RESEARCH ARTICLE
Complex spa io empo al oscilla ions eme ge
om ans e se ins abili ies in la ge-scale
b ain ne wo ks
Pau ClusellaID
1
*, Gus a o Deco
2,3
, Mo en L. K ingelbach
4,5
, Giulio Ru ini
6
, Jo di Ga cia-
Ojal oID
1
1Depa men o Medicine and Li e Sciences, Uni e si a Pompeu Fab a, Ba celona, Spain, 2Compu a ional
Neu oscience G oup, Cen e o B ain and Cogni ion, Depa men o In o ma ion and Communica ion
Technologies, Uni e si a Pompeu Fab a, Ba celona, Spain, 3Ins i ucio
´Ca alana de Rece ca i Es udis
A anc¸a s (ICREA), Ba celona, Spain, 4Depa men o Psychia y, Uni e si y o Ox o d, Ox o d, Uni ed
Kingdom, 5Cen e o Music in he B ain, Depa men o Clinical Medicine, Aa hus Uni e si y, Aa hus,
Denma k, 6B ain Modeling Depa men , Neu oelec ics, Ba celona, Spain
*[email p o ec ed]
Abs ac
Spa io empo al oscilla ions unde lie all cogni i e b ain unc ions. La ge-scale b ain models,
cons ained by neu oimaging da a, aim o ace he p inciples unde lying such mac oscopic
neu al ac i i y om he in ica e and mul i-scale s uc u e o he b ain. Despi e subs an ial
p og ess in he ield, many aspec s abou he mechanisms behind he onse o spa io empo-
al neu al dynamics a e s ill unknown. In his wo k we es ablish a simple amewo k o he
eme gence o complex b ain dynamics, including high-dimensional chaos and a elling
wa es. The model consis s o a complex ne wo k o 90 b ain egions, whose s uc u al con-
nec i i y is ob ained om ac og aphy da a. The ac i i y o each b ain a ea is go e ned by
a Jansen neu al mass model and we no malize he o al inpu ecei ed by each node so i
amoun s he same ac oss all b ain a eas. This assump ion allows o he exis ence o an
homogeneous in a ian mani old, i.e., a se o di e en s a iona y and oscilla o y s a es
in which all nodes beha e iden ically. S abili y analysis o hese homogeneous solu ions
un eils a ans e se ins abili y o he synch onized s a e, which gi es ise o di e en ypes
o spa io empo al dynamics, such as chao ic alpha ac i i y. Addi ionally, we illus a e he
ubiqui y o his ou e owa ds complex spa io empo al ac i i y in a ne wo k o nex gene a ion
neu al mass models. Al ogeh e , ou esul s un eil he bi u ca ion landscape ha unde lies
he eme gence o unc ion om s uc u e in he b ain.
Au ho summa y
Moni o ing b ain ac i i y wi h echniques such as elec oencephalog am (EEG) and unc-
ional magne ic esonance imaging ( MRI) has e ealed ha no mal b ain unc ion is
cha ac e ized by complex spa io empo al dynamics. This beha io is well cap u ed by
la ge-scale b ain models ha inco po a e s uc u al connec i i y da a ob ained wi h MRI-
based ac og aphy me hods. None heless, i is no ye clea how hese complex dynamics
PLOS COMPUTATIONAL BIOLOGY
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1010781 Ap il 12, 2023 1 / 34
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OPEN ACCESS
Ci a ion: Clusella P, Deco G, K ingelbach ML,
Ru ini G, Ga cia-Ojal o J (2023) Complex
spa io empo al oscilla ions eme ge om
ans e se ins abili ies in la ge-scale b ain
ne wo ks. PLoS Compu Biol 19(4): e1010781.
h ps://doi.o g/10.1371/jou nal.pcbi.1010781
Edi o : Bo is S. Gu kin, E
´cole No male Supe
´ ieu e,
College de F ance, CNRS, FRANCE
Recei ed: Decembe 1, 2022
Accep ed: Ma ch 24, 2023
Published: Ap il 12, 2023
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eme ge om he in e play o he di e en b ain egions. In his pape we show ha com-
plex spa io empo al dynamics, including a elling wa es and high-dimensional chaos
can a ise in simple la ge-scale b ain models h ough he des abiliza ion o a synch onized
oscilla o y s a e. Such ans e se ins abili ies a e akin o hose obse ed in chemical eac-
ions and u bulence, and allow o a semi-analy ical ea men ha unco e s he o e all
dynamical landscape o he sys em. O e all, ou wo k es ablishes and cha ac e izes a gen-
e al ou e owa ds spa io empo al oscilla ions in la ge-scale b ain models.
In oduc ion
The in e play be ween spiking neu ons ac oss he b ain p oduces collec i e hy hmic beha io
a mul iple equencies and spa ial esolu ions [1,2]. This oscilla o y neu al ac i i y is unda-
men al o p ope cogni i e unc ion [3,4], and is e lec ed in a ple ho a o spa io empo al
phenomena in eco ded signals [5–8]. A he mic oscopic scale, compu a ional and ma hema -
ical models ha e success ully cap u ed some o hese eme gen p ope ies by analyzing he col-
lec i e beha io o ne wo ks o coupled neu ons [1,9–11]. In la ge scales, he ask becomes
inc easingly challenging, as one needs o model se e al popula ions o neu ons, which
inc eases he ma hema ical complexi y and he compu a ional cos o he p oblem. Mesoscale
neu al-mass models (NMMs) allow o o e come his si ua ion by cap u ing he neu al ac i i y
o la ge numbe s o neu ons wi h a ew equa ions [12–17].
Recen ad ances on neu oimaging allow o cha ac e ize he s uc u al o ganiza ion o he
b ain as a complex ne wo k [18–20]. In his ep esen a ion, each node o he ne wo k co e-
sponds o a b ain egion composed o densely in e -connec ed neu ons, and edges ac oss
nodes ep esen pai wise in e ac ions ac oss dis an egions. Combining NMMs wi h connec-
omics da a one can c ea e la ge-scale b ain models whose dynamical p ope ies e lec he
p inciples unde lying mac oscopic neu al ac i i y [21–24]. This amewo k has been used,
o ins ance, o un eil he na u e o es ing-s a e luc ua ions [25–30], in es iga e he ela ion
be ween s uc u al and unc ional connec i i y [31–35], and cha ac e ize ansi ions be ween
b ain s a es [36–39]. In clinical applica ions, la ge-scale b ain models allow o s udying mac-
oscopic aspec s o neu opha ologies such as epilepsy o Alzheime disease [40–43], and simu-
la e he use non-in asi e b ain s imula ion p o ocols o po en ial ea men s [44,45].
Despi e all his p og ess, many aspec s abou he mechanisms h ough which la ge-scale
b ain models ep oduce mac oscopic neu al dynamics a e s ill unknown. Fo ins ance, synap-
ic delays be ween dis an egions [25,46], noise [25,29], and he e ogenei ies [28] a e usually
acknowledged as a sou ce o dynamical complexi y. None heless, spa io empo al beha io can
also a ise om homogeneous de e minis ic sys ems wi h ins an aneous in e ac ions [30,35,
44]. In pa icula , Fo es e e . al. [35] ecen ly in es iga ed he dynamics o a la ge-scale b ain
model composed o weakly-coupled Jansen’s NMMs. By means o a phase- educ ion app oxi-
ma ion, hey un eil phase-locked s a es eme ging om a ins abili y o he synch onized s a e
(see also [33]). Fo a bi a y coupling alues, howe e , bi u ca ion s udies o whole-b ain ne -
wo ks a e a he limi ed o nume ical in es iga ions [30,44]. O he s udies show ha sys ems
o NMMs coupled h ough simpli ied ne wo k opologies migh display a elling wa es and
e en chao ic dynamics [47–50]. Whe he hese esul s ansla e o i egula b ain ne wo ks
emains, so a , unexplo ed.
In his pape we cha ac e ize he onse o spa io empo al dynamics in a simple la ge-scale
b ain model wi hou he e ogenei ies, noise, no delays. In close analogy o pa e n- o ma ion
mechanisms in eac ion-di usion sys ems [51–55], coupling among b ain egions alone is
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T ans e se ins abili ies in la ge-scale b ain ne wo ks
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Funding: PC, GD, GR, and JGO ha e ecei ed
unding om he Fu u e and Eme ging
Technologies P og amme (FET) o he Eu opean
Union’s Ho izon 2020 esea ch and inno a ion
p og amme (p ojec NEUROTWIN, g an
ag eemen No 101017716). JGO also
acknowledges inancial suppo om he Spanish
Minis y o Science and Inno a ion and FEDER
(g an PID2021-127311NB-I00), by he “Ma ia de
Maez u” P og amme o Uni s o Excellence in
R&D (g an CEX2018-000792-M), and by he
Gene ali a de Ca alunya (ICREA Academia
p og amme). The unde s had no ole in s udy
design, da a collec ion and analysis, decision o
publish, o p epa a ion o he manusc ip .
