Small variation in dynamic functional connectivity in cerebellar networks

Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Iza o Fe nandez-I iondoa,An onio Jimenez-Ma ina,b,Ibai Diezc,d,Paolo Boni azia,e,S ephan P. Swinnen ,g,Miguel A. Muñozh,j and Jesus M. Co esa,e,i,j aBioc uces-Bizkaia Heal h Resea ch Ins i u e, Ba akaldo, Spain bBiomedical Resea ch Doc o a e P og am, Uni e si y o he Basque Coun y (UPV/EHU), Leioa, Spain cFunc ional Neu ology Resea ch G oup, Depa men o Neu ology, Massachuse s Gene al Hospi al, Ha a d Medical School, Bos on, MA 02115, USA dGo don Cen e , Depa men o Nuclea Medicine, Massachuse s Gene al Hospi al, Ha a d Medical School, Bos on, MA 02115, USA eIKERBASQUE: The Basque Founda ion o Science, Bilbao, Spain Mo emen Con ol and Neu oplas ici y Resea ch G oup. KU Leu en, Leu en, Belgium gLeu en B ain Ins i u e (LBI). KU Leu en, Leu en, Belgium hIns i u o de Física Teó ica y Compu acional Ca los I, Uni e sidad de G anada, Facul ad de Ciencias, G anada, Spain iDepa amen o Cell Biology and His ology. Uni e si y o he Basque Coun y (UPV/EHU), Leioa, Spain jEqual las -au ho con ibu ion A R T I C L E I N F O Keywo ds: Dynamic unc ional connec i i y S uc u al connec i i y An e io Ce ebellum Pos e io Ce ebellum Res ing s a e Abs ac B ain ne wo ks can be defined and explo ed h ough hei connec i i y. He e, we analyzed he e- la ionship be ween s uc u al connec i i y (SC) ac oss 2,514 egions ha co e he en i e b ain and b ains em, and hei dynamic unc ional connec i i y (DFC). To do so, we ocused on a combina ion o wo me ics: he fi s assesses he deg ee o SC-DFC simila i y –i.e. he ex en o which he dy- namic unc ional co ela ions can be explained by s uc u al pa hways–; and he second is he in insic a iabili y o he DFC ne wo ks o e ime. O e all, we ound ha ce ebella ne wo ks ha e a smalle DFC a iabili y han o he ne wo ks in he b ain. Mo eo e , he in e nal s uc u e o he ce ebellum could be clea ly di ided in wo dis inc pos e io and an e io pa s, he la e also connec ed o he b ains em. The mechanism o main ain small a iabili y o he DFC in he pos e io pa o he ce ebel- lum is consis en wi h ano he o ou findings, namely, ha his s uc u e exhibi s he highes SC-DFC simila i y ela i e o he o he ne wo ks s udied, i.e. s uc u e cons ains he a ia ion in dynamics. By con as , he an e io pa o he ce ebellum also exhibi s small DFC a iabili y bu i has he lowes SC-DFC simila i y, sugges ing a diffe en mechanism is a play. Because his s uc u e connec s o he b ains em, which egula es sleep cycles, ca diac and espi a o y unc ioning, we sugges ha such c i ical unc ionali y d i es he low a iabili y in he DFC. O e all, he low a iabili y de ec ed in DFC expands ou cu en knowledge o ce ebella ne wo ks, which a e ex emely ich and complex, pa icipa ing in a wide ange o cogni i e unc ions, om mo emen con ol and coo dina ion o execu i e unc ion o emo ional egula ion. Mo eo e , he associa ion be ween such low a iabili y and s uc u e sugges s ha diffe en ia ed compu a ional p inciples can be applied in he ce ebellum as opposed o o he s uc u es, such as he ce eb al co ex. In oduc ion Unde s anding he ela ionship be ween diffe en classes o connec i i y is undamen al in ne wo k neu oscience [ 1 ]. To da e, diffe en s a egies exis o ob ain s uc u al con- nec i i y (SC) ma ices om magne ic esonance imaging (MRI), whe e each en y o he ma ix ep esen s he numbe o whi e-ma e s eamlines be ween pai s o b ain egions ob ained om diffusion-weigh ed images (DWIs) [ 2 ]. Simi- la ly, he e is conside able in o ma ion as o how o cons uc unc ional connec i i y (FC) ma ices, ob ained by assessing he simila i y in he dynamics o gi en pai s o b ain egions om a sequence o unc ional images employing di e se me - ics, e.g., pai wise Pea son co ela ions o synch oniza ion measu es [ 2 ]. Howe e , despi e hese significan ad ances, a p o ocol o define o p edic one connec i i y class om ano he emains o be defined. ∗Co esponding au ho [email p o ec ed] (J.M. Co es) ORCID(s): 0000-0002-9059-8194 (J.M. Co es) One issue ha complica es ma ching SC o FC is he ac ha hey in ol e e y diffe en ime scales e en hough hey deal wi h ne wo k connec i i y be ween he same b ain e- gions. Hence, SC is p ac ically in a ian in he pe iod o e which he FC is calcula ed ( ypically a maximum o 10 min- u es), he la e known o a y o e sho ime scales o e en a ew seconds, exhibi ing a ich dynamic epe oi e (see [ 3 ] and e e ences he ein). When conside ing sho ime scales, he simples manne o assess and quan i y he empo al a ia ion in "dynamic unc ional connec i i y" (DFC) is by conside - ing a sliding window analysis, whe e he o al window leng h is di ided in se e al in e als o a fixed du a ion. One FC is hen calcula ed o each ime window and in his way, he DFC is ep esen ed as a ime o de ed sequence o FC ma i- ces. Impo an ly, i has been shown ha he SC can be in e ed om he DFC when he ime window o calcula e i is in- fini ely la ge [ 4 ]. In o he wo ds, pai wise co ela ions –when Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 1 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks a e aged o e sufficien ly long ime pe iods– me ely eflec he unde lying SC ma ix (see also [ 5 ]). This ac has become e en clea e h ough he obse a ion ha unc ional ne wo ks in he es ing s a e can be de i ed om he spec um o eigen- modes –o ha monics– o he SC ma ix (ac ually, om i s associa ed Laplacian ma ix [ 6 ]). By con as , when unc- ional ne wo ks a e ob ained o e much sho e ime windows, he dynamics o he b ain ope a ing on a fixed SC ne wo k gene a es a la ge epe oi e o diffe en a ying unc ional ne wo ks. Howe e , how such dynamic pa e ns a e ela ed o he b ain’s unc ion and disease emains la gely unclea , despi e he subs an ial ad ances achie ed o e ecen yea s [4,7–11]. The ela ionship be ween he empo ally-in a ian SC and he highly empo al-sensi i e DFC can be assessed by compa ing he wo g aphs a he le el o indi idual links, a s a egy which equi es – o symme ical ma ices– 2∕2 compa isons (whe e  is he numbe o nodes in he ne - wo k). Al e na i ely, he e we ollow ano he and mo e e - ficien s a egy ha in ol es es ablishing a compa ison a a modula o communi y le el [ 12 ]. In pa icula , using a s anda d algo i hm o communi y de ec ion [ 13 , 14 ], mod- ules can be iden ified om ei he s uc u al and/o unc ional ma ices, and he wo ypes o ne wo ks a e hen compa ed using he same module- ep esen a ion o he wo classes o ne wo ks. Ou hypo hesis is ha i we assume ha seg ega ed unc ions a e associa ed wi h dis inc s uc u al modules, i- sualizing he unc ional modules in e ms o he s uc u al ones should help define and highligh how s ongly s uc u al cons ain s affec unc ion, and ice e sa. In his s udy, we assessed how SC cons ain s affec DFC a he module le el, and we ound ha DFC ce ebella mod- ules we e much less a iable han hose in he ce eb um. We show ha he small a iabili y ound is d i en by wo diffe - en mechanisms, one media ed by he cons ain imposed by he SC and he o he , possibly ela ed o ex e nal influences ha affec unc ion. The small a iabili y in he ce ebella DFC ne wo ks de ec ed in his s udy migh eflec ha diffe - en compu a ional