Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: “simple” models, “Hodge-coupled” models, and “order-coupled” (Dirac) models. Our framework is based on topology and discrete differential geometry, as well as gradient systems and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.

1
M.
Barahona
and
L. M.
Pecora
, “
Synchronization in small-world systems
,”
Phys. Rev. Lett.
89
,
054101
(
2002
).
2
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Sciences
(
Cambridge University Press
,
2003
).
3
S. H.
Strogatz
,
Sync: The Emerging Science of Spontaneous Order
(
Penguin UK
,
2004
).
4
M.
Breakspear
,
S.
Heitmann
, and
A.
Daffertshofer
, “
Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model
,”
Front. Hum. Neurosci.
4
,
190
(
2010
).
5
S. H.
Strogatz
, “Spontaneous synchronization in nature,” in Proceedings of International Frequency Control Symposium (IEEE, 1997), pp. 2–4.
6
M.
Rohden
,
A.
Sorge
,
M.
Timme
, and
D.
Witthaut
, “
Self-organized synchronization in decentralized power grids
,”
Phys. Rev. Lett.
109
,
064101
(
2012
).
7
T.
Nishikawa
and
A. E.
Motter
, “
Comparative analysis of existing models for power-grid synchronization
,”
New J. Phys.
17
,
015012
(
2015
).
8
Z.
Néda
,
E.
Ravasz
,
T.
Vicsek
,
Y.
Brechet
, and
A.-L.
Barabási
, “
Physics of the rhythmic applause
,”
Phys. Rev. E
61
,
6987
(
2000
).
9
Y.
Kuramoto
, “Self-entrainment of a population of coupled non-linear oscillators,” in International Symposium on Mathematical Problems in Theoretical Physics, edited by H. Araki (Springer, Berlin, 1975), pp. 420–422.
10
A.
Arenas
,
A.
Díaz-Guilera
,
J.
Kurths
,
Y.
Moreno
, and
C.
Zhou
, “
Synchronization in complex networks
,”
Phys. Rep.
469
,
93
153
(
2008
).
11
A.
Jadbabaie
,
N.
Motee
, and
M.
Barahona
, “On the stability of the Kuramoto model of coupled nonlinear oscillators,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 5, pp. 4296–4301.
12
D. J.
Watts
and
S. H.
Strogatz
, “
Collective dynamics of “small-world” networks
,”
Nature
393
,
440
442
(
1998
).
13
F.
Battiston
,
G.
Cencetti
,
I.
Iacopini
,
V.
Latora
,
M.
Lucas
,
A.
Patania
,
J.-G.
Young
, and
G.
Petri
, “
Networks beyond pairwise interactions: Structure and dynamics
,”
Phys. Rep.
874
,
1
92
(
2020
).
14
F.
Battiston
,
E.
Amico
,
A.
Barrat
,
G.
Bianconi
,
G. F.
de Arruda
,
B.
Franceschiello
,
I.
Iacopini
,
S.
Kéfi
,
V.
Latora
,
Y.
Moreno
,
M. M.
Murray
,
T. P.
Peixoto
,
F.
Vaccarino
, and
G.
Petri
, “
The physics of higher-order interactions in complex systems
,”
Nat. Phys.
17
,
1093
1098
(
2021
).
15
C.
Bick
,
E.
Gross
,
H. A.
Harrington
, and
M. T.
Schaub
, “
What are higher-order networks?
,”
SIAM Rev.
65
,
686
731
(
2023
).
16
S.
Yu
,
H.
Yang
,
H.
Nakahara
,
G.
Santos
,
D.
Nikolić
, and
D.
Plenz
, “
Higher-order interactions characterized in cortical activity
,”
J. Neurosci.
31
,
17514
17526
(
2011
).
17
A.
Patania
,
G.
Petri
, and
F.
Vaccarino
, “
The shape of collaborations
,”
EPJ Data Sci.
6
,
1
16
(
2017
).
18
A. R.
Benson
,
R.
Abebe
,
M. T.
Schaub
,
A.
Jadbabaie
, and
J.
Kleinberg
, “
Simplicial closure and higher-order link prediction
,”
Proc. Natl. Acad. Sci. U.S.A.
115
,
E11221
E11230
(
2018
).
19
J. L.
Juul
,
A. R.
Benson
, and
J.
Kleinberg
, “
Hypergraph patterns and collaboration structure
,”
Front. Phys.
11
,
1301994
(
2024
).
20
J.
Grilli
,
G.