Compe ing in e es s: I ha e ead he jou nal’s
policy and he au ho s o his manusc ip ha e he
ollowing compe ing in e es s: GR is a co- ounde
o Neu oelec ics, a Company ha manu ac u es
ES and EEG echnology. The emaining au ho s
don’ ha e any con lic o in e es .
enough o spon aneously des abilize an unde lying synch onized s a e, he eby gene a ing
complex oscilla o y beha io . Ou analysis consis s o wo pa s: Fi s we show ha , gi en a
common no maliza ion on he incoming inpu o each egion [56,57], he ne wo k possesses
an in a ian homogeneous mani old, i.e., a se o s a es in which he beha io o each node is
iden ical ac oss all he ne wo k. These s a es a e desc ibed by a low-dimensional sys em: a sel -
coupled e sion o he NMM used o he e olu ion o each b ain egion. Bi u ca ion analysis
o he sys em e eals how di e en sys em pa ame e s modi y he onse o synch onized oscil-
la o y s a es wi hin he mani old.
Second, we employ he Mas e S abili y Func ion (MSF) o malism [58–62] o in es iga e
he s abili y o he homogeneous s a es o he e ogeneous pe u ba ions. The synch onized
oscilla o y solu ion o he sys em u ns ou o be ans e sally uns able in a la ge egion o
pa ame e space, gi ing ise o complex spa io empo al dynamics including a elling wa es,
mul is abili y, and high-dimensional chaos.
In o de o illus a e he ubiqui y o his mechanism, we mainly use he Jansen NMM o
he single-node dynamics [15,16,63], bu also b ie ly e iew he onse o chao ic dynamics
using a nex gene a ion NMM [17]. Mo eo e , we also show ha , e en when he no maliza ion
condi ion is no ul illed, he bi u ca ion diag am o he sys em emains simila o ha o he
simpli ied model. Ou wo k ex ends p e ious indings on weakly-coupled NMMs [33,35] and
simpli ied ne wo k opologies [47–50] o a comp ehensi e cha ac e iza ion o he eme ging
spa io empo al beha io .
Resul s
The la ge-scale b ain model
We build a la ge-scale b ain model using a s uc u al connec i i y (SC) ne wo k comp ised o
N= 90 b ain egions de ined by he Au oma ed Ana omical Labeling pa cella ion (AAL-90)
[64], which includes 76 co ical and 14 subco ical egions. Each b ain egion is ep esen ed
by a ne wo k node, and pai wise in e ac ions be ween egions a e gi en by a ow-no malized
connec i i y ma ix ~
W¼ ð~
wijÞwhe e i,j= 1, . . .,N. The connec ion weigh s ~
wij a e non-nega-
i e quan i ies ha indica e he synap ic s eng h (a e age numbe o connec ions) om egion
j o egion i, and ha e been ob ained om ac og aphy da a om 16 human subjec s (see [65]
and also Me hods). Al hough ne wo ks ob ained om DTI a e usually symme ic p io ow-
no maliza ion, ou app oach can also be applied o asymme ic ne wo ks (see Me hods).
In o de o model he dynamics o each b ain egion we use, o mos o he pape , Jansen’s
model o a co ical column [15,16,63]. Acco ding o his model, he beha io o each b ain
egion is gi en by a sys em o six o dina y di e en ial equa ions (Eq (9) in Me hods) ha
accoun o he in e ac ions be ween a popula ion o exci a o y py amidal neu ons (PNs),
a popula ion o inhibi o y in e neu ons (INs), and ecu en connec ions wi hin py amidal
neu ons ( PNs). Wi hin each egion, he PNs ecei e wo sou ces o ex e nal inpu : a baseline
i ing a e p, which we assume cons an and iden ical ac oss he ne wo k, and he incoming i -
ing a es om he PNs o o he b ain egions, modula ed by a coupling pa ame e �. Hence,
long ange connec ions es ablished by he s uc u al connec i i y ma ix a e all assumed exci -
a o y, whe eas inhibi ion ac s only locally wi hin each ne wo k node.
O e all, we ob ain a la ge-scale b ain model composed o 90 ×6 = 540 ODEs (see Eq (10)
in Me hods). Despi e i s high dimensionali y, his sys em is ela i ely simple o analyze, as i
does no include noise no ime delays and i s pa ame e s a e assumed o be iden ical ac oss
b ain egions. Simila la ge-scale b ain models based on he Jansen sys em ha e been analyzed
in p e ious wo ks, he main di e ence being he connec i i y da a used o he unde lying
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T ans e se ins abili ies in la ge-scale b ain ne wo ks
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ne wo k opology [35,44]. Ou wo k ex ends he esul s o hese p e ious s udies, p o iding a
comp ehensi e bi u ca ion landscape o he ne wo k.
Bi u ca ion diag am
Following a cus oma y app oach in he li e a u e [30,33,35,44,45,56,57], ou de ini ion
o he s uc u al connec ome ma ix ~
Wes ablishes ha all ne wo k nodes ecei e he same
amoun o inpu , i.e., PN
j¼1~
wij ¼1 o all i= 1, . . .,N. This condi ion ensu es ha he sys em
has homogeneous (uni o m) s a es, i.e., s a es in which all ne wo k nodes e ol e iden ically. I
he sys em is ini ialized in such a s a e, i will emain he e o e e unless pe u bed: ma he-
ma ically speaking, hese s a es lie on an in a ian mani old. The exis ence o homogeneous
solu ions does no p e en he exis ence o he e ogeneous s a es, in which he ajec o ies o di -
e en nodes e ol e di e en ly. I a homogeneous s a e p o es o be uns able o a bi a y (he -
e ogeneous) pe u ba ions, complex spa io empo al dynamics migh a ise.
In his sec ion we in es iga e he homogeneous s a es o he la ge-scale b ain model sys em-
a ically, as a means o un eil he eme gence o non- i ial he e ogeneous pa e ns. This analy-
sis equi es wo s eps: Fi s , we iden i y he homogeneous s a es o he sys em and hei
s abili y o uni o m pe u ba ions, i.e. we s udy he homogeneous in a ian mani old o he
sys em. Then, we analyze he s abili y o hese s a es o a bi a y pe u ba ions using he Mas-
e S abili y Func ion (MSF) o malism [58].
Dynamics o he homogeneous in a ian mani old. I all b ain egions beha e iden i-
cally, hen each ne wo k node ollows he dynamics o he sel -coupled Jansen model gi en by
Eq (18) in Me hods. This sys em de ines he homogeneous in a ian mani old o he sys em,
hence i s s able s a es co espond o hose homogeneous s a es o he ull model (Eq 10) ha
a e s able o uni o m pe u ba ions. In his sec ion we analyze he dynamics o Eq (18) alone.
The e o e, he e m s abili y in his sec ion e e s o s abili y wi hin he homogeneous mani old
only. We unco e he di e en a ac o s o he sel -coupled sys em wi h he help o he bi u -
ca ion analysis so wa e AUTO-07p [66], using he common ex e nal inpu p�0 and he cou-
pling s eng h ��0 as con ol pa ame e s. Despi e bo h pa ame e s being posi i e quan i ies,
we also include nega i e alues o pin he bi u ca ion diag ams in o de o e eal he en i e
bi u ca ion s uc u e.
We s a by ecalling he dynamics o he uncoupled Jansen model ( o de ails eade s can
e e o [63]). Fig 1(a) shows he alue o he mean memb ane po en ial o he PN popula ion
co esponding o di e en solu ions o sys em (18) o ixed �= 0 and a ying baseline inpu p.
Fo small posi i e p, wo s able s eady s a es coexis (da k pink cu es), each o hem leading o
a di e en ype o s able pe iodic s a e (black cu es) upon inc easing he baseline inpu . Fi s ,
he high-ac i i y ixed poin (uppe pink b anch) unde goes a supe c i ical Hop bi u ca ion
(HBþ
1) a p�90, which co esponds o he onse o alpha oscilla o y ac i i y (�10Hz). This
pe iodic s a e pe sis s un il p�315, whe e i anishes h ough a second supe c i ical Hop -
bi u ca ion (HBþ
2) leading again o a s able high-ac i i y s eady s a e. Second, he low-ac i i y
s a iona y s a e o psmall (lowe da k pink b anch) anishes h ough a saddle-node in a
in a ian cycle (SNIC) bi u ca ion o p�114, gi ing ise o ini e ampli ude oscilla ions a a
he a ange (�4Hz). This spiky oscilla o y ac i i y anishes a p�137 h ough a old (o sad-
dle-node) bi u ca ion o limi -cycles (FLC
1
).
The addi ion o coupling (� > 0) modi ies his bi u ca ion scena io. Fo ins ance, Fig 1(b)
shows he bi u ca ion diag am o �= 4. The igu e shows ha coupling elimina es he supe -
c i ical Hop bi u ca ion HBþ
1, so ha o small alues o ponly he low-ac i i y ixed poin is
s able. As in he uncoupled case, his s eady s a e anishes a p�111 h ough a SNIC bi u ca-
ion. Howe e , now he b anch o oscilla o y s a es a ising om he SNIC connec s bo h he
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he a (111 ≲p≲150) and alpha (134 ≲p≲351) equency bands, which coexis in a small
egion bounded by wo FLCs. Hence, o small bu non-ze o coupling he alpha oscilla o y
s a e eme ges h ough he old o cycles FLC
2
ins ead o a Hop bi u ca ion, and, as in he
uncoupled case, i disappea s h ough he HBþ
2supe c i ical Hop bi u ca ion o la ge inpu
(p�351).