p inciples a e ac i e in he ce ebellum as opposed o o he b ain ci cui s. Ma e ials and me hods Pa icipan s A popula ion o heal hy subjec s ( = 48 ) we e ec ui ed om he gene al popula ion in he icini y o Leu en and Hassel (Belgium) h ough ad e isemen s on websi es, an- nouncemen s a mee ings and he use o flye s a isi s o o ganiza ions and public ga he ings (PI: S ephan Swinnen). The pa icipan ’s ages anged be ween 20 and 51 yea s (mean 33.9 and s anda d de ia ion 9.79), and none o he pa icipan s had a his o y o oph halmological, neu ological, psychia ic o ca dio ascula diseases ha could po en ially influence he imaging o clinical measu es. All he pa icipan s p o ided hei in o med consen be o e en ollmen on he s udy, in ag eemen wi h he local e hics commi ee o biomedical esea ch. Image acquisi ion Ana omical da a: A high- esolu ion T1 image was ac- qui ed wi h a 3D magne iza ion p epa ed apid acquisi ion g adien echo (MPRAGE): epe i ion ime (TR) = 2,300 ms, echo ime (TE) = 2.98 ms, oxel size = 1×1×1.1 mm 3 , slice hickness = 1.1 mm, field o iew (FOV) = 256 × 240 mm 2 , 160 con iguous sagi al slices co e ing he en i e b ain and b ains em. Di usion weigh ed imaging (DWI): A DWI SE-EPI (di - usion weigh ed single sho spin-echo echo-plana imaging [EPI]) sequence was acqui ed wi h he ollowing pa ame e s: TR = 8,000 ms, TE = 91 ms, oxel size = 2.2×2.2×2.2 mm 3 , slice hickness = 2.2 mm, FOV = 212 × 212 mm 2 , 60 con- iguous sagi al slices co e ing he en i e b ain and b ains em. A diffusion g adien was applied along 64 non-collinea di- ec ions wi h a 𝑏 alue o 1,000 s/mm 2 . Addi ionally, one se o images was acqui ed wi hou diffusion weigh ing (𝑏= 0 s/mm2). Res ing s a e unc ional da a: Acqui ed wi h a g adien EPI sequence o e a 10 min session using he ollowing pa- ame e s: 200 whole-b ain olumes wi h TR/TE = 3,000∕30 ms, flip angle = 90◦ , in e -slice gap = 0.28 mm, oxel size = 2.5 × 3 × 2.5 mm 3 , 80 × 80 ma ix, slice hickness = 2.8 mm, 50 oblique axial slices, in e lea ed in descending o - de .Du ing es ing s a e acquisi ion, all pa icipan s ecei ed he ins uc ions o keep hei eyes open and o no hink o any hing in pa icula . Image p ep ocessing Di usion images: We applied a p e-p ocessing pipeline simila o ha employed p e iously [ 15 – 22 ] using he FSL (FMRIB so wa e Lib a y 5.0) and he Diffusion Toolki . Fi s , an eddy cu en co ec ion was applied o o e come he a i ac s p oduced by a ia ion in he di ec ion o he g adien fields in he MR scanne , oge he wi h he a i ac s p oduced by head mo emen s. Specifically, he pa icipan ’s head mo emen was ex ac ed om he ans o ma ion ap- plied a he s ep o eddy cu en co ec ion. The mo emen in o ma ion was also used o co ec he g adien di ec ions p io o enso es ima ion. F om he co ec ed da a, a local fi ing o he diffusion enso pe oxel was ob ained using he d i i ool inco po a ed in o FSL and finally, fibe assignmen was achie ed wi h a con inuous acking algo i hm [ 23 ]. We hen compu ed he ans o ma ion om he Mon eal Neu- ological Ins i u e (MNI) space o he indi idual-pa icipan diffusion space and chose he ne wo k nodes o calcula e he SC using a unc ional pa i ion (see below). Func ional images: We applied a p e-p ocessing pipeline simila o p e ious wo k [ 15 – 18 , 24 – 26 ] using FSL AFNI (h p://a ni.nimh.nih.go /a ni/). A slice- ime co ec ion was fi s applied o he MRI da a se and hen, he each olume was aligned o he middle olume o co ec o head mo e- men a i ac s. A e in ensi y no maliza ion, we eg essed ou he mo emen ime cou ses, he a e age ce eb ospinal Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 2 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks fluid (CSF) signal, he a e age whi e ma e signal and he global signal 1 . A bandpass fil e was hen applied be ween 0.01 and 0.08 Hz [ 27 ], and he p ep ocessed unc ional da a we e spa ially no malized o he MNI152 b ain empla e, wi h an iso opic 2 oxel size o 3 mm. All he oxels we e hen spa ially smoo hed wi h a 6 mm ull wid h a hal maxi- mum iso opic Gaussian ke nel. Finally, and in addi ion o head mo emen co ec ion, we pe o med sc ubbing, h ough which ime poin s wi h a ame-wise displacemen >0.5 we e in e pola ed wi h a cubic spline [ 28 ]. We finally emo ed he effec o head mo emen using he global ame displacemen as a co a ia e. Calcula ion o SC and DFC Bo h he SC and DFC we e buil using = 2,514 nodes, iden ified a e unning an unsupe ised me hod o clus e he unc ional da a [ 29 ]. On a e age, each clus e –which in his s udy coincides wi h one node in he ne wo k– con ained abou 20 oxels o 3 × 3 × 3 mm 3 . One SC ma ix o 2,514 × 2,514 dimensions was ob ained o each subjec by coun ing he numbe o whi e ma e s eamlines connec ing all possi- ble 2,514 × 2,514 pai s o nodes. Thus, he elemen ma ix (𝑖, 𝑗) o SC is gi en by he s eamline numbe be ween nodes 𝑖 and 𝑗 , wi h 𝑖, 𝑗 = 1,…, . Gi en he lack o di ec ionali y o he s eamlines he SC is a symme ic ma ix. To calcula e he popula ion SC ma ix, deno ed by pSC, we fi s bina ized he indi idual SC ma ices and hen ook he o e all a e age o he pa icipan s. Wi h espec o he unc ional ne wo ks, a e a e aging all he oxel ime se ies wi hin each ne wo k node, we ex- ac ed a single ime se ies o each o he 2,514 nodes. By di iding he o al ime se ies leng h  in  non-o e lapping windows o leng h 𝛿 , we ob ained × ma ices, whe e  is he numbe o pa icipan s and  he numbe o ime windows. DFC ≡{FC𝑤} 𝑤=1 was defined as he empo al sequence o squa ed ma ices FC 𝑤 , each one o dimension × and calcula ed o e a fixed window 𝑤 by assessing he pai wise Pea son co ela ion coefficien be ween all-node ime se ies wi hin he ime window 𝑤 . The popula ion DFCs, deno ed by pDFC, we e buil by a e aging each one o all he pa icipan s. No e ha he DFC is a enso , al hough we can also e e o he wo objec s DFC and FC 𝑤 a he module le el, simply ex ac ing om hem he wi hin he module con ibu ions, which we will deno e as DFC 𝑚 and FC 𝑚 𝑤 , espec i ely. The o me is ano he enso composed o a sequence o ma ices o dimensions 𝑚×𝑚 and he la e is a squa ed ma ix o dimension 𝑚×𝑚 . Fo bo h cases, ∑𝑀 𝑚=1 𝑁𝑚=𝑁. 1We also epea ed ou analyses wi hou global signal eg ession. 