdiffusionmodelswithnoise (Honeyetal.,2009;Saggioetal.,2016),ormorecomplexnon-linear modelswithtimedelays(Honeyetal.,2009;Cabraletal.,2011).Rather, itsuggeststhatlocalgraymatterconnectivitymayalsoplayanimpor- tantroleintheemergenceofrestingstates.Strongreverberatinglateral loops presentingray mattermicrocircuitslikely influencetheemer- genceandsustenanceoflocalizedcorticalactivityrecordedinEEGand fMRI, andispresentinsomeotherwork(Honeyetal.,2009).These loopsareincorporatedintotheconnectomeharmonicsframeworkvia thelocalconnectivitykernelandourresultsdemonstratethattheincor- porationofbothlocalandlong-rangeconnectionsprovidesaninterest- ingbasistoaddresstheroleofgrayversuswhitematterinlargescale corticaldynamics.Furtherrefinementofthelocalgraymatterconnec- tionstoincluderegion-specificconnectivityprofilesshallbeexamined inthefutureinordertoassessspecificallywhichaspectsofgraymatter connectivityinfluencelargescalefunctionalnetworks.

Relationtodiseaseconditions

Interestingly,ourgradualremovalofinter-hemisphericwhitematter connectionscanberelatedtothedegenerationoccurringinthecorpus callosumasobservedincohortsofpatientswithHuntington’sdisease (McColganetal.,2017).Inthiscondition,lateralcallosaldegeneration isassociatedwithpre-manifeststagesofthedisease,andevolveswith thediseaseprogressiontowhitematterdisruptioninmedialareas.Strik- ingly,functionalstudiesshowthattheDMNremainsunaffectedbythe diseaseuntilmuchlaterstageswhenthepatientbecomesstronglysymp- tomatic(Poudeletal.,2014),anobservationwhichisconsistentwith ourpredictionabouttheeffectsofcallosectomy.Similarly,ourdisrup- tionofgraymatterconnectivity,herereferredtoasanisotropy,could berelatedtolaterphasesofnormalaging,whereingraymatteratrophy isobserved(Sala-Llonchetal.,2015),andmayinturninducedegener- ativedisorders(Thompsonetal.,2003;Whitwelletal.,2010).Inthose conditions,graymatterdisruptionsareoftenlocalizedtospecificbrain regions,whileinourstudytheyareappliedtothewholecortex.Nev- ertheless,theinvestigationofthesediseasespecificdisruptionsthrough theprismofgraphspectraltheoryispromising(Sihagetal.,2020)and willbethesubjectoffuturestudies.

Otherusesofgraphspectraltheoryinneurosciences

Other frameworks utilizing the graph Laplacian on brain struc- tural connectivityrelyoncoarser resolutionconnectomes(Rajetal., 2012; Huang et al., 2018; Preti and Van De Ville, 2019; Tewarie et al., 2019), defined by atlases of larger brain regions, each of whichusuallyspanningseveralcubiccentimeters(Desikanetal.,2006; Destrieux etal., 2010; Glasser etal., 2016).Employing suchatlases alleviatestheproblemof”isolatedcomponents” becauseisolatedsub- networksdonotemergegiventhechosensizeofbrainareas.Anim- portant limitationof theparcellatedapproach isthat ityieldssharp boundariesbetweencoarsebrainregions,suchthattwoneighbouring points onthecortexcanbe affiliatedwithtwodifferent brainareas, andresultinginwholebrainconnectivitymatricesreflectingonlythe whitemattertracksandnotadequatelytakingintoaccountgraymat- terconduction.Theeffectoflocalgraymatterversuslong-rangewhite matter connectivity on large-scalebrain dynamics usingcoarse con- nectomeshavenonethelessbeeninvestigatedusingneuralfieldmodels (Proixetal.,2016).Ithasbeensuggestedthatwhitematterconnectivity shapesthecorticalslowrhythmswhilefasterdynamicsaredependent on short-rangeconnections.Hence,furtherextensionofthese studies totheuseofhigh-resolutionconnectomessuchasutilizedhereandin (Atasoyetal.,2016,2017a)canbe animportantdirection forfuture research.