Fu he inc ease o �leads o a o al disappea ance o bis abili y be ween oscilla o y s a es.
Fig 1(c) shows an example o his si ua ion o �= 50. Upon inc easing he ex e nal inpu p,
he low ac i i y ixed poin leads o ini e-ampli ude oscilla ions h ough he SNIC bi u ca ion
(p�84.68). This oscilla o y s a e becomes he only a ac o o a wide ange o p, un il i an-
ishes a he supe c i ical Hop bi u ca ion HBþ
2(p�330), beyond which ajec o ies con e ge
o he high-ac i i y ixed poin . The equency o his single s able oscilla o y s a e (see Fig
1(d)) shows wo dis inc pla eaus ha domina e o mos alues o p, loca ed a a ound he
he a (3–5Hz) and alpha (7–9Hz) le els.
O e all, he diag ams in Fig 1(a) and (c) e eal a change on he onse o alpha oscilla o y
ac i i y, oge he wi h a loss o bis abili y as �inc eases. These changes can be be e unde -
s ood om he wo-pa ame e bi u ca ion analysis shown in Fig 1(e) and ( ). These diag ams
illus a e he di e en egions o s abili y o Eq (18) o a ying pand �, wi h panel ( ) showing
a zoomed e sion o panel (e) o small coupling. The ed con inuous cu e in Fig 1(e) and ( )
co esponds o he Hop bi u ca ion HBþ
1gi ing aise o alpha ac i i y o low alues o �(see
Fig 1(a)). This bi u ca ion becomes subc i ical (dashed ed line) h ough a Bau in (o gene al-
ized Hop ) codimension-2 bi u ca ion a p�40, ��1.8. A his poin , he FLC
2
appea s
Fig 1. Bi u ca ion diag ams o he homogeneous s a es subjec o uni o m pe u ba ions. (a-c) One-pa ame e bi u ca ion diag ams o Eq (18)
ob ained by a ying pand wi h ixed �= 0 (a), �= 4 (b), and �= 50 (c). Colo ed cu es indica e s able ixed poin s (da k pink), uns able ixed poin s
(ligh pink), ex ema o s able limi -cycles (black), and ex ema o uns able limi -cycles (g ey). Dashed black e ical lines indica e he ele an
bi u ca ion poin s ci ed in he ex . (d) F equency o he s able oscilla o y s a e by a ying pand wi h ixed �= 50. (e, ) Two-pa ame e bi u ca ion
diag am o Eq (18) depending on he ex e nal inpu pand he coupling s eng h �. Panel ( ) is a zoomed e sion o (e). Cu es indica e di e en
bi u ca ion ypes: supe c i ical Hop (HB
+
, con inuous ed), subc i ical Hop (HB
−
, dashed ed), saddle-node (SN, b own), saddle-node in a in a ian
cycle (SNIC, da k blue), saddle-node o limi -cycles (FLC, g een), and homoclinic (Hom., o ange) The ligh -blue egion indica es he exis ence o a
single pe iodic s a e. The ligh -blue egion wi h s ipped black pa e n indica es he coexis ence be ween a limi -cycle and a ixed poin . The da k-blue
egion indica es he coexis ence o wo s able pe iodic s a es. All esul s ob ained by analyzing he sys em o Eq (18) using he bi u ca ion analysis
so wa e AUTO-07p (sc ip s o one-pa ame e bi u ca ions a ailable a www.gi hub.com/pclus/ ans e se-ins abili ies).
h ps://doi.o g/10.1371/jou nal.pcbi.1010781.g001
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(g een line), and join s he FLC
1
a a cusp (p�165, ��10) a which bo h bi u ca ions anish.
The e o e, wi hin he da k-blue egion delimi ed by he wo FLCs (g een cu es) and he
SNIC bi u ca ions (da k-blue cu e) wo s able oscilla o y s a es coexis . Beyond his egion,
and o a wide ange o � alues, he dynamical landscape o he sys em becomes simple and
coincides wi h ha shown in Fig 1(c) (see Fig 1(e) o �= 50). This scena io changes o e y
high coupling, when he SNIC bi u ca ion u ns o a SN h ough a saddle-node sepa a ix loop
(SNSL) codimension-2 bi u ca ion [67]. F om his poin , a homoclinic bi u ca ion (Hom.)
bounds he egion o oscilla o y dynamics, which appea o a bi a y low p. Pa allel o he
homoclinic line, wo addi ional b anches o FLC appea , leading o a e y na ow egion o
oscilla o y bis abili y. Since hey ha e a mino e ec on he o e all bi u ca ion landscape,
we do no depic hem in Fig 1(e). Finally, u he inc ease o �ceases all oscilla o y ac i i y
h ough he HBþ
2( ed con inuous cu e).
In summa y, he dynamics o he sys em wi hin he homogeneous in a ian mani old can
be di ided in h ee main egions:
• Low ac i i y ixed poin , ep esen ed by he whi e egion o he le o he SNIC bi u ca ion
(da k blue cu e) in Fig 1(e).
• High ac i i y ixed poin , ep esen ed by he whi e egion o he igh and abo e o he HBþ
2
line ( ed cu e) in Fig 1(e).
• Oscilla o y ac i i y a he he a o alpha anges (o bis abili y be ween bo h), mos ly delim-
i ed by he HBþ
2and he SNIC bi u ca ion cu es (ligh blue shadowed egion) in Fig 1(e).
O e all, he bi u ca ion analysis o Eq (18) unco e s he ich dynamical epe oi e o he
homogeneous mani old o he ull sys em (Eq 10). None heless, we ema k again ha his anal-
ysis only conce ns homogeneous s a es, i.e., he di e en egions o s abili y and ins abili y
ou lined so a only assume pe u ba ions ha ac iden ically a each ne wo k node. In he nex
sec ion we assess he s abili y o he homogeneous solu ions o he e ogeneous pe u ba ions.
T ans e se s abili y. The analysis o sys em (18) discussed abo e e eals he loci o all
homogeneous s a es o he ne wo k model ha a e s able o uni o m pe u ba ions. In o de o
de e mine he s abili y o hese s a es o a bi a y non-uni o m pe u ba ions, we ollow a well-
es ablished app oach ha can be applied o ei he ixed poin s o pe iodic s a es (see Me hods).
This echnique, analogous o he one used in he s udy o Tu ing bi u ca ions in complex ne -
wo ks [54,68,69], consis s on decomposing an a bi a y pe u ba ion ec o on he basis
gi en by he eigen ec o s o a sui able ma ix ep esen ing he way he nodes a e coupled. This
p o ides a dispe sion ela ion o he g ow h a e o he pe u ba ions. In ou case, ins ead o
he Laplacian ma ix used in di usi ely coupled sys ems, we diagonalize he no malized s uc-
u al connec i i y ma ix ~
W. As we will see, s able ixed poin s in he homogeneous mani old
emain always s able o he e ogeneous pe u ba ions in he ull-model. Hence, ou ocus will
be on he s abili y o he limi -cycle solu ions, o which his decomposi ion echnique is
known as he Mas e S abili y Func ion (MSF) [58,59,61].
Fig 2(a) shows he i s 5 eigen ec o s o ~
Win e ms o hei componen s ac oss he 90
b ain egions. The i s eigenmode is homogeneous, and i s associa ed eigen alue is always
Λ
1
= 1 (see Me hods). Pe u ba ions along his di ec ion a e he ones al eady accoun ed by he
analysis o Eq (18) in he p e ious sec ion. The subsequen eigenmodes a e he e ogeneous and
hei associa ed eigen alues a e be ween -1 and 1 (see Me hods). Pe u ba ions along hese
di ec ions a e ans e se o he homogeneous in a ian mani old. Despi e s emming om an
i egula ne wo k opology, he eigen ec o s exhibi a well de ined spa ial s uc u e, which
can be aced back o he well-known exponen ial decay o in e - egion connec i i y wi h
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Euclidean dis ance obse ed in b ain connec i i y da a ob ained wi h di usion enso imaging
(DTI) [70,71].
Following he echnique explained in Me hods, we ound ha no ans e se ins abili ies
a ise om he homogeneous ixed poin s. The e o e nex we ocus on he s abili y o he limi -
cycle solu ions by means o he Mas e S abili y Func ion (MSF) [58,59,61]. The g ow h a e
o an in ini esimal pe u ba ion o a pe iodic s a e is gi en by he eal pa o he co esponding
Floque exponen s [72], which we deno e by μ. The MSF p o ides he la ges g ow h a e μo a
pe u ba ion ac ing along each eigenmode Φ
(α)
as a unc ion o i s associa ed eigen alue Λ
α
,
ma¼MSFðLaÞ:ð1Þ
This ela ion is analogous o dispe sion ela ions in spa ially ex ended sys ems, whe e he
g ow h a e o a pe u ba ion is gi en as a unc ion o he pe u ba ion wa enumbe [55]. In
he Jansen model, he MSF needs o be compu ed nume ically (see Me hods sec ion o a
de ailed explana ion o i s de i a ion and nume ical compu a ion).