2 No ice ha al hough he o iginal unc ional oxel was no iso opic, howe e , he p ocessed images a e ans o med in o he MNI152 empla e, whe e oxels a e now iso opic wi h a size o 3 mm. Adap ing high-pass il e ing o e y sho ime windows Con en ional FC app oaches ypically wo k wi h long- ime se ies. Howe e , o ela i ely sho ime windows, he calcula ion o he FC 𝑤 ma ix equi es adap ing he lowes bound (LB) used o band-pass fil e ing o he ime se ies using he equa ion LB−1 =𝛿× TR , whe e 𝛿 is he window leng h [ 30 ]. Fo example, o 𝛿= 8 and TR=3 seconds (as used he e), he high-pass fil e has o be adap ed by aking LB = 0.042 a he han 0.01, as used he e in he pipeline o p e-p ocess he unc ional da a. S uc u al modules used as a empla e o eo de ing DFC Maximizing he modula i y o he pSC ma ix by employ- ing he algo i hm desc ibed in [ 31 ], we ob ained a subdi ision o he s uc u al ne wo k in o  non-o e lapping modules ha maximizes he numbe o wi hin-module connec i i y whils minimizing ha be ween modules [ 14 ]. We nex used his pa i ion o eo de he elemen s o he unc ional ma i- ces, and o assess he link- o-link —o pai wise— simila i y be ween he SC and FC 𝑤 ma ices. This compa ison was epea ed o all he unc ional ma ices ob ained in he diffe - en windows. In his way, we achie ed a common s uc u al o ganiza ion o he modules, which was used as a empla e o eo de all he unc ional ma ices. As ound p e iously [ 12 ], his is a e y con enien s a egy o highligh he simi- la i ies and diffe ences be ween bo h ypes o ne wo ks a a "mesoscopic" le el. Assessmen o SC-DFC simila i y A e eo de ing all he unc ional ne wo ks using he s uc u al modules, he SC-DFC simila i y was assessed a he le el o he modules. As such, we fi s ex ac ed he  squa ed ma ices pSC 𝑚 and pFC 𝑚 𝑤 om he o iginal pSC and pFC 𝑤 ma ices. Fo each module and window 𝑤 we calcula ed he Pea son co ela ion as a simila i y measu e: m 𝑤=𝜌( pSCm, pFCm 𝑤) , whe e  pSCm and  pFCm 𝑤 ep esen he ec o -wise ep esen a ion o ma ices pSC 𝑚 and pFC 𝑚 𝑤 , e- spec i ely. Finally, we a e aged he simila i y o e windows o he same size, i.e.: m=< m 𝑤>𝑤. Assessmen o DFC a iabili y Fo a gi en window leng h, we ob ained a se ies o con- secu i e pFC 𝑚 𝑤 ma ices using he  s uc u al modules. Fo each o he 𝑚= 1,…, modules and 𝑤= 1,…, windows, we assessed he a iabili y o e he diffe en ime windows by calcula ing hei pai wise spec al dis ance [ 32 ]: Δ𝑚 𝑤,𝑤′= 𝑚 ∑ 𝑢=1 | | |𝜆𝑢(pFC𝑚 𝑤) − 𝜆𝑢(pFC𝑚 𝑤′)| | |,(1) whe e 𝑤, 𝑤′= 1,…, a e wo gene ic windows, and 𝜆1(𝐺)≤ 𝜆2(𝐺)≤𝜆3(𝐺)≤... ≤𝜆𝑁𝑚(𝐺) o he wo g aphs 𝐺= {pFC𝑚 𝑤,pFC𝑚 𝑤′}a e he se s o eigen alues. Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 3 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks No e ha o he me ics can be used o assess he DFC a ia ions, such as he empo al a iance o he connec i - i y ma ix ac oss he ime windows [ 33 , 34 ] o i s empo al s anda d de ia ion [ 35 , 36 ]. He e, ou choice elies on he spec al p ope ies o he g aphs, namely ha he eigen alues o he wo isomo phic g aphs a e iden ical. Thus, Eq. (1) is one possible way o quan i y he isomo phic-sepa abili y in pai s o g aphs. Labelling o he ana omical egions The ana omical iden ifica ion o each module (c . Tables 1and S1) was achie ed by calcula ing he pe cen age o e lap be ween each module and each egion in he bi-la e alized au oma ed ana omical labeling (AAL) b ain a las [ 37 ], pool- ing he le and igh sides o each ana omical s uc u e in he o iginal a las in o he same egion. Only egions wi h an o e lap abo e 5% we e epo ed. Ce ebella p ojec ions owa ds es ing s a e ne wo ks To u he unde s and he unc ional oles o he ce e- bella modules, we p ojec ed hem on o a highly specialized ce ebella a las [ 38 , 39 ], con aining in o ma ion abou he p ojec ions om each ce ebella egion in he a las owa ds each es ing-s a e ne wo k. We finally calcula ed he pe - cen age o e lap be ween ou modules and each egion in he ce ebella a las. SC ma ices using p obabilis ic ac og aphy We also calcula ed he SC ma ices using p obabilis ic ac og aphy, as implemen ed in FSL. We fi s es ima ed a p obabilis ic model o compu e he fibe o ien a ion using bed- pos [ 40 ]. We hen calcula ed he connec i i y ma ices (one pe subjec ) using he p ob ackx2 unc ion and 100 pa hways pe oxel. Connec i i y ma ices we e calcula ed by defining a h eshold o he p obabili y o define a non-ze o connec- ion such ha bo h ma ices, he popula ion-de e minis ic one used o all he o he analyses and he popula ion-p obabilis ic one, had app oxima ely he same numbe o non-ze o ele- men s, i.e.: he wo ma ices achie ed he same link-densi y as when 97.6% o all p obabilis ic connec ions we e fixed o ze o. O e lapping ime-windows o he calcula ion o DFC Al hough he esul s p esen ed he e conside ed non-o e lapping ime windows o he calcula ion o DFC, we also assessed DFC using sliding windows wi h an o e lap o 96% be ween hem [41]. Templa es o Res ing S a e Ne wo ks We also calcula ed he wo me ics analyzed he e (DFC a iabili y and SC-DFC simila i y) o classic es ing s a e ne wo ks (RSNs). In pa icula , we made use o he em- pla es de eloped p e iously [ 42 ] o he ou ollowing RSNs: De aul Mode Ne wo k (DMN), Ven al A en ion Ne wo k (VAN), Do sal A en ion Ne wo k (DAN) and Soma o-Mo o Ne wo k (SMN). Synch oniza ion me ics o DFC In addi ion, o assess DFC by calcula ing he pai wise Pea son’s co ela ion be ween node ime se ies, we also es- ablished synch oniza ion me ics. These we e calcula ed by fi s ob aining he Hilbe ans o ma ion o he node ime se- ies 𝑥(𝑡) ep esen ed by 𝑥(𝑡)≡ {𝑥}(𝑡) o build he complex analy ical signal as 𝑥(𝑡)+𝑖 𝑥(𝑡) , ep esen ed as 𝐴(𝑡) exp (𝑖𝜃(𝑡)) , whe e 𝐴(𝑡) and 𝜃(𝑡) ep esen he ins an aneous ampli ude and he ins an aneous phase o he analy ical signal, espec i ely. We hen ob ained he FC 𝑤 ma ices in wo mo e diffe en o ms: (1) calcula ing he pai wise Pea son’s co ela ion be- ween he ime se ies o he ins an aneous ampli ude along diffe en ime windows; and (2) by calcula ing he phase lock- ing alue (PLV) [ 43 – 46 ], defined o any pai o nodes i and j as: PLV𝑤(𝑖, 𝑗) = 1 𝛿| | | | |∑ 𝑡∈𝑤 𝑒𝑖(𝜃𝑖(𝑡)−𝜃𝑗(𝑡))| | | | | ,(2) whe e he a e age is aken in a ime window o leng h , and |⋅| indica es he modulus o a complex numbe . The PLV akes alues in he in e al [0,1], wi h 0 co esponding o he case whe e he e is no phase synch ony and 1 whene e he wo phases o he wo signals a e always iden ical. Resul s A popula ion o young heal hy pa icipan s (  = 48) was s udied he e, acqui ing diffusion and es ing-s a e images o each pa icipan (see pipeline in figu e 1). We fi s di ided he popula ion SC ma ix, ep esen ed by he pSC, h ough mod- ula i y maximiza ion, esul ing in = 14 non-o e lapping modules (see figu e 2) wi h a modula i y index o 0.7324. Modules 6 and 13 had 2 and 1 egions, espec i ely, and he e o e, nei he o hese modules we e conside ed in he ollowing quan i a i e analyses. A e calcula ing one pFC 𝑤 ma ix pe ime window 𝑤 , we eo de ed all o hem acco ding o he s uc u al modules and calcula ed he link- o-link co ela ion o all he modules sepa a ely, m 𝑤 , as a measu e o he simila i y be ween he pSC 𝑚 and he diffe en pFC 𝑚 𝑤 (figu e 2). In pa icula , we fi s conside ed 𝛿= 8 , equi alen o a ime du a ion o 24 seconds, and a e aged he measu emen ac oss all he ime windows. Mo eo e , we s udied he a iabili y in pFC 𝑤 ac oss he ime windows, in his case measu ing he pai wise spec al dis ance Δ𝑚 𝑤,𝑤′ wi hin each module 𝑚 o he ime windows 𝑤 and 𝑤′ , and a e aging his o e all pai s (𝑤, 𝑤′) . The anal- ysis o he SC-DFC simila i y ac oss he diffe en modules e ealed he exis ence o an ou lie , module 1, which had a simila i y alue o 𝑟𝑚>0.7 , con as ing wi h he es o he modules whose simila i y was less han 0.55 (figu e 3). The ana omy o module 1 is shown in Table 1and s ikingly, i is mainly o med by pos e io ce ebella s uc u es. By con- as , he lowes SC-DFC simila i y was epo ed o module 14, wi h 𝑟𝑚<0.4 o all ime-windows. Impo an ly, his Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 4 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks module is o med by a diffe en pa o he ce ebellum (i s an e io pa : see Table 1), oge he wi h o he non-ce ebella s uc u es like he b ains em, he usi o