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 .
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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