Fig 2(b) shows he MSFs o he homogeneous limi -cycle solu ion o di e en p alues.
Each ci cle indica es he eal pa o he la ges Floque exponen co esponding o a
Fig 2. T ans e se ins abili ies o oscilla o y s a es. (a) Fi s 5 eigen ec o s o ~
Win a spa ial ep esen a ion o he b ain (supe io iew). Each ne wo k
node has been colo ed acco ding o hei con ibu ion o he co esponding eigen ec o ΦðaÞ¼ ð�ðaÞ
1;...; �ðaÞ
NÞ. (b) Mas e S abili y Func ion o sys em
(10) showing he dependence o he la ges Floque exponen μwi h espec o he s uc u al connec i i y eigen alues Λ
α
o h ee di e en alues o he
inpu p.�= 50 in all cases. Ci cles co espond o he eigen alues o ~
W, whe eas black cu es a e ob ained by con inuously uning Λ. (c,d) Resul s om
nume ical simula ions o Eq (10) wi h (c) p= 280 and �= 50 and (d) p= 210 and �= 50. (e) Comple e bi u ca ion diag am o he homogeneous s a es.
Colo s o lines and egions as in Fig 1(e) and ( ), wi h he egion o ans e se ins abili y (i.e., μ
2
>0) shaded in pink and delimi ed wi h a black cu e.
Black ci cles co espond o nume ical simula ions in which hσi>10
−5
. Simula ions ini ialized close o a homogeneous s a e. ( ,g) Bi u ca ion diag am
ob ained om di ec simula ions o he sys em wi h ini ial condi ions close o homogeneous ( ) and andom (g). Con inuous cu es as in Fig 1(e) and
( ). Regions colo ed acco ding he dynamical classi ica ion gi en by he wo la ges Lyapuno exponen s (see Me hods).
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pe u ba ion applied along he α h eigen ec o compu ed acco ding o he MSF. Since we a e
conside ing a s able limi -cycle solu ion gi en by he analysis o sys em (18), he la ges g ow h
a e co esponding o he uni o m eigen ec o (i.e., Λ
1
= 1) is μ
1
= 0. The o he exponen s
migh be posi i e o nega i e depending on bo h he sys em pa ame e s and Λ
α
. Fo ins ance,
o p= 280 (g een ci cles) he dispe sion ela ion is nega i e o all he s uc u al eigen alues
−1<Λ
α
<1. The e o e, in his case we expec small inhomogeneous pe u ba ions o he
homogeneous s a e o decay exponen ially. In con as , o p= 210 (blue ci cles) some o he
connec i i y ma ix eigen ec o s ha e posi i e g ow h a es μ
α
, hus small pe u ba ions
should gi e ise o he e ogeneous pa e ns. Indeed, Fig 2(c) and 2(d) show he esul s o in e-
g a ing nume ically Eq (10) o p= 280 and 210, espec i ely. The ini ial condi ions co e-
spond o a uni o m s a e plus a small andom pe u ba ion. In ag eemen wi h he esul s
gi en by he s abili y analysis, a e a sho ansien (no shown), he dynamics o p= 280
alls back o he homogeneous s a e, whe eas a he e ogeneous spa io empo al pa e n a ises o
p= 210.
The MSF dispe sion ela ions ep esen ed in Fig 2(b) show ha he posi i e g ow h a es
appea i s h ough he second la ges s uc u al-connec i i y eigen alue, Λ
2
. We hus ex en-
si ely analyze he loci o uns able di ec ions by checking whe he μ
2
has a posi i e eal pa o
he en i e egion o exis ence o oscilla o y ac i i y. Fig 2(e) displays he egion whe e μ
2
>0
(pu ple) supe imposed on he bi u ca ion diag am o he homogeneous mani old, calcula ed
in he p e ious sec ion and shown o iginally in Fig 1( ). Rema kably, he egion o ans e se
ins abili y occupies a la ge po ion o he pa ame e space. Mo eo e , he co esponding alues
o �co e a ange compa able o he o he coupling pa ame e s o he sys em, C
1
,. . .,C
4
2[0,
135].
In o de o alida e he eme gence o he e ogeneous dynamics in he sys em we pe o m
nume ical simula ions o sys em (10) o di e en alues o pand �, s a ing om ini ial condi-
ions close o a homogeneous s a e. A each ime s ep we compu e he s anda d de ia ion
ac oss ne wo k nodes,
sð Þ≔1
NX
N
i¼1ð ið Þ �
ð ÞÞ2
" #1
2
whe e �
ð Þ ¼ 1
NX
N
i¼1
ið Þ:ð2Þ
This quan i y anishes in he homogeneous s a es, whe eas i is posi i e in he e ogeneous
s a es. Black ci cles in Fig 2(b) indica e he pa ame e alues o which hσ( )i>10
−5
in he sim-
ula ions, showing a comple e o e lap wi h he esul s coming om he linea s abili y analysis
(pu ple egion in he igu e).
Finally, we cha ac e ize he ype o dynamical s a es a ising in he egion o ans e se ins a-
bili y, by compu ing he wo la ges Lyapuno exponen s (LE, see Me hods), λ
1
and λ
2
, in
independen nume ical simula ions wi h a ying pand �. Using his ool we can classi y he
a ac o s o he sys em depending on he sign o he wo exponen s (see Me hods). Fig 2( )
and 2(g) show he esul ing nume ical bi u ca ion diag ams co esponding o ini ial condi-
ions close o a homogeneous s a e (Fig 2( )) o en i ely andom ini ial condi ions (Fig 2(g)).
In bo h cases he egion o ans e se ins abili y can be oughly di ided in wo pa s: one
domina ed by pe iodic he e ogeneous oscilla ions (la ge �, blue), and one displaying chao ic
dynamics (small �, ed) wi h a leas wo uns able di ec ions. Addi ionally, simula ions ini ial-
ized a andom (Fig 2(g)) show ha he chao ic egime ex ends much beyond he ans e se
ins abili y egion, hus unco e ing a coexis ence egion be ween spa io empo al chaos and
homogeneous s a es. In he nex sec ions we in es iga e he p ope ies o hese di e en
dynamical egimes in de ail.
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Pe iodic a elling wa es
The simples ins ance o he e ogeneous dynamics in sys em (10) is a pe iodic egime (blue
egion in Fig 2( ) and 2(g)). In his egime, he dynamics o each b ain egion is pu ely pe i-
odic, bu wi h a non-ze o phase di e ence be ween egions, i.e., he e is phase-locking be ween
nodes. He e we e eal ha a mul iple o such s a es migh exis o a single choice o pa ame e
alues.
Fig 3(a) and 3(b) show he di e ence be ween each node’s phase ϕ
j
and he collec i e phase
C(see Me hods) o wo simula ions wi h ixed p= 230 and �= 50 and di e en ini ial condi-
ions. In bo h cases he ini ial condi ions a e se close o he homogeneous (uns able) limi -
cycle, plus a small andom pe u ba ion. The i s case (simula ion A, Fig 3(a)) e eals a wa e
pa e n ha a els om he le o he igh hemisphe e, whe eas he second (simula ion B,
Fig 3(b)) displays a a elling wa e ha goes om he pa ie al o he on al egion, (see also S1
and S2 Mo ies).
Following [30] we ob ain he di ec ion and speed o he wa e p opaga ion by means o con-
s ained na u al elemen di e en ia ion (see Me hods, and also [30,73]). The esul ing ec o
ield (Fig 3(c) and 3(d)) p o ides be e indica ion on he speci ic ype o pa e n shown by
each simula ion, oge he wi h he wa e p opaga ion speed o each egion. These p opaga ion
speeds a e he e ogeneous in space, as expec ed om he i egula b ain connec ome, wi h al-
ues anging be ween 1 and 8m/s (see Fig 3(e)), in close ag eemen o esul s om non-in asi e
eco dings, bu one o de o magni ude la ge han hose obse ed in in asi e eco dings
[74]. This disag eemen migh be due o he ac ha ou s is a model o in e -co ical wa e
Fig 3. Mul is abili y be ween di e en ypes o pe iodic a elling wa es. (a,b) Di e ence be ween each node’s phase ϕ
j
and he collec i e phase C
co esponding o simula ions wi h p= 230 and �= 50, in he spa ial ep esen a ion o he b ain ne wo k ( om le o igh : on ola e al, supe io , and
on al iews). Panels (a) and (b) co espond o wo di e en ini ial condi ions. (c,d) P opaga ion di ec ion ec o s ζ
j
/z
j
co esponding o he phase
pa e ns o (a) and (b). Colo indica es he p opaga ion speed ν
j
. (e, ) Swa m plo o p opaga ion speed z
j
(e) and local pola iza ion a
j
( ). The ci cles
co espond o indi idual b ain egions. Black squa es show he a e age o e he en i e ne wo k, and e o ba s indica e s anda d de ia ion. (g)
Con ibu ion o each s uc u al connec i i y eigenmode o he g ow h a e o he pe u ba ions in simula ions A and B (see Eq (29) in Me hods).