m nucleus and pa o he lingual co ex. In addi ion o he SC-DFC simila i y, we quan ified he amoun o DFC a iabili y and ound ha module 11 exhib- i ed he highes a iabili y o e ime (Table 1), a module composed o se e al co ical s uc u es ha include he cal- ca ine, middle and in e io empo al, lingual and p ecuneus. By assessing bo h me ics simul aneously, 𝑚 and Δ𝑚 , modules 1 and 14 had conside ably smalle Δ𝑚 alues han module 11 (figu e 3A), indica ing ha he ce ebella s uc- u es displayed much less DFC a iabili y o e ime han o he s uc u es in he ce eb um (figu e 3A shows he mean alues ac oss windows and he his og ams o all he possible alues a e shown in figu e 3B). Se e al b ain slices om hese h ee modules a e shown in Figu e 3C and he ana omical compo- si ion o he es o he modules analyzed is gi en in Table S1. To u he demons a e he obus ness o ou findings, ha ce ebella modules 1 and 14 ulfil a diffe en ia ed ole in e ms o DFC a iabili y and SC-DFC simila i y, we pe - o med se e al con ol-analyses unde diffe en condi ions ha included a ying some image-p ep ocessing s eps, o using diffe en pa ame e s in ou modeliza ion o diffe en me ics o calcula e he DFC (see Me hods o de ails). Fi s , we epea ed he same analysis bu conside ing p obabilis ic a he han de e minis ic ac og aphy (figu e S1A). Second, we pe o med a sliding window analysis o calcula e he DFC using 96% o e lapping windows a he han non-o e lapping windows (figu e S1B). Thi d, we p ep ocessed all he unc- ional images wi hou global signal eg ession and epea ed he same analyses (figu e S1C). Finally, we calcula ed di - e en me ics o assessing DFC a iabili y bu based on he analy ical signal, which is he e o e mo e closely ela ed o s anda d synch oniza ion s udies. In pa icula , we assessed he DFC by calcula ing he pai wise co ela ions be ween ime se ies o ins an aneous ampli ude (figu e S1D) and in- s an aneous phase (figu e S1E), ha is, he PLV. Indeed, a simila equi alence be ween diffe en ways o cons uc unc- ional ma ices has been also epo ed elsewhe e [ 47 ]. In all hese si ua ions, bo h ce ebella modules 1 and 14 p ese ed hei diffe en ial ole ela i e o he o he modules in he ce e- b um. We also asked whe he ou findings ob ained o a win- dow o 𝛿= 8 we e obus when a ying he leng h o he window (figu e 4). Specifically, we ob ained all he mea- su emen s again o he ollowing window leng hs: 𝛿= {4,5,7,8,10,25} , and also o 𝛿= , equi alen o a single ime window wi h a leng h equi alen o he en i e ime se ies. In his la e case, we only add essed he SC-DFC simila i y because DFC a iabili y could no be assessed in a single ime window (see Table S2). O e all, we ound wo ce e- bella modules ha p ese ed hei oles i espec i e o he window leng h chosen and o o he con ol condi ions (figu e S1), modules 1 and 14 in he pos e io and an e io pa o he ce ebellum, espec i ely. By con as , when we looked a module 11 o e diffe en window leng hs, he module wi h he highes DFC a iabili y, we no iced ha his beha io was mo e a iable ac oss ime windows and ha modules 2 and 11 in e changed hei posi ions. We nex asked which b ain a eas in modules 1 and 14 we e s uc u ally and unc ionally connec ed (Table 2), and we ound s ong mu ual-connec i i y be ween he an e io and pos e io pa s o he ce ebellum, as seen p e iously [ 48 , 49 ]. Mo eo e , he an e io egion also p ojec s o he mo o co ex, as has been well es ablished [ 50 – 52 ]. By quan- i ying he o e lap o modules 1 and 14 wi h he ce ebella egions p ojec ing o diffe en RSNs (see Me hods), we ound ha module 1 p ojec ed owa ds he DMN (22.8%) and he on opa ie al ne wo k (19.2%), while module 14 did so o- wa ds he SMN (17.7%). Mo eo e , bo h modules p ojec ed simila ly o he VAN, wi h a 13.3% o e lap o module 1 and 10.5% o module 14. Thus, hese esul s show in a di - e en manne ha bo h ce ebella modules a e diffe en ye complemen a y o each o he , wi h module 1 pa icipa ing in high o de cogni i e ne wo ks and module 14 in soma omo o ne wo ks, while bo h pa icipa e in mul imodal in eg a ion ne wo ks like hose associa ed wi h en al a en ion [53]. Finally, we assessed he SC-DFC simila i y and DFC a iabili y in classical RSNs [ 42 , 54 – 56 ]. In pa icula , we analyzed he DMN, SMN, VAN and DAN ne wo ks (figu e 5), which all had much less SC-DFC simila i y han ou modules. This p obably eflec s he ac ha he RSNs we e buil pu ely om unc ional da a, whe eas we combined unc ional and s uc u al da a oge he o ob ain he final modules. Mo eo e , we also ound ha he DMN had he highes DFC a iabil- i y when compa ed wi h he o he RSNs and also wi h ou modules. Discussion We ha e assessed he e he ela ionship be ween SC and DFC by combining s uc u al and unc ional b ain ne wo ks, and assessing hei modula o ganiza ion. We ha e buil ne - wo ks wi h high spa ial esolu ion, co e ing he en i e b ain, using 2,514 nodes ha each ha e an a e age size o 0.54 cm 3 . Th ough modula i y maximiza ion o he popula ion SC ma ix, we ob ained 14 non-o e lapping s uc u al mod- ules ha we e used o eo de he unc ional ma ices. This was done unde he assump ion ha i seg ega ed unc ions a e associa ed wi h dis inc s uc u al modules, isualizing he unc ional ma ices ollowing hei s uc u al finge p in s would cla i y how SC cons ains unc ion, a undamen al ques ion ha has ye o be esol ed. We analyzed he SC-DFC simila i y in combina ion wi h he amoun o DFC a iabili y o e ime o all he p e iously iden ified modules, and cha ac e ized each module in e ms o hese wo me ics. This allowed us o iden i y h ee ex- Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 5 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks eme cases: 1, a ully co ical module wi h he highes DFC a iabili y; 2, a module in he pos e io ce ebellum ha had he highes SC-DFC simila i y while main aining low DFC a iabili y; 3, a module in he an e io ce ebellum connec ed o he b ains em ha had he lowes SC-DFC simila i y bu also, ha e ained small DFC a iabili y. The e o e, ce ebel- la ne wo ks appea o ha e small DFC a iabili y. Module 1, loca ed a he pos e io ce ebellum, has abou wice he simila i y be ween SC and DFC han he es o he modules, indica ing ha s uc u e cons ains he dynamic connec i i y [57,58]. Howe e , how module 14 associa es weak SC-DFC simi- la i y wi h low DFC a iabili y is mo e challenging o unde - s and. On he one hand, module 14 includes he b ains em in addi ion o he an e io pa o he ce ebellum, he o me ha ing s ong connec i i y o many o he pa s o he b ain and body h ough majo ac s like he co icospinal, lemnis- cus and spino halamic ac s [ 59 ]. Thus, by looking a he in a-module simila i ies be ween SC and DFC, as pe o med he e, i is possible ha we igno ed ele an aspec s o con- nec i i y om his module o he es o he b ain, he eby unde es ima ing he in a-module SC-DFC simila i y. Mo e- o e , i is well known ha he b ains em plays a c i ical ole in egula ing sleep cycles, as well as ca diac and espi a o y unc ion. Pe haps, such c i ical unc ions a e no compa ible wi h la ge DFC a iabili y, as occu s in co ical ne wo ks, al hough u u e esea ch will be needed o ully explain hese obse a ions. Fu u e s udies should also e i y ou esul s on la ge sample