Differentmethodsforconnectomereconstruction

Different approaches can also be utilized in reconstructing a weightedconnectomefromtractographystreamlines(Smithetal.,2015;

Donahueetal.,2016;HornandBlankenburg,2016).Forexample,in (HornandBlankenburg,2016)eachfiberweightdecaysexponentially withdistanceoftrackboundstocorticalsurface.Inthisstudy,wehave onlyexcludedfibertrackswhoseendpointsarefartherthan5mmfrom thecorticalsurface.Weacknowledgethatstreamlineend-pointslocal- izationareassociatedwithsomedegreeofspatialerror,duetothelim- itationsoftractographyandaccumulationoferrorsduringstreamline propagation(Jonesetal.,2013).Somedegreeofspatialsmoothingon thehigh-resolutionconnectomecouldbeperformedtoalleviatethisspa- tialerror,andwillbeincorporatedtofuturereleasesoftheframework.

Itmustalsobenotedthathigh-resolutionconnectomesarecomputation- allyexpensivetocreateandmanipulate.Specifically,sincetheeigende- compositionofamatrixwithnentrieshasacomputationalcomplexity of(𝑛³),itbecomesquicklyintractableformatrixsizesabove30,000 vertices.Forreference,thealgorithmtook1to5minutestoconverge usingthe20,484-by-20,484matrixcorrespondingtofsaverage5onalab- oratoryworkstation.

In line with previous studies (Besson et al., 2014; Donahue etal.,2016),wefoundtheintersectionbetweenwhitematterandgray mattercorticalsurfacetoyieldthemosteffectiveconnectivitymatrix, sincetheintersectionbetween thewhite-matterfibertracksandthis corticalmesharemorereliableandanatomicallymoremeaningfulthan theirintersectionwiththepialsurface.Theuseoftemplatedatasetsfur- therintroducespossibleerrorsintheintersectionbetweenthecortical surfaceandthetracks.These templatesallowedourfindingstogen- eralizewellatthegrouplevel,butwiththedrawbackoflosinginter- individualspecificity,which maybecrucialformedicalapplications.

Thisfurtherhighlightsanimportantdifferencewiththeoriginalwork fromAtasoyetal.(2016)whousedsubjectspecifictractographystream- lineswiththeircorticalsurfacemeshesinthesamenativespace(n=10 subjects).Instead,weusednormalizedtractographystreamlinesfrom the169subjectsoftheGibbsdataset(HornandBlankenburg,2016) withtemplatecorticalmeshesprovidedinFreeSurfer.Tracksandsur- facewereregisteredinMNIcoordinatesandvisuallycheckedforproper alignmentwiththe20.7millionstreamlinesfromtheGibbsdataset.The track-meshintersectionalgorithmpresented hereresultsinaconnec- tomewithmorethan10millionentries,usingaminimaltracklength constraintof10mm.Ontheonehand,thisapproachhastheadvantage ofa muchlarger cohortandhigherquality tractographystreamlines thathavebeenvalidatedacrossseveralwhitematteratlases(Hornand Blankenburg,2016).Ontheotherhand,ouruse oftemplatecortical surfacemeshes,whosegeometryintemplatespacemaynotperfectly correspondtothenormalizedstreamlineboundsinnativespace,repre- sentsasensiblepitfall.Forfuturestudies,itwillbeimportanttofavor theuseofcorticalsurfacesinnativespacesothatstreamlineboundsare perfectlyalignedtothecorticalmeshaswedidfortheHCPsubjectin SupplementaryMaterials.