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bo h he pe iodic and chao ic dynamics. Second, he ampli ude co ela es mildly wi h he
node s eng h a he chao ic egion ( ed squa es in Fig 6( )), bu appea s independen om s
i
close o he onse o oscilla o y ac i i y (blue ci cles in Fig 6( )) Finally, Fig 6(g) shows a nega-
i e co ela ion be ween he phase di e ence o each b ain egion and hei co esponding
node s eng h: Regions wi h lowe inpu end o oscilla e i s , and he hubs end o be he las .
In he pe iodic egime (e.g., p= 320) his ela ion ansla es o a a elling wa e in which oscil-
la ions sp ead om ou e o he inne egion o he b ain (see S4 Mo ie). In he chao ic egime
(p= 170) he ole o he node s eng h on he dynamics is s ill p ominen (blue squa es in Fig
6(g)), al hough much blu ed by he i egula beha io o he sys em, as shown by he la ge
e o ba s (see also S5 Mo ie).
O e all, he non-no malized ne wo k opology adds a new laye o complexi y in he
model, as homogeneous s a es no longe exis . None heless, many ea u es o he dynamics
can be aced back di ec ly o he dis ibu ion o node s eng hs s
i
. Mo eo e , he ag eemen
be ween diag ams in Fig 6(a) and 6(b) and hose o Fig 2( ) and 2(g) indica e ha he onse o
chao ic dynamics is mos ly e ained in he ow-no malized ne wo k opology ~
W.
Spa io empo al chaos in a la ge-scale b ain model wi h nex gene a ion
NMMs
We ha e seen so a ha i egula spa io empo al dynamics a ise in ne wo ks o coupled
NMMs om ans e se ins abili ies o oscilla o y s a es. We expec his o be a gene al mecha-
nism in b ain ne wo ks. In o de o illus a e his ubiqui y, we now analyze a la ge-scale b ain
model composed o coupled nex gene a ion NMM (NG-NMM). These models a e de i ed
om an exac mean- ield heo y o quad a ic in eg a e-and- i e neu ons, and he e o e, he
co esponding i ing a e equa ions can be aced back o he dynamics o single neu ons [17].
He e we conside a py amidal-in e neu onal ne wo k gamma (PING) se up, in which he local
dynamics wi hin each b ain egion p oduces gamma ac i i y h ough he in e play be ween
exci a o y and inhibi o y neu ons (see Me hods, and also [83]). In o de o ob ain a bi u ca ion
diag am, we p oceed as we ha e done abo e o he Jansen sys em: i s we in es iga e he
homogeneous mani old o he sys em, and hen apply he MSF o malism.
The mani old o homogeneous ajec o ies co esponds again o a sel -coupled e sion o
he model (Eq (38) in Me hods). Fig 7(a) shows he bi u ca ion diag am co esponding o
homogeneous ajec o ies o he sys em, using as con ol pa ame e s he ex e nal- o-exci a o y
baseline inpu η
e
and he coupling s eng h �. Fo weak coupling (��1) and low η
e
he sys em
emains in a single low-ac i i y ixed poin (whi e a ea in he plo ) co esponding o a s a e o
asynch onous dynamics. Upon inc easing he cons an ex e nal inpu η
e
, such s eady s a e
gi es aise o as oscilla o y ac i i y (blue-shaded a ea) h ough a supe c i ical Hop bi u ca-
ion ( ed con inuous line). Fo la ge alues o η
e
he ixed poin eco e s s abili y h ough a
subc i ical Hop (dashed ed cu e), gi ing aise o a bis able s a e be ween gamma ac i i y
and asynch ony (o ange-shaded a ea). Fo e en la ge alues o η
e
gamma ac i i y inally an-
ishes h ough a saddle-node o limi cycles (ou side igu e ange).
As i happened wi h he Jansen sys em, inc easing �leads o a change on he onse o oscil-
la o y ac i i y. Following a ypical scena io in oscilla o y sys ems (see, e.g., [84–88]), in a iny
egion o he pa ame e space (see panel (b)) h ee codimension-2 bi u ca ions coexis : a Bog-
dano -Takens (BT), a saddle-node sepa a ix loop (SNSL), and a cusp o saddle nodes. The
Hop bi u ca ion anishes a a Bogdano -Takens (BT) poin , and a saddle-node sepa a ix
loop (SNSL) gi es aise o a SNIC b anch (da k blue line). The e o e, o mos alues o �,
gamma ac i i y a ises h ough a in ini e-pe iod (SNIC) bi u ca ion. Also, an inc ease o he
coupling causes he egion o bis abili y be ween he oscilla o y s a es and he ixed-poin o
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inc ease (o ange-shaded a ea). O e all, homogeneous oscilla o y ac i i y –wi h o wi hou
bis abili y– domina es he bi u ca ion diag am. An example o such homogeneous gamma
ac i i y is displayed in Fig 7(c), co esponding o η
e
= 5 and �= 30 (black iangle in Fig 7(a)).
By applying he MSF o malism on hese oscilla o y s a es, we un eil a egion o ans e se
ins abili y (pink-shaded egion in Fig 7(a)), which eme ges o la ge alues o �. As a esul ,
simula ions ini ialized close o he homogeneous s a e wi hin his pa ame e egion exhibi
i egula spa io empo al pa e ns, as he one depic ed in Fig 7(d) (η
e
= 5, �= 40 i.e, black
squa e in Fig 7(a)). No ice ha he equency o he oscilla ions becomes almos wice as ha
o he (uns able) homogeneous s a e, which in his case is app oxima ely 46Hz. This is a di e -
en ia ing ea u e wi h espec o he Jansen model, in which he e ogeneous ac i i y is as e
han ha o he unde lying homogeneous s a e, bu always close o he ypical alues displayed
by he model.
O e all, he la ge-scale b ain model wi h NG-NMMs displays a mechanism owa ds he
onse o i egula s a es analogous o he Jansen case, despi e he di e en na u e o he wo
models [89]. Fo ins ance, we did no include synap ic dynamics in Eq (36), i.e., neu ons
ecei e ins an aneous del a-like pulses. The onse o spa io empo al chaos shown he e sup-
po s hus he gene ali y o ans e se ins abili i es in he oscilla o y dynamics o coupled
NMMs.
Discussion
The idea ha sys ems composed o simple de e minis ic subuni s can gi e ise o complex spa-
io empo al beha io aces back o he seminal wo k o Alan Tu ing [51]. Tha wo k showed
ha a homogeneous equilib ium in a sys em o di usi ely coupled uni s migh lose s abili y
due o he coupling be ween neighbou ing si es, he eby p oducing spa ial pa e ns. Decades
o esea ch ha e ex ended his simple ye powe ul amewo k o co e a wide ange o possi-
bili ies, including ins abili ies a ising om uni o m oscilla o y s a es [52,53,90] and pa e n
o ma ion in complex ne wo ks [54]. Bo h o hese ex ensions a e embodied in he ield o col-
lec i e synch oniza ion, in which he Mas e S abili y Func ion p o ides a p ope o malism o
analyze he s abili y o homogeneous oscilla o y s a es [58–62]. In b ain ne wo ks, he s abili y
o synch onized s a es has been analyzed only in simpli ied ne wo k opologies [48,50] o by
means o phase- educ ion app oxima ions ha only apply o weak coupling [33,35]. In his
pape we ha e s udied a gene al scena io, ha e eals ans e se ins abili ies o homogeneous
s a es as an ubiqui ous mechanism o he onse o a elling wa es and high-dimensional
chaos in la ge-scale b ain models.
In compu a ional neu oscience, he spon aneous eme gence o pa e ns h ough ins abili-
ies o a uni o m s a e has been emphasized mos ly in he con ex o neu al ields [91–95]. Neu-
al ields a e models de ined in a con inuous spa ial suppo , whe e he synap ic coupling is a
smoo h unc ion o he dis ance be ween egions. Hence, hese ype o models canno cap u e
he ine mac oscopic o ganiza ion o he b ain connec ome ep esen ed by complex ne wo ks,
and a e hus mo e adequa e o model local in a-co ical dynamics [96,97]. None heless, he
assessmen o ans e se ins abili ies is gene al enough o apply o bo h con inuous spa ial sup-
po wi h simpli ied in e ac ion ules and neu al mass models in e ac ing h ough complex
ne wo ks. The main di e ence be ween he wo cases lays on he decomposi ion o he
pe u ba ion ec o . In neu al ields and egula ne wo k opologies [50], as in he Tu ing
amewo k, s abili y analysis o homogeneous s a es is a ained by decomposing a spa ial pe -
u ba ion in Fou ie space. Ins ead, in complex ne wo ks composed o coupled NMM, he
MSF equi es he diagonaliza ion o he s uc u al connec i i y ma ix. In e es ingly, some
s udies show ha spec al analysis o whole-b ain ne wo ks enables he cha ac e iza ion o
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unc ional and es ing b ain ac i i y [98–100]. Ou wo k p o ides hus a ma hema ical ame-
wo k o u he explo e he ela ion be ween s uc u al connec i i y (in e ms e.g. o g aph
spec a) and unc ional ac i i y da a.