sizes. I is impo an o emphasize ha in e ms o SC-DFC simila i y, ou unsupe ised me hod iden ifies wo di isions wi hin he ce ebellum, he pos e io and an e io egions. This u ns ou o be a well-known and s anda d ana omical and unc ional di ision o he ce ebellum in humans and ani- mals [ 48 ]. Mo eo e , while he classical ce ebellum di ision g ouped he an e io lobules om 1 o 5 [ 52 ], ou esul s include lobule 6 in bo h he an e io and pos e io ce ebel- lum, in ag eemen wi h unc ional MRI s udies ha indica ed ce ebella lobules 4-6 pa icipa e in senso imo o asks [ 60 ]. We ound ha module 1, he pos e io ce ebellum, pa - icipa ed in high-o de cogni i e unc ions, as eflec ed by a s ong o e lap wi h he DMN and he on opa ie al ne wo k, in ag eemen wi h p e ious da a [ 61 ]. This is also consis- en wi h he ac ha he ce ebellum a eas C us 1 and 2, included in module 1, connec o he halamus and hen, o p e on al a eas. In ela ion o module 14, he an e io pa o he ce ebellum, we ound p ojec ions owa ds he soma omo- o ne wo k, in ag eemen wi h [ 61 ]. Mo eo e , bo h hese modules p ojec ed o mul imodal in eg a ion hubs like he VAN [53]. In he es ing s a e, he DFC is domina ed by he DMN, oge he wi h a en ional and senso y ne wo ks [ 62 , 63 ]. We show he e ha DMN a iabili y was indeed he highes o all he modules and ne wo ks s udied, which migh sugges why diffe en DFC pa e ns in he DMN can encode mul iple b ain s a es [ 64 ], and also why he DFC is mo e limi ed in se e al pa hological condi ions [ 65 ]. Al hough ou esul s we e ob- ained in he es ing s a e and he e o e, he pa icipan s we e no pe o ming any specific ask in he MRI scanne , hese findings migh also explain ha when a mo o ask is mo e difficul o pe o m o i is simply a new ask o he subjec , pos e io ce ebella ac i a ion occu s oge he wi h p e on al ac i i y o enhance cogni i e moni o ing o he indi idual’s pe o mance [60,66–69]. I is well-known ha ce ebella ne wo ks ha e an ex- emely ich and complex ana omy and unc ionali y [ 70 ], connec ing o he b ains em and ce eb al hemisphe es, and pa icipa ing in a la ge a ie y o cogni i e unc ions like mo emen coo dina ion, bimanual coo dina ion pe o mance [ 71 ], execu i e unc ion, isual-spa ial cogni ion, language p ocessing and emo ional egula ion [ 72 , 73 ]. Howe e , as a as we know he small a iabili y o he DFC wi hin he ce ebellum has no been epo ed p e iously. Finally, he ex ao dina y cons ain o he DFC o he SC in ce ebella module 1 migh indica e dis inc ope a ional and compu a ional p inciples in he ce ebellum. Classically, ce e- bella a chi ec u e has been modeled in a eed o wa d manne , unlike he highly ecu en ci cui s ound in he ce eb al co - ices (see [ 74 ] and e e ences he ein). This phenomenon enables he ce ebellum o linea ly in eg a e diffe en inpu s om o he sys ems in o de o gene a e ou pu s acco ding o p e iously lea ned in o ma ion pa e ns, ollowing eed o - wa d e o -co ec ion compu a ions [ 75 , 76 ]. Pe haps, such compu a ional machine y makes he ce ebellum’s in o ma- ion p ocessing mo e eliable, in acco dance wi h he low a iabili y in i s DFC, al hough u he esea ch is needed o shed ligh on hese findings. Decla a ion o Compe ing In e es The au ho s decla e ha hey ha e no compe ing in e es s. CRediT au ho ship con ibu ion s a emen Iza o Fe nandez-I iondo: Pe o med he analyses, Made he figu es, D a ed he fi s manusc ip , W o e he manusc ip . An onio Jimenez-Ma in: Pe o med he analysis, Made he figu es, W o e he manusc ip . Ibai Diez: P ep ocessed he images, W o e he manusc ip . Paolo Boni azi: W o e he manusc ip . S ephan P. Swinnen: Acqui ed he da a, W o e he manusc ip . Miguel A. Muñoz: Supe ised he esea ch, Equal las -au ho con ibu ion, W o e he manusc ip . Jesus M. Co es: D a ed he fi s manusc ip , Supe ised he e- sea ch, Equal las -au ho con ibu ion, W o e he manusc ip . Acknowledgemen s A.J.M. acknowledges financial suppo om a p edoc o al g an om he Basque Go e nmen (PRE _ 2019 _ 1 _ 0070). J.M.C. and P.B. acknowledge finan- cial suppo om Ike basque (The Basque Founda ion o Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 6 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Science) and om he Minis e io Economia, Indus ia y Com- pe i i idad (Spain) and FEDER (g an DPI2016-79874-R o J.M.C., g an SAF2015-69484-R o P.B.). J.M.C. acknowl- edges financial suppo om he Depa men o Economi- cal De elopmen and In as uc u e o he Basque Coun y (Elka ek P og am, KK-2018/00032 and KK-2018/00090). S.P.S was suppo ed by he he FWO Resea ch Founda ion Flande s (G089818N), he Excellence o Science unding compe i ion (EOS; 30446199) and he KU Leu en Special Resea ch Fund (g an C16/15/070). M.A.M acknowledges financial suppo om he Spanish Minis y and Agencia Es- a al de in es igación (AEI) h ough g an FIS2017-84256-P (Eu opean Regional De elopmen Fund), as well as he Con- seje ía de Conocimien o, In es igación y Uni e sidad, Jun a de Andalucía and Eu opean Regional De elopmen Fund (ERDF), e . SOMM17/6105/UGR o financial suppo . Re e ences [1] Alex Fo ni o, And ew Zalesky, and Edwa d Bullmo e. Fundamen als o B ain Ne wo k Analysis. Academic P ess, 2016. [2] RC C addock, S Jbabdi, CG Yan, JT Vogels ein, FX Cas ellanos, A Di Ma ino, C Kelly, K Hebe lein, S Colcombe, and MP Milham. Imaging human connec omes a he mac oscale. Na Me hods, 10:524–539, June 2013. [3] Ma ia Giulia P e i, Thomas AW Bol on, and Dimi i Van De Ville. The dynamic unc ional connec ome: S a e-o - he-a and pe spec i es. Neu oImage, 160:41–54, Oc obe 2017. [4] H.-J. Pa k and K. F is on. S uc u al and Func ional B ain Ne - wo ks: F om Connec ions o Cogni ion. Science, 342(6158):1238411– 1238411, No embe 2013. [5] Ma ijn P. an den Heu el and Hilleke E. Hulshoff Pol. Explo ing he b ain ne wo k: a e iew on es ing-s a e MRI unc ional connec i i y. Eu opean Neu opsychopha macology: The Jou nal o he Eu opean College o Neu opsychopha macology, 20(8), Augus 2010. [6] S A asoy abd I Donnelly and JPea son. Human b ain ne wo ks unc ion in connec ome-specific ha monic wa es. Na Comm, 7:10340, 2016. [7] G. Deco, A. Ponce-Al a ez, D. Man ini, G. L. Romani, P. Hagmann, and M. Co be a. Res ing-S a e Func ional Connec i i y Eme ges om S uc u ally and Dynamically Shaped Slow Linea Fluc ua ions. Jou nal o Neu oscience, 33(27):11239–11252, July 2013. [8] Ad ián Ponce-Al a ez, Gus a o Deco, Pa ic Hagmann, Gian Luca Romani, Dan e Man ini, and Mau izio Co be a. Res ing-S a e Tem- po al Synch oniza ion Ne wo ks Eme ge om Connec i i y Topology and He e ogenei y. PLOS Compu a ional Biology, 11(2):e1004100, Feb ua y 2015. [9] Joana Cab al, Mo en L. K ingelbach, and Gus a o Deco. Func ional connec i i y dynamically e ol es on mul iple ime-scales o e a s a ic s uc u al connec ome: Models and mechanisms. Neu oImage, 160:84– 96, Oc obe 2017. [10] Mu a Demi aş, C is ian To nado , Ca les Falcón, Ma ina López-Solà, Rosa He nández-Ribas, Jesús Pujol, José M. Menchón, Pe a Ri e , Na cis Ca done , Ca les So iano-Mas, and Gus a o Deco. Dynamic unc ional connec i i y e eals al e ed a iabili y in unc ional con- nec i i y among pa ien s wi h majo dep essi e diso de : Dynamic Func ional Connec i i y in Majo Dep ession. Human B