RelationtoSphericalHarmonicsandSurfaceLaplacian

Connectomeharmonicscanalsobeviewedasanextensionofspher- icalharmonics,whichcorrespondtotheeigenfunctionsoftheLaplace operatorappliedtoasphere.Intheabsenceof anylong-rangewhite matterconnectivity, connectome harmonicsof each hemisphere,de- riving onlyfrom thegraymatter connectivity, correspondtospheri- calharmonicsmappedtothecorticalsurface.Similardecompositions ofcorticalactivitypatternsintosphericalharmonicshavebeenprevi- ouslybroughtby(Jirsaetal.,2002;Robinsonetal.,2016),whereby eachbrainhemisphere ismodeled asaperfectsphere.A similarap- proachbasedonsphericalharmonicshasalsobeenutilizedinelectroen- cephalogram(EEG)andmagnetoencephalography(MEG)sourcerecon- struction,wherethescalpismodelledasthehomogeneoussurfaceof asphereonwhichtheelectricalsignalspropagateisotropicallyfroma neuronalsource(NunezandSrinivasan,2006;Petrov,2012;Carvalhaes

anddeBarros,2015).Whileconceptuallysimilar,theconnectomehar- monics frameworkextendsothersphericalharmonic approachesand embedsthefullstructuralconnectivityof thehumanbrainbytaking intoaccountbothlocalgraymatterandlong-rangewhitemattercon- nectivities.

Lastly,wepointoutthatasconnectomeharmonicsprovideanexten- sionoftheFourierbasisandsphericalharmonicstothehumanconnec- tome,theyalsoyieldafrequency-specificfunctionalbasiswhichenables therepresentionofanypatternofcorticalactivity.Thisimportantprop- ertyofconnectomeharmonicshasbeenutilizedforthespatialfrequency analysisoffunctionalmagneticresonanceimaging(fMRI)data,which hasdemonstratedcrucialdifferencesinthesignaturesofdifferentbrain states(Atasoyetal.,2017a,2017b,2018).

5. Dataandcodeavailability

The data used in this study is publicly available. Tractogra- phy streamlines are part of the open source Gibbs connectome dataset (https://www.nitrc.org/projects/gibbsconnectome/) and cor- tical surface meshes are provided by FreeSurfer software suite (https://surfer.nmr.mgh.harvard.edu/).The frameworkfor construct- ingconnectomeharmonicsis integratedtotheopensource SCRIPTS pipeline, in the branch ConnectomeHarmonics (https://github.com/

ins-amu/scripts/).

Templateconnectomeharmonics𝜓𝑘∈𝐾=1…100areprovidedin .mat formatalongsideintermediategraphmeasuresforthecorticalsurface meshescvs_avg35_inMNI152,fsaverage4andfsaverage5,usingtheGibbs connectomewithdefaultframeworkparametersfromTable1.Thefiles arehostedonZenodoathttps://zenodo.org/record/4027989. Conflictofinterest

The authors declare the absence of conflict of interest. Funding sourceshadnoroleinstudydesign,dataanalysis,decisiontopublish, orpreparationofthemanuscript.

CRediTauthorshipcontributionstatement

SébastienNaze:Conceptualization,Methodology,Software,Valida- tion,Formalanalysis,Writing-originaldraft,Writing-review&editing, Visualization,Projectadministration.TimothéeProix:Conceptualiza- tion,Methodology,Software,Validation,Writing-originaldraft,Writ- ing-review&editing.SelenAtasoy:Conceptualization,Methodology, Validation,Writing-review&editing.JamesR.Kozloski:Conceptu- alization,Resources,Writing-review&editing,Fundingacquisition.

Acknowledgements

ThisworkwaspartlyfundedbytheCHDIFoundation.S.A.issup- portedbytheBeatriudePinos-MarieCuriePostdoctoralFellowship andbytheEuropeanResearchCounselConsolidatorgrant:CAREGIV- ING”615539”.

Supplementarymaterial

Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,at10.1016/j.neuroimage.2020.117364

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