An impo an di e ence be ween la ge-scale neu al dynamics and classical pa e n o ma-
ion lies in he na u e o coupling. Classical pa e n o ming sys ems usually in ol e di usi e
coupling, o which he homogeneous mani old o he sys em coincides wi h he dynamics o
he uncoupled uni s [55]. In con as , he long- ange b ain connec ions ep esen ed by s uc-
u al connec i i y ma ices co espond o myelina ed ne e ac s ac oss b ain egions. Hence,
he in e ac ions be ween nodes a e media ed by chemical synapses d i en by he i ing a e o
p e-synap ic egions. As a esul , he homogeneous mani old o he sys em is gi en by a sel -
coupled e sion o he single-node model, and he e o e he coupling s eng h modi ies he
uni o m s a es o he sys em in a non- i ial manne . We ha e shown, o ins ance, ha in
bo h Jansen and nex gene a ion NMMs, he e is a change on he onse o synch onized ac i -
i y om a Hop o a SNIC bi u ca ion upon inc easing he coupling pa ame e �. Addi ionally,
coupling elimina es all o ms o bis abili y in he homogeneous s a es o he Jansen model,
whe eas i enla ges he egion o coexis ence be ween homogeneously a ac ing limi -cycles
and s eady s a es in he NG-NMM.
Mos o he heo y o pa e n o ma ion is based on s udying ins abili ies om ixed poin s.
Al hough we also analyzed his case, we did no ind ins abili ies a ising om s a iona y s a es.
Ins ead, he linea s abili y o oscilla o y solu ions e ealed he ubiqui y o ans e se ins abili-
ies as a ou e o he eme gence o spa io empo al chaos in la ge-scale b ain models. This
mechanism explains he onse o de e minis ic chaos ou lined in p e ious wo ks by means o
di ec nume ical simula ions [44,49]. Mo eo e , he nume ically compu ed Mas e S abili y
Func ions (Fig 2(b)) show ha he onse o uns able modes occu s h ough eigen alues a bi-
a ily close o he ze o eigenmode. This scena io is e y close o ha o he Benjamin-Fei
ins abili y in he Ginzbu g-Landau sys em, which was s udied by Ku amo o as a main ou e o
u bulence in chemical eac ion sys ems [52,53,60].
The eme gence o high-dimensional chaos in la ge-scale b ain ne wo ks is in line wi h
ecen s udies ha highligh he p ominence o u bulen dynamics as a signa u e o heal hy
b ain ac i i y [8,71,101–103]. These complex luc ua ions a e obse ed in blood-oxygen-
le el-dependen (BOLD) signals ob ained om MRI scans, which a e cha ac e ized by slow
(below 1Hz) luc ua ions. These a e in he same ime scale as he chao ic slow modula ions o
alpha hy hms ha we obse ed in ou b ain ne wo k composed o Jansen NMMs. Rema k-
ably, expe imen al s udies ha e also ound a nega i e co ela ion be ween he ampli ude o
alpha hy hms and BOLD ac i i y [104–107]. Thus, we conjec u e ha he chao ic dynamics
gene a ed by ans e se ins abili ies migh cap u e he spa io empo al luc ua ions o BOLD
signals obse ed in eco dings.
Ano he impo an ea u e o he chao ic dynamics in he Jansen model is he coexis ence
o wo equency anges, he a (�5Hz) and alpha (�10Hz), in some egions o he pa ame e
space (see Fig 4(c)–4(e)). These al e na ing bu s s o ac i i y be ween he wo hy hms a e con-
sis en wi h he mul i equency and ansien beha io o oscilla ions in EEG eco dings.
La ge-scale b ain models aim o ace he basic p inciples behind neu al ac i i y. The e o e,
one migh need o accoun o o he sou ces o complexi y no included in his wo k. Fo
ins ance, despi e being a common choice in he li e a u e [30,33,35,44,45,56,57], in some
cases i would be p e e able o use non-no malized opologies. We ha e shown ha ega dless
o his simpli ica ion, complex spa io empo al pa e ns and high-dimensional chaos exis
al eady in he ow-no malized sys em. Mo eo e , he nume ically-de i ed bi u ca ion diag am
o he non-no malized sys em shown in Fig 6(a) and 6(b) quali a i ely ma ches he dynamical
landscape unco e ed by he analysis o he homogeneous s a es o he simpli ied model.
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Addi ionally, he e we ha e conside ed long- ange exci a o y coupling a ge ing only py a-
midal neu ons. The MSF o malism can also be applied in he case whe e long- ange exci a ion
a ge s bo h exci a o y and inhibi o y uni s. A simple explo a ion o his se up using Jansen
NMMs indica ed ha no ans e se ins abili ies eme ged in his si ua ion ( esul s no shown).
This is in ag eemen wi h [50], which shows ha sel -sus ained a eling wa es eme ge in a
ing o Wilson-Cowan uni s when c oss-exci a ion only a ec s exci a o y popula ions bu an-
ish i in e neu ons a e also a ge ed.
Finally, he assump ion o homogeneous dynamics ac oss b ain egions migh be chal-
lenged by p e ious indings ha indica e hie a chical he e ogenei ies in synap ic s eng hs and
ime scales [108,109]. None heless, a deepe unde s anding o ans e se ins abili ies in sim-
pli ied sys ems p o ides a solid g ound o analyze he eme gence and p opaga ion o spa io-
empo al neu al ac i i y in comp ehensi e models including such he e ogenei ies. To his end,
u u e wo k should conside quan i a i e compa isons be ween he dynamics eme ging om
ans e se ins abili ies and dynamical da a om EEG o MRI eco dings. This migh help o
alida e some o he a o emen ioned modeling choices as well as p o iding ele an es ima-
ions o �, usually gi en by i ing he model o a ailable da a (see, e.g. [8,29]).
Ma e ials and me hods
S uc u al connec i i y da a
The s uc u al connec i i y o ou NMM ne wo k has been ob ained om di usion enso
imaging (DTI) da a o 16 heal hy subjec s, collec ed and analyzed in a p e ious s udy [65]. The
di e en b ain egions we e de ined using he AAL90 pa cella ion [64]. A 90 ×90 s uc u al
connec i i y ma ix Wwas ob ained a e aging he connec ome o he 16 subjec s. Fo mo e
de ails abou da a collec ion and p epossessing we e e he eade s o [65]. Fo de ails on he
u he no maliza ion used in mos o his pape , see Sec ion en i led “No malized connec i -
i y” below.
The Jansen NMM model
Single egion. Jansen’s NMM (also known as Jansen-Ri model) desc ibes he dynamical
ac i i y o a co ical column o neu ons [15,16,63]. Following p inciples de i ed om ea lie
wo ks [13,110], he model assumes popula ions o py amidal neu ons (PN) and inhibi o y
in e neu ons (IN), Recu en connec ions a e only p esen in he PN popula ion, whe eas
inhibi o y neu ons solely ecei e inpu s om py amidal neu ons Fo simplici y he model
assumes ha ecu en connec ions wi hin he PN popula ion a e media ed h ough neu ons
ha do no ecei e di ec inpu om inhibi o y neu ons, and can he e o e be in e p e ed as
a hi d independen popula ion, which we call ecu en py amidal neu ons ( PN). Finally,
py amidal neu ons also ecei e exci a o y ex e nal s imuli om o he b ain egions, modelled
h ough a i ing a e a iable I( ).
In he Jansen model he exci a o y and inhibi o y pos -synap ic po en ials (PSP) a e gi en
by
heð Þ ¼ Aa ea Hð Þ
hið Þ ¼ Bb eb Hð Þð5Þ
whe e His he Hea iside s ep unc ion, Aand Ba e he PSP ampli udes, and aand bquan i y
he synap ic ime scales. As a esul , a neu al popula ion ecei ing an exci a o y i ing a e o
( ) gene a es an exci a o y pos -synap ic po en ial (ePSP) desc ibed by he second-o de
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di e en ial equa ion
€
yð Þ ¼ Aa ð Þ 2a_
yð Þ a2yð Þ:ð6Þ
Analogously, a inhibi o y i ing a e gene a es a inhibi o y pos -synap ic po en ial (iPSP)
acco ding o
€
yð Þ ¼ Bb ð Þ 2b_
yð Þ b2yð Þ:ð7Þ
On he o he hand, a popula ion wi h mean memb ane po en ial ( ) gene a es a i ing a e
acco ding o he sigmoid unc ion
Sigmð Þ≔2e0
1þe ð 0 Þ:ð8Þ
The model assumes ha he sel -dynamics o neu ons wi hin a popula ion can be a e aged
ou . Thus, he en i e sys em e ol es as de e mined by he e olu ion o he a e age PSPs o he
di e en popula ions gi en by Eqs (6) and (7). As a esul , he in e ac ions be ween he h ee
popula ions (PN, IN, and PN) lead o he 6-dimensional Jansen model:
_
y0ð Þ ¼ y3ð Þ
_
y1ð Þ ¼ y4ð Þ
_
y2ð Þ ¼ y5ð Þ
_
y3ð Þ ¼ Aa Sigm½y1ð Þ y2ð Þ� 2ay3ð Þ a2y0ð Þ
_
y4ð Þ ¼ Aa Ið ÞþC2Sigm½C1y0ð Þ�g 2ay4ð Þ a2y1ð Þ
_
y5ð Þ ¼ BbC4Sigm½C3y0ð Þ� 2by5ð Þ b2y2ð Þ;
ð9Þ
whe e y
0
accoun s o he ePSP gene a ed by he PNs, y
1
is he sum o he ePSP gene a ed by
he PNs and he ex e nal exci a o y inpu s, and y
2
is he iPSP gene a ed by he INs. Finally,
he mean memb ane po en ial o he py amidal popula ion is gi en by ≔y
1
−y
2
, which we
use as he main obse able h oughou his pape .