ain Mapping, 37(8):2918–2930, Augus 2016. [11] Mu a Demi aş, Ca les Falcon, Alan Tucholka, Juan Domingo Gispe , José Luis Molinue o, and Gus a o Deco. A whole-b ain compu a ional modeling app oach o explain he al e a ions in es ing-s a e unc ional connec i i y du ing p og ession o Alzheime ’s disease. Neu oImage: Clinical, 16:343–354, Janua y 2017. [12] Ibai Diez, Paolo Boni azi, Iñaki Escude o, Bea iz Ma eos, Miguel A. Muñoz, Sebas iano S amaglia, and Jesus M. Co es. A no el b ain pa - i ion highligh s he modula skele on sha ed by s uc u e and unc ion. a Xi :1410.7959 [q-bio], pages 1–2, Ap il 2015. a Xi : 1410.7959. [13] M. E. J. Newman. Spec al me hods o communi y de ec ion and g aph pa i ioning. Physical Re iew E, 88(4):042822, Oc obe 2013. [14] M. E. J. Newman. Modula i y and communi y s uc u e in ne wo ks. P oceedings o he Na ional Academy o Sciences, 103(23):8577–8582, June 2006. [15] Paolo Boni azi, Asie E amuzpe, Ibai Diez, Iñigo Gabilondo, Ma hieu P. Boisgon ie , Lisa Pauwels, Sebas iano S amaglia, S ephan P. Swinnen, and Jesus M. Co es. S uc u e- unc ion mul i- scale connec omics e eals a majo ole o he on o-s ia o- halamic ci cui in b ain aging. Human B ain Mapping, 39(12):4663–4677, Decembe 2018. [16] Ca men Alonso-Mon es, Ibai Diez, Lakhda Remaki, Iñaki Escud- e o, Bea iz Ma eos, Y es Rosseel, Daniele Ma inazzo, Sebas iano S amaglia, and Jesus M. Co es. Lagged and ins an aneous dynam- ical influences ela ed o b ain s uc u al connec i i y. F on ie s in Psychology, 6:5–9, July 2015. [17] T. A. Amo , R. Russo, I. Diez, P. Bha a h, M. Zi o ich, S. S amaglia, J. M. Co es, L. de A cangelis, and D. R. Chial o. Ex eme b ain e en s: Highe -o de s a is ics o b ain es ing ac i i y and i s ela ion wi h s uc u al connec i i y. EPL (Eu ophysics Le e s), 111(6):68007, Sep embe 2015. [18] Ibai Diez, Da id D ijkoningen, Sebas iano S amaglia, Paolo Boni azi, Daniele Ma inazzo, Jolien Gooije s, S ephan P. Swinnen, and Jesus M. Co es. Enhanced p e on al unc ional–s uc u al ne wo ks o suppo pos u al con ol defici s a e auma ic b ain inju y in a pedia ic popula ion. Ne wo k Neu oscience, 1(2):116–142, June 2017. [19] Julia M. K oos, Isabella Ma inelli, Ibai Diez, Jesus M. Co es, Sebas- iano S amaglia, and Luca Ge a do-Gio da. Pa ien -specific compu a- ional modeling o co ical sp eading dep ession ia diffusion enso imaging. In e na ional Jou nal o Nume ical Me hods in Biomedical Enginee ing, 33(11):e2874, No embe 2017. [20] Daniele Ma inazzo, Ma io Pellico o, Guo ong Wu, Leona do Angelini, Jesús M. Co és, and Sebas iano S amaglia. In o ma ion T ans e and C i icali y in he Ising Model on he Human Connec ome. PLoS ONE, 9(4):e93616, Ap il 2014. [21] Ja ie Rase o, Ma io Pellico o, Leona do Angelini, Jesus M. Co es, Daniele Ma inazzo, and Sebas iano S amaglia. Consensus clus e ing app oach o g oup b ain connec i i y ma ices. Ne wo k Neu oscience, 1(3):242–253, Oc obe 2017. [22] S. S amaglia, M. Pellico o, L. Angelini, E. Amico, H. Ae s, J. M. Co és, S. Lau eys, and D. Ma inazzo. Ising model wi h conse ed magne iza ion on he human connec ome: Implica ions on he ela- ion s uc u e- unc ion in wake ulness and anes hesia. Chaos: An In e disciplina y Jou nal o Nonlinea Science, 27(4):047407, Ap il 2017. [23] Susumu Mo i, Ba ba a C ain, V.P. Chacko, and Pe e an zijl. Mo i S, C ain BJ, Chacko VP, an Zijl PCM. Th ee dimensional acking o axonal p ojec ions in he b ain by magne ic esonance imaging. Annal Neu ol 45: 265-269. Annals o neu ology, 45:265–9, Ma ch 1999. [24] Ibai Diez, Asie E amuzpe, Iñaki Escude o, Bea iz Ma eos, Albe o Cab e a, Daniele Ma inazzo, E nes o J. Sanz-A igi a, Sebas iano S a- maglia, Jesus M. Co es Diaz, and o he Alzheime ’s Disease Neu- oimaging Ini ia i e. In o ma ion Flow Be ween Res ing-S a e Ne - wo ks. B ain Connec i i y, 5(9):554–564, No embe 2015. [25] Ve ónica Mäki-Ma unen, Ibai Diez, Jesus M. Co es, Dan e R. Chial o, and Mi a Villa eal. Dis up ion o ans e en opy and in e -hemisphe ic b ain unc ional connec i i y in pa ien s wi h diso - de o consciousness. F on ie s in Neu oin o ma ics, 7:5–10, 2013. [26] Sebas iano S amaglia, Leona do Angelini, Guo ong Wu, Jesus M. Co és, Luca Faes, and Daniele Ma inazzo. Syne ge ic and edun- dan in o ma ion flow de ec ed by unno malized g ange causali y: applica ion o es ing s a e m i. IEEE T ansac ions on Biomedical Enginee ing, 63(12):2518–2524, Decembe 2016. [27] D. Co des, V. M. Haugh on, K. A anakis, J. D. Ca ew, P. A. Tu ski, C. H. Mo i z, M. A. Quigley, and M. E. Meye and. F equencies con ibu ing o unc ional connec i i y in he ce eb al co ex in " es ing- s a e" da a. AJNR. Ame ican jou nal o neu o adiology, 22(7):1326– Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 7 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks 1333, Augus 2001. [28] Chao-Gan Yan, B ian Cheung, Cla e Kelly, S an Colcombe, R. Came on C addock, Ad iana Di Ma ino, Qingyang Li, Xi-Nian Zuo, F. Xa ie Cas ellanos, and Michael P. Milham. A comp ehensi e assessmen o egional a ia ion in he impac o head mic omo e- men s on unc ional connec omics. Neu oImage, 76:183–201, Augus 2013. [29] R. Came on C addock, G.And ew James, Paul E. Hol zheime , Xiaop- ing P. Hu, and Helen S. Maybe g. A whole b ain MRI a las gene a ed ia spa ially cons ained spec al clus e ing. Human B ain Mapping, 33(8):1914–1928, Augus 2012. [30] Ibai Diez and Jo ge Sepulc e. Neu ogene ic p ofiles delinea e la ge- scale connec i i y dynamics o he human b ain. Na u e Communica- ions, 9(1):1–10, Sep embe 2018. [31] M. E. J. Newman. Finding communi y s uc u e in ne wo ks using he eigen ec o s o ma ices. Physical Re iew E, 74(3), Sep embe 2006. a Xi : physics/0605087. [32] I ena Jo ano ić and Zo an S anić. Spec al dis ances o g aphs. Linea Algeb a and i s Applica ions, 436(5):1425–1435, Ma ch 2012. [33] Changchun He, Yanchi Chen, Tao ong Jian, Heng Chen, Xiaonan Guo, Jia Wang, Lijie Wu, Hua u Chen, and Xujun Duan. Dynamic unc ional connec i i y analysis e eals dec eased a iabili y o he de aul -mode ne wo k in de eloping au is ic b ain. Au ism Res, 11:1479–1493, 2018. [34] Wei Liao, Jiao Li, Gong-Jun Ji, Guo-Rong Wu, Zhiliang Long, Qiang Xu, Xujun Duan, Qian Cui, Bha a B Biswal, and Hua u Chen. Endless fluc ua ions: Tempo al dynamics o he ampli ude o low equency fluc ua ions. IEEE T ans Med Imaging, 38:2523–2532, 2019. [35] M Falahpou , WK Thompson, AE Abbo , A Jahedi, ME Mul ey, M Da ko, TT Liu, and RA Mülle . Unde connec ed, bu no b o- ken? dynamic unc ional connec i i y m i shows unde connec i i y in au ism is linked o inc eased in a-indi idual a iabili y ac oss ime. B ain Connec , 6:403–414, 2016. [36] H Chen, JS Nomi, LQ Uddin, X Duan, and H Chen. In insic unc ional connec i i y a iance and s a e-specific unde -connec i i y in au ism. Hum B ain Mapp, 38:5740–5755, 2017. [37] N. Tzou io-Mazoye , B. Landeau, D. Papa hanassiou, F. C i ello, O. E a d, N. Delc oix, B. Mazoye , and M. Jolio . Au oma ed Ana om- ical Labeling o Ac i a ions in SPM Using a Mac oscopic Ana omical Pa cella ion o he MNI MRI Single-Subjec B ain. Neu oImage, 15(1):273–289, Janua y 2002. [38] Jö n Died ichsen and Ewa Zo ow. Su ace-based display o olume- a e aged ce ebella