Ne wo k model. We conside a ne wo k composed o Nnodes, each ep esen ing a b ain
egion. The dynamics o each node ollows he Jansen model o a co ical column. The e o e,
he 6 equa ions go e ning he dynamics o node i ead [35]
_
y0;ið Þ ¼ y3;ið Þ
_
y1;ið Þ ¼ y4;ið Þ
_
y2;ið Þ ¼ y5;ið Þ
_
y3;ið Þ ¼ Aa Sigm½y1;ið Þ y2;ið Þ� 2ay3;ið Þ a2y0;ið Þ
_
y4;ið Þ ¼ Aa Iið ÞþC2Sigm½C1y0;ið Þ�g 2ay4;ið Þ a2y1;ið Þ
_
y5;ið Þ ¼ BbC4Sigm½C3y0;ið Þ� 2by5;ið Þ b2y2;ið Þ:
ð10Þ
The quan i y I
i
accoun s o he incoming signals om he es o he ne wo k o o he laye s
no ep esen ed in he model. In he b ain model we conside ha he di e en egions a e
coupled only h ough exci a ion, hus inhibi ion ac s only locally. Also, all egions ecei e
an ex e nal inpu om subco ical egions no ep esen ed in ou model, in he o m o a
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common cons an i ing a e p. Al oge he , I
i
( ) akes he o m o he sum o wo independen
e ms:
Iið Þ ¼ pþ�X
N
j¼1
~
Wij Sigm½y1;jð Þ y2;jð Þ� ;ð11Þ
whe e �is he coupling s eng h, and ~
W¼ ð~
wijÞis he ow-no malized s uc u al connec i i y
ma ix (see nex sec ion). We s udy he sys em unde he in luence o he ex e nal d i ing i ing
a e pand he coupling s eng h �, lea ing all he o he pa ame e s ixed, as de ined in Table 1.
As men ioned abo e, om he six a iables ha cha ac e ize he dynamics o each b ain egion,
we moni o he mean memb ane po en ial o he py amidal neu ons,
i
≔y
1,i
−y
2,i
.
No malized connec i i y. Gi en a N×Ns uc u al connec i i y (SC) ma ix W, a
de ailed ma hema ical analysis o sys em (10) is gene ally un easible. In o de o allow o a
semi-analy ical ea men , we conside a no malized e sion o he opology. Such no maliza-
ion is usually employed in la ge-scale b ain models [30,33,35,44,45,56,57]. Le D= (d
ij
) be
a diagonal N×Nma ix whose non-ze o en ies a e he sum o incoming connec ions o each
node:
dij ¼
sii¼j
0i6¼ j
(ð12Þ
whe e
si¼X
N
j¼1
wij ð13Þ
is he in-s eng h o node i. Then we conside he ow-no malized SC ma ix
~
W¼ ð~
wijÞ≔D1Wð14Þ
whose elemen s a e ~
wij ≔wij=si, i.e., ~
Wis ob ained di iding each ow o Wby i s sum. The e-
o e, he sum o he elemen s on each ow equals uni y i.e.,
X
N
j¼1
~
wij ¼1:ð15Þ
Table 1. Pa ame e s o he Jansen sys em.
Pa ame e Meaning Value
AMaximal ampli ude o exci a o y pos -synap ic po en ials 3.25mV
BMaximal ampli ude o inhibi o y pos -synap ic po en ials 22mV
aCha ac e is ic decay ime o ePSP 100s
−1
bCha ac e is ic decay ime o iPSP 50s
−1
C
1
,C
2
,C
3
,C
4
Synap ic s eng h (a e age numbe o synapses) be ween popula ions 135, 108, 33.75, 33.75
e
0
Hal o he maximum i ing a e 2.5Hz
0
Po en ial whe e hal he maximum i ing a e is achie ed 6mV
Neu onal exci abili y 0.56mV
−1
~wij Connec i i y weigh s om da a
pCons an baseline i ing a e o py amidal neu ons no ixed (Hz)
�Coupling s eng h no ixed
h ps://doi.o g/10.1371/jou nal.pcbi.1010781. 001
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Ma ices wi h his no maliza ion a e some imes called igh s ochas ic ma ices [111]. An
impo an p ope y o igh s ochas ic ma ices ha we use in ou analysis is ha hei la ges
eigen alue is exac ly Λ
1
= 1, which co esponds o a uni o m eigen ec o ϕ
(1)
≔(1, . . ., 1)
T
. By
he by he Ge shgo in ci cle heo em [112], all o he eigen alues a e bounded wi hin he uni
ci cle.
Finally, we ema k ha since he o iginal s uc u al connec ome W, is symme ic, so is
he ma ix Z≔D1
2WD1
2. Hence Zis diagonalizable and has eal eigen alues. Using ha
W¼D~
Wwe ob ain
Z≔D1
2~
WD1
2:ð16Þ
The e o e, Zand ~
Wa e simila (in he ma hema ical sense), i.e., hey sha e he same eigen al-
ues. The ac ha he eigen alues o ~
Wa e eal (due o he symme y o W) is no s ic ly neces-
sa y o ca y ou analysis based on he MSF, bu i simpli ies i .
Homogeneous s a es
A ow-no malized connec i i y ma ix ensu es ha all he di e en uni s in he ne wo k
ecei e he same amoun o inpu , al hough dis ibu ed di e en ly ac oss he di e en
nodes. Using such ype o connec i i ies, i is always possible o ind homogeneous o uni o m
solu ions -ei he s a iona y o ime dependen - in which all nodes o he ne wo k beha e
iden ically.
Le y
0
,. . .,y
5
be he a iables ha cha ac e ize he dynamics o each b ain egion in a
homogeneous s a e. Imposing hen y
m,i
( ) = y
m
( ) o m= 0, . . ., 5 and i= 1, . . .,None inds
ha he incoming inpu o each b ain egion in (10) eads
Iið Þ ¼ pþ�X
N
j¼1
~
wij Sigm½y1;jð Þ y2;jð Þ� ¼ pþ�Sigm½y1ð Þ y2ð Þ� ;ð17Þ
hus i does no depend on he node index ianymo e. Replacing his exp ession o he inpu in
he Jansen model (10) one inds ha , in a homogeneous s a e, he equa ions o he e olu ion
o each ne wo k node ead
_
y0ð Þ ¼ y3ð Þ
_
y1ð Þ ¼ y4ð Þ
_
y2ð Þ ¼ y5ð Þ
_
y3ð Þ ¼ Aa Sigm½y1ð Þ y2ð Þ� 2ay3ð Þ a2y0ð Þ
_
y4ð Þ ¼ Aa pþ�Sigm½y1ð Þ y2ð Þ�þC2Sigm½C1y0ð Þ�g 2ay4ð Þ a2y1ð Þ
_
y5ð Þ ¼ BbC4Sigm½C3y0ð Þ� 2by5ð Þ b2y2ð Þ:
ð18Þ
The e o e, his low-dimensional sys em de e mines all homogeneous s a es o he coupled sys-
em (10). Mo eo e , his sys em also e ains he s abili y o such homogeneous s a es subjec o
uni o m pe u ba ions, i.e., pe u ba ions ha ac iden ically a each b ain egion, and he e-
o e do no change he homogeneous cha ac e o he ajec o ies. In o he wo ds, Eq (18)
de ine an in a ian mani old o Eq (10). None heless, s able s a es in he homogeneous in a i-
an mani old migh s ill be uns able o he e ogeneous pe u ba ions, i.e., pe u ba ions ans-
e se o he mani old.
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T ans e se s abili y
The ollowing s abili y analysis is common in he s udy o dynamical sys ems in complex ne -
wo ks, specially (bu no only), in he con ex o di usi e coupling [54,68,69]. The same ech-
nique can be applied o bo h, homogeneous ixed poin s and homogeneous limi -cycles. In he
la e case i is known as he Mas e S abili y Func ion (MSF) [58] (see also [59,61] o in o-
duc o y e iews). In his sec ion we explain such s abili y analysis on he Jansen model, bu i
can be easily ex ended o o he sys ems. We use bold symbols o Nand 6Ndimensional ec-
o s and ma ices, and egula symbols o 6-dimensional ec o s, 6-dimensional ma ices,
and scala quan i ies. All scala quan i ies excep he ime and he coupling s eng h �ha e a
subsc ip .