imaging da a. PLOS ONE, 10(7):e0133402, 2015. [39] Jö n Died ichsen, Joshua H. Bals e s, Jona han Fla ell, Emma Cus- sans, and Na ende Ramnani. A p obabilis ic MR a las o he human ce ebellum. Neu oImage, 46(1):39–46, 2015. [40] TEJ Beh ens, HJ Be g, S Jbabdi, MFS Rushwo h, and MW Wool ich. P obabilis ic Diffusion T ac og aphy wi h Mul iple Fib e O ien a ions: Wha Can We Gain? Neu oimage, 34:144–155, 2007. [41] En ique C. A. Hansen, Demian Ba aglia, And eas Spiegle , Gus- a o Deco, and Vik o K. Ji sa. Func ional connec i i y dynamics: modeling he swi ching beha io o he es ing s a e. Neu oImage, 105:525–535, Janua y 2015. [42] BTT Yeo, FM K ienen, J Sepulc e, MR Sabuncu, D Lashka i, M Hollinshead, JL Roffman, JW Smolle , L Zollei, JR Polimeni, B Fis- chl, H Liu, and RL Buckne . The o ganiza ion o he human ce eb al co ex es ima ed by in insic unc ional connec i i y. J Neu ophysiol, 106:1125–1165, 2011. [43] Joana Cab al, Diego Vidau e, Paulo Ma ques, Rica do Magalhães, Ped o Sil a Mo ei a, José Miguel Soa es, Gus a o Deco, Nuno Sousa, and Mo en L. K ingelbach. Cogni i e pe o mance in heal hy olde adul s ela es o spon aneous swi ching be ween s a es o unc ional connec i i y du ing es . Scien i ic Repo s, 7(1):5135, July 2017. Numbe : 1 Publishe : Na u e Publishing G oup. [44] Se gul Aydo e, Dimi ios Pan azis, and Richa d M. Leahy. A no e on he phase locking alue and i s p ope ies. Neu oImage, 74:231–244, July 2013. [45] Alex A enas, Albe Díaz-Guile a, Ju gen Ku hs, Yami Mo eno, and Changsong Zhou. Synch oniza ion in complex ne wo ks. Physics Repo s, 469(3):93–153, Decembe 2008. [46] A kady Piko sky, M.G. Rosenblum, and Jue gen Ku hs. Synch oniza- ion: A Uni e sal Concep In Nonlinea Sciences. Ame ican Jou nal o Physics, 12, June 2002. [47] Mango Pede sen, Ami Omid a nia, And ew Zalesky, and G aeme D. Jackson. On he ela ionship be ween ins an aneous phase synch ony and co ela ion-based sliding windows o ime- esol ed MRI con- nec i i y analysis. Neu oImage, 181:85 – 94, 2018. [48] O La sell and J Jansen. The Compa a i e Ana omy and His ology o he Ce ebellum: : The Human Ce ebellum, Ce ebella Connec ions, and Ce ebella Co ex. The Uni e si y o Minneso a P ess, 1972. [49] RL Buckne , FM K ienen, A Cas ellanos, JC Diaz, and BTT Yeo. The o ganiza ion o he human ce ebellum es ima ed by in insic unc ional connec i i y. J Neu ophysiol, 106:2322–2345, 2011. [50] F Debae e, N Wende o h, S Sunae , P Van Hecke, and SP Swinnen. Ce ebella and p emo o unc ion in bimanual coo dina ion: pa ame ic neu al esponses o spa io empo al complexi y and cycling equency. Neu oimage, 21:1416–1427, 2004. [51] Da id D ijkoningen, Inge Leunissen, Ka en Caeyenbe ghs, Wou e Hoogkame , S e an Sunae , Jacques Duysens, and S ephan P. Swinnen. Regional olumes in b ain s em and ce ebellum a e associa ed wi h pos u al impai men s in young b ain-inju ed pa ien s. Human B ain Mapping, 36(12):4897–4909, Decembe 2015. [52] CJ S oodley and JD Schmahmann. E idence o opog aphic o ganiza- ion in he ce ebellum o mo o con ol e sus cogni i e and affec i e p ocessing. Co ex J De o ed S udy Ne Sys Beha , 46:831–844, 2010. [53] Jo ge Sepulc e, Me R. Sabuncu, Thomas B. Yeo, Hesheng Liu, and Kei h A. Johnson. S epwise Connec i i y o he Modal Co ex Re- eals he Mul imodal O ganiza ion o he Human B ain. Jou nal o Neu oscience, 32(31):10649–10661, Augus 2012. Publishe : Socie y o Neu oscience Sec ion: A icles. [54] M. E. Raichle, A. M. MacLeod, A. Z. Snyde , W. J. Powe s, D. A. Gusna d, and G. L. Shulman. A de aul mode o b ain unc ion. P o- ceedings o he Na ional Academy o Sciences, 98(2):676–682, Janua y 2001. [55] Michael D. Fox, Ab aham Z. Snyde , Jus in L. Vincen , Mau izio Co be a, Da id C. Van Essen, and Ma cus E. Raichle. The human b ain is in insically o ganized in o dynamic, an ico ela ed unc ional ne wo ks. P oceedings o he Na ional Academy o Sciences o he Uni ed S a es o Ame ica, 102(27):9673–9678, July 2005. [56] CF Beckmann, M DeLuca, JT De lin, and SM Smi h. In es iga ions In o Res ing-S a e Connec i i y Using Independen Componen Anal- ysis. Philos T ans R Soc Lond B Biol Sci, 360:1001–1013, 2005. [57] Maedbh King, Ca los R. He nandez-Cas illo, Russell A. Pold ack, Richa d B. I y, and Jö n Died ichsen. Func ional bounda ies in he human ce ebellum e ealed by a mul i-domain ask ba e y. Na u e Neu oscience, 22(8):1371–1378, Augus 2019. [58] Jö n Died ichsen, Maedbh King, Ca los He nandez-Cas illo, Ma y Se eno, and Richa d B. I y. Uni e sal T ans o m o Mul iple Func- ionali y? Unde s anding he Con ibu ion o he Human Ce ebellum ac oss Task Domains. Neu on, 102(5):918–928, June 2019. [59] SG Waxman. Clinical Neu oana omy 27/E. McG aw-Hill Educa ion - Eu ope, 2013. [60] CJ S oodley, EM Vale a, and JD Schmahmanna. Func ional opog a- phy o he ce ebellum o mo o and cogni i e asks: an MRI s udy. Neu oimage, 59:1560–1570, 2012. [61] Ch is ophe Habas, Ni a Kamda , Daniel Nguyen, Ka he ine P a e , Ch is ian F. Beckmann, Vinod Menon, and Michael D. G eicius. Dis- inc ce ebella con ibu ions o in insic connec i i y ne wo ks. The Jou nal o Neu oscience: The O icial Jou nal o he Socie y o Neu- oscience, 29(26):8586–8594, July 2009. [62] X Di and BB Biswal. Dynamic B ain Func ional Connec i i y Mod- ula ed by Res ing-S a e Ne wo ks. B ain S uc Func , 220:37–46, 2015. [63] Fik e Işik Ka ahanoğlu and Dimi i Van De Ville. T ansien b ain ac i i y disen angles MRI es ing-s a e dynamics in e ms o spa- ially and empo ally o e lapping ne wo ks. Na u e Communica ions, Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 8 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks 6:7751, July 2015. [64] P Lin, Y Yang, J Gao, N Pisapia, S Ge, X Wang, CS Zuo, JJ Le i , and C Niu. Dynamic De aul Mode Ne wo k Ac oss Diffe en B ain S a es. Sci Rep, 7:46088, 2017. [65] M Demi as, C To nado , C Falcon, M Lopez-Sola, R He nandez- Ribas, J Pujol, JM Menchon, P Ri e , N Ca done , C So iano-Mas, and G Deco. Dynamic Func ional Connec i i y Re eals Al e ed Va iabili y in Func ional Connec i i y Among Pa ien s Wi h Majo Dep essi e Diso de . Hum B ain Mapp, 37:2918–2930, 2016. [66] F Debae e, N Wende o h, S Sunae , P Van Hecke, and SP Swinnen. Changes in b ain ac i a ion du ing he acquisi ion o a new bimanual coodina ion ask. Neu opsychologia, 42:855–867, 2004. [67] CJ S oodley and JD Schmahmann. Func ional opog aphy in he human ce ebellum: a me a-analysis o neu oimaging s udies. Na Neu osci, 44:489–501, 2009. [68] J Gooije s, IA Bee s, G Albouy, K Beeckmans, K Michiels, S Sunae , and SP Swinnen. Mo emen p epa a ion and execu ion: diffe en- ial unc ional ac i a ion pa e ns a e auma ic b ain inju y. B ain, 139:2469–2485, 2016. [69] T San os Mon ei o, IAM Bee s, MP Boisgon ie , J Gooije s, L Pauwels, S Chala i, B King, G Albouy, and SP Swinnen. Rela i e co ico- subco ical shi in b ain ac i i y bu p ese ed aining-induced neu al modula ion in olde adul s du ing bimanual mo o lea ning. Neu obiol Aging, 58:54–67, 2017. [70] JD Schmahmann. An eme ging concep . he ce ebella con ibu ion o highe unc ion. A ch Neu ol, 48:1178–1187, 1991. [71] Ma hieu P. Boisgon ie , Bo is Che al, Pe e an Rui enbeek, Koen Cuype s, Inge Leunissen, S e an Sunae , Ra Meesen, Hamed Zi- a i Adab, Oli ie Renaud, and S ephan P. Swinnen. Ce ebella g ay ma e explains bimanual coo dina ion pe o mance in child en and olde adul s. Neu obiology o Aging, 65:109–120, 2018. [72] HC Leine . Sol ing he mys e y o he human ce ebellum. Neu opsy- chol Re , 20:229–235, 2010. [73] M No oozian. The ole o he ce ebellum in cogni ion: beyond coo - dina ion in he cen al ne ous sys em. Neu ol Clin, 32:1081–1104, 2014. [74] Egidio D‘Angelo and S e ano Casali. Seeking a unified amewo k o ce ebella unc ion and dys unc ion: om ci cui ope a ions o cogni ion. F on ie s in Neu al Ci cui s, 6:116, 2013. [75] JF Medina and MD Mauk. Compu e simula ion o ce ebella in o - ma ion p ocessing. Na Neu osci, 3:1205–1211, 2000. [76] T Ohyama, WL No es, M Mu phy, and MD Mauk. Wha he ce ebel- lum compu es. T ends Neu osci, 26:222–227, 2003. Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 9 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Figu e 4: Robus ness o he h ee ele an module-scena ios o di e en window leng hs. The cha ac e is ics o modules 1 and 14 ha e he highes and lowes alue o 𝑚 , espec i ely, whils keeping low Δ𝑚 , p ese ed independen ly on he alue o window leng h 𝛿 . Howe e , module 11 in igu e 3 had he highes Δ𝑚 alue and when changing 𝛿 , he oles swi ch be ween module 2 and 11. Impo an ly, he wo in a ian modules 1 and 14 a e bo h pa s o he ce ebellum. The window leng h 𝛿 is gi en in ime poin s, each one co esponding o a ime du a ion o TR = 3 sec. Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 16 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Figu e 5: S uc u e- unc ion modules s classical es ing s a e ne wo ks (RSNs). Modules m1-m14 we e ob ained om he s uc u al connec i i y ma ix. By con as , RSNs a e pu ely unc ional modules, as e lec ed by he weak SC-DFC simila i y (pink). Mo eo e , he DMN had he la ges DFC a iabili y, which sugges s di e en compu a ional p inciples o his ne wo k (no eally cons ained o b ain s uc u e). Abb e ia ions: De aul Mode Ne wo k (DMN), Ven al A en ion Ne wo k (VAN), Soma omo o Ne wo k (SMN), Do sal A en ion Ne wo k (DAN). Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 17 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Figu e S1: The oles o modules 1 and 14 a e p ese ed in di e en con ol condi ions, when calcula ing SC ma ices by p obabilis ic ac og aphy ( A ), wi h 96%-o e lapping sliding windows ( B ), wi hou GSR emo al ( C ), and calcula ing FC ma ices using he ins an aneous ampli ude o he complex analy ical signal ( D ), and he ins an aneous phase, also known as he phase locking alue ( E ). ( A-E ): In all panels, module 1 had he highes SC-DFC simila i y and low DFC a iabili y alues, while module 14 had he lowes SC-DFC simila i y alue and also low DFC a iabili y alues. S ikingly, module 1 and 14 we e localized in he pos e io and an e io pa s o he ce ebellum, espec i ely. (C,E): * di e en scale. Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 18 o 19 Small a ia ion in dynamic unc ional connec i i y in ce ebella ne wo ks Iza o Fe nandez-I iondo ecei ed he Bachelo ’s deg ee in Physics in 2019 om he Uni e si y o he Basque Coun y (UPV). She is cu en ly s udy- ing a mas e ’s deg ee in compu e enginee ing and in elligen sys ems a he UPV in San Sebas ian, aiming o comple e his wi h a machine lea ning p ojec applied o s uc u al and unc ional b ain ne wo ks. An onio Jimenez-Ma in is a PhD s uden a he Biomedical Resea ch P og am o he Uni e si y o he Basque Coun y. His esea ch is ca ied ou in he Compu a ional Neu oimaging Lab a Bioc uces Bizkaia Heal h Resea ch Ins i u e. He ob ained his deg ee in Telecommunica ion Technology Engi- nee ing a he Uni e si y o G anada (2015) and his MSc in Biomedical Enginee ing a he Uni e - si y o he Basque Coun y (2018). His esea ch in e es s ocus on b ain connec i i y in heal h and disease. Ibai Diez ecei ed his Bachelo ’s deg ee in Telecom- munica ion Enginee ing in 2009 om Deus o Uni- e si y, Spain, and his PhD in 2015 om Uni e si y o he Basque Coun y (UPV/EHU, Spain). He is cu en ly a pos doc o al esea che in he Neu ol- ogy Depa men a Massachuse s Gene al Hospi al – Ha a d Medical School. His esea ch in e es include s uc u al and unc ional b ain connec i i y, biomedical da a analysis and unc ional in eg a ion in he b ain. Paolo Boni azi is a Tenu ed Ike basque Resea che a he Bioc uces-Bizkaia Heal h Resea ch Ins i u e in Bilbao (Spain), wi hin he Compu a ional Neu- oimaging Lab. He is a Sys em Neu oscien is wi h a deg ee in Physics (uni e si y o Pe ugia, I aly) and a PhD in Neu oscience (SISSA, T ies e, I aly). A e comple ing wo pos doc o al posi ions in he UK (wi h P o . Hugh Robinson) and F ance (wi h D Rosa Cossa ), he became a Resea ch Associa e a he Tel A i Uni e si y in he g oup o P o . A i Ba zilai and he la e P o . Eshel Ben-Jacob. His esea ch in e es s a e mul i-disciplina y, ocusing on he s uc u e and unc ion o b ain ci cui s, span- ning om mic oci cui s o b ain ne wo ks, b idging expe imen al and compu a ional app oaches, he la e inspi ed by complex ne wo ks. S ephan P. Swinnen s a ed his esea ch ca ee in mo emen con ol a he Uni e si y o Cali o nia a Los Angeles (UCLA, 1983-85) unde he di ec ion o P o . R. A. Schmid . He comple ed his PhD a KU Leu en in 1987 and he was awa ded a F ancqui Resea ch P o esso posi ion (2013-2016). He cu - en ly di ec s he Mo emen Con ol & Neu oplas- ici y Resea ch G oup a KU Leu en (Belgium). His esea ch in e es s ocus on mechanisms unde lying mo emen con ol and neu oplas ici y in no mal and pa hological condi ions, using mul idisciplina y ap- p oaches ocusing on he s udy o b ain unc ion, s uc u e, connec i i y and neu ochemicals. Miguel A. Muñoz is a Full P o esso in Physics a he Uni e si y o G anada (Spain). He is an expe in s a is ical mechanics and has wo ked on, among o he issues, non-equilib ium phase ansi ions, c i - ical and collec i e phenomena and s ochas ic p o- cesses. In pa icula , he helped de elop he he- o y o "sel -o ganized c i icali y" and app oaches owa ds he s udy o non-equilib ium phenomena, ne wo k heo y and complex sys ems. His esea ch in e es s span om undamen al p inciples o s a- is ical mechanics o in e disciplina y p oblems in e olu iona y biology, heo e ical ecology and sys- ems neu oscience. Jesus M. Co es is an Ike basque Resea ch P o esso and he head o he Compu a ional Neu oimaging Lab a he Bioc uces-Bizkaia Heal h Resea ch Ins i- u e in Bilbao (Spain). He eaches B ain Connec i - i y and Neu oimaging on he M.Sc. o Biomedical Enginee ing. He ob ained a Ph.D. in Physics in 2005 and comple ed h ee pos doc o al posi ions in The Ne he lands (supe ised by P o . Be Kappen), USA (supe ised by P o . Te y Sejnowski) and UK (supe ised by P o . Ma k an Rossum). Among many o he me i s, he pa icipa ed in he eam headed by P o . Mazahi T. Hasan ha achie ed he miles one o ecei ing he fi s p ojec unded in Spain by he B ain Ini ia i e. He also go Ph.D. wi h Dis inc ion. His a ea o esea ch now ocuses on b ain connec i i y, neu oimaging and machine lea ning me hods applied o heal hy and pa holog- ical condi ions. Fe nandez-I iondo e al.: P ep in submi ed o Else ie Page 19 o 19