Le yð0Þ≔ðyð0Þ
0;. . . ;yð0Þ
5ÞTbe a solu ion o (18). Then, he 6N-dimensional ec o
yð0Þ¼ ðyð0Þ
0;...;yð0Þ
0;. . .
z}|{
N
;yð0Þ
0;. . . ;yð0Þ
5ÞTð19Þ
is a homogeneous solu ion o (10), ei he s a iona y o pe iodic. Le us conside an a bi a y
small pe u ba ion
δy¼ ðdy0;1;...;dy5;1;. . . ;dy0;N;. . . ;dy5;NÞTð20Þ
whe e each componen δy
k,j
is he pe u ba ion ac ing on he a iable ko node jin he sys em
(10). Expanding he eloci y ields o (10) up o i s o de a ound y
(0)
one ob ains he linea
e olu ion o he pe u ba ion ec o as
d
d δyð Þ ¼ Jðyð0ÞÞδyð Þð21Þ
whe e Jis he ull 6N×6NJacobian o sys em (10), e alua ed a y
(0)
. No ice ha Jcan be w i -
en as
Jðyð0ÞÞ ¼ Jðyð0ÞÞ�INþ�Kðyð0ÞÞ� ~
Wð22Þ
whe e �deno es he K onecke p oduc , Jis he 6 ×6 Jacobian o he uncoupled Jansen Eq
(9),I
N
is he N×Niden i y ma ix, and K= (k
ij
) is a 6 ×6 ma ix de ined by
kij ¼
Aa Sigm0ðy1y2Þi¼5;j¼2
Aa Sigm0ðy1y2Þi¼5;j¼3
0o he wise :
8
>
>
>
<
>
>
>
:ð23Þ
One could, in p inciple, e alua e nume ically he eigen alues and eigen ec o s o his Jaco-
bian in o de o ob ain he s abili y p ope ies o he sys em. None heless, he e is a simple
and mo e in o ma i e app oach based on exp essing he pe u ba ion ec o δyin an adequa e
coo dina e sys em.
Le ΦðaÞ¼ ð�ðaÞ
1;...; �ðaÞ
NÞTbe a no malized eigen ec o o ~
Wassocia ed wi h he eigen alue
Λ
α
o α= 1, . . .,N, so ha
~
WΦðaÞ¼LaΦðaÞ:ð24Þ
The se o eigen ec o s {Φ
(1)
,. . .,Φ
(N)
} cons i u e a basis o he ec o space RN, hus we can
exp ess he pe u ba ion δyo he homogeneous solu ion as a linea combina ion o such
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ec o s. In his new basis, he pe u ba ion ac ing on a iable ko he j h node eads
dyk;jð Þ ¼ X
N
a¼1
uðaÞ
kð Þ�ðaÞ
jð25Þ
whe e uðaÞð Þ ¼ ðuðaÞ
0ð Þ;...;uðaÞ
5ð ÞÞTa e he coo dina es o he pe u ba ion ec o exp essed
in he new basis. This can also be exp essed in ec o ial o m using he K onecke ope a o �
as
δyð Þ ¼ X
N
a¼1
uðaÞð Þ�ΦðaÞ:ð26Þ
Now i is necessa y o pe o m some calcula ions using he enso p oduc �. Fo simplici y
we d op some dependences: J=J(y
(0)
), K=K(y
(0)
), and u
(α)
=u
(α)
( ). Applying he change o
coo dina es on Eq (21) and using Eq (22), we ha e ha
d
d δy¼d
d X
N
a¼1
uðaÞ�ΦðaÞ
¼X
N
a¼1
d
d uðaÞ
� ��ΦðaÞ
¼ ðJ�INþ�K�~
WÞX
N
a¼1
uðaÞ�ΦðaÞ
¼ ðJ�INÞX
N
a¼1
uðaÞ�ΦðaÞ
!þ�ðK�~
WÞX
N
a¼1
uðaÞ�ΦðaÞ
!
¼X
N
a¼1ðJ�INÞðuðaÞ�ΦðaÞÞþ�X
N
a¼1ðK�~
WÞðuðaÞ�ΦðaÞÞ
¼X
N
a¼1ðJuðaÞÞ�ðINΦðaÞÞþ�X
N
a¼1ðKuðaÞÞ�ð ~
WFðaÞÞ
¼X
N
a¼1ðJuðaÞÞ�ΦðaÞþ�X
N
a¼1ðKuðaÞÞ�ðLaΦðaÞÞ
¼X
N
a¼1ðJuðaÞÞ�ΦðaÞþ�X
N
a¼1ðLaKuðaÞÞ�ΦðaÞ
¼X
N
a¼1ðJþ�LaKÞuðaÞ�ΦðaÞ;
ð27Þ
whe e we ha e used he diagonaliza ion o he connec i i y ma ix Eq (24). Now, making
use o he linea independence o he eigen ec o s ΦðaÞgN
a¼1one ob ains ha he e olu ion o
u
(α)
( ) becomes independen o each α= 1, . . .,N h ough he ela ion
_
uðaÞ¼ ðJþ�LaKÞuðaÞ;ð28Þ
whe e Jðyð0Þ;LÞ≔Jþ�LaKis a amily o 6 ×6 Jacobians ha depend on he homogeneous
s a e o he sys em y
(0)
and on he s uc u al connec i i y eigen alues Λ
α
. I y
(0)
is a ixed poin ,
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T ans e se ins abili ies in la ge-scale b ain ne wo ks
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1010781 Ap il 12, 2023 24 / 34
he s abili y o he homogeneous solu ion simpli ies o s udying he eigen alues (and eigen ec-
o s) o he Jacobian J, which is a unc ion o he connec i i y ma ix eigen alues Λ
α
. I y
(0)
=
y
(0)
( ) is a pe iodic solu ion, hen Jis a pe iodic ma ix and Floque heo y applies [72]. In
his con ex , he g ow h a e o a pe u ba ion ac ing a he limi -cycle solu ion is de e mined
by he eal pa o he Floque exponen s co esponding o J, which mus be de e mined
nume ically. P ac ically speaking, o each alue o Λ
α
, we compu e he eal pa o he la ges
Floque exponen o Jas he la ges Lyapuno exponen o he linea sys em Eq (28). The
code is a ailable in gi hub (www.gi hub.com/pclus/ ans e se-ins abili ies). Finally, in o de o
analyze he con ibu ion o each s uc u al eigenmode αon he g ow h a e o a speci ic pe u -
ba ion ec o δywe compu e he quan i y
ca¼makuðaÞk:ð29Þ
Lyapuno exponen s
Lyapuno exponen s (LE) p o ide he g ow h a e o small pe u ba ions ac ing on a ime-
e ol ing ajec o y o a dynamical sys em [77,113]. To compu e hese quan i ies in p ac ice we
employ a dynamical algo i hm based on QR-decomposi ions, using Householde e lec ions
[77]. The algo i hm is embedded in he a ailable so wa e (www.gi hub.com/pclus/ ans e se-
ins abili ies). The e a e as many Lyapuno exponen s as sys em dimensions, and hey a e usu-
ally so ed om la ges o smalles : λ
1
�λ
2
�λ
3
�. . . �λ
6N
. Using he wo la ges Lyapuno
exponen s λ
1
and λ
2
, we can classi y he sys em s a e in 5 ypes:
• Fixed poin s, co esponding o bo h LE being nega i e (λ
1
,λ
2
<0).
• Pe iodic dynamics, iden i ied by a ze o LE and he es being nega i e (λ
1
= 0, λ
2
<0).
• Quasipe iodic dynamics, i.e., a egime in which he sys em e ol es wi h a leas wo incom-
mensu a e cha ac e is ic equencies. In his case bo h LE a e ze o (λ
1
= 0, λ
2
= 0).
• Chao ic dynamics, wi h a single posi i e LE (λ
1
>0, λ
2
�0).
• Hype chaos, wi h mo e han one posi i e LE (λ
1
,λ
2
>0).
Since he Lyapuno exponen s a e compu ed nume ically, we need o impose a h eshold o
disce n be ween ze o and non-ze o alues. We ound ha a alue o |λ
k
|<10
−4
was a eason-
able cu -o .
Ano he use ul applica ion o Lyapuno exponen s is o compu e he dimensionali y o
he a ac o , which is ac al in chao ic s a es. The Kaplan-Yo ke o mula [80] p o ides an
app oxima ion o he ac al dimension o an a ac o as
DKY ¼jþPj
i¼1li
jljþ1j;ð30Þ
whe e jis he LE o which
X
j
i¼1
li�0and X
jþ1
i¼1
li<0:ð31Þ
PLOS COMPUTATIONAL BIOLOGY
T ans e se ins abili ies in la ge-scale b ain ne wo ks
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