Barabás
,
M. J.
Michalska-Smith
, and
S.
Allesina
, “
Higher-order interactions stabilize dynamics in competitive network models
,”
Nature
548
,
210
213
(
2017
).
21
M.
Mayfield
and
D.
Stouffer
, “
Higher-order interactions capture unexplained complexity in diverse communities
,”
Nat. Ecol. Evol.
1
,
0062
(
2017
).
22
A.
Sanchez-Gorostiaga
,
D.
Bajić
,
M. L.
Osborne
,
J. F.
Poyatos
, and
A.
Sanchez
, “
High-order interactions distort the functional landscape of microbial consortia
,”
PLoS Biol.
17
,
e3000550
(
2019
).
23
F.
Baccini
,
F.
Geraci
, and
G.
Bianconi
, “
Weighted simplicial complexes and their representation power of higher-order network data and topology
,”
Phys. Rev. E
106
,
034319
(
2022
).
24
G.
Petri
,
M.
Scolamiero
,
I.
Donato
, and
F.
Vaccarino
, “
Topological strata of weighted complex networks
,”
PLoS One
8
,
e66506
(
2013
).
25
G.
Petri
,
P.
Expert
,
F.
Turkheimer
,
R.
Carhart-Harris
,
D.
Nutt
,
P. J.
Hellyer
, and
F.
Vaccarino
, “
Homological scaffolds of brain functional networks
,”
J. R. Soc. Interface
11
,
20140873
(
2014
).
26
D.
Horak
and
J.
Jost
, “
Spectra of combinatorial Laplace operators on simplicial complexes
,”
Adv. Math.
244
,
303
336
(
2013
).
27
C.
Kuehn
and
C.
Bick
, “
A universal route to explosive phenomena
,”
Sci. Adv.
7
,
eabe3824
(
2021
).
28
G.
Ferraz de Arruda
,
G.
Petri
,
P. M.
Rodriguez
, and
Y.
Moreno
, “
Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs
,”
Nat. Commun.
14
,
1375
(
2023
).
29
M. T.
Schaub
,
A. R.
Benson
,
P.
Horn
,
G.
Lippner
, and
A.
Jadbabaie
, “
Random walks on simplicial complexes and the normalized Hodge 1-Laplacian
,”
SIAM Rev.
62
,
353
391
(
2020
).
30
M. T.
Schaub
and
S.
Segarra
, “Flow smoothing and denoising: Graph signal processing in the edge-space,” in 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP) (IEEE, 2018), pp. 735–739.
31
T.
Carletti
,
F.
Battiston
,
G.
Cencetti
, and
D.
Fanelli
, “
Random walks on hypergraphs
,”
Phys. Rev. E
101
,
022308
(
2020
).
32
A. P.
Millán
,
R.
Ghorbanchian
,
N.
Defenu
,
F.
Battiston
, and
G.
Bianconi
, “
Local topological moves determine global diffusion properties of hyperbolic higher-order networks
,”
Phys. Rev. E
104
,
054302
(
2021
).
33
L.
Neuhäuser
,
A.
Mellor
, and
R.
Lambiotte
, “
Multibody interactions and nonlinear consensus dynamics on networked systems
,”
Phys. Rev. E
101
,
032310
(
2020
).
34
L.
DeVille
, “
Consensus on simplicial complexes: Results on stability and synchronization
,”
Chaos
31
,
023137
(
2021
).
35
I.
Iacopini
,
G.
Petri
,
A.
Baronchelli
, and
A.
Barrat
, “
Group interactions modulate critical mass dynamics in social convention
,”
Commun. Phys.
5
,
64
(
2022
).
36
M.
Lucas
,
I.
Iacopini
,
T.
Robiglio
,
A.
Barrat
, and
G.
Petri
, “
Simplicially driven simple contagion
,”
Phys. Rev. Res.
5
,
013201
(
2023
).
37
I.
Iacopini
,
G.
Petri
,
A.
Barrat
, and
V.
Latora
, “
Simplicial models of social contagion
,”
Nat. Commun.
10
,
2485
(
2019
).
38
G.
Ferraz de Arruda
,
M.
Tizzani
, and
Y.
Moreno
, “
Phase transitions and stability of dynamical processes on hypergraphs
,”
Commun. Phys.
4
,
24
(
2021
).
39
S.
Chowdhary
,
A.
Kumar
,
G.
Cencetti
,
I.
Iacopini
, and
F.
Battiston
, “
Simplicial contagion in temporal higher-order networks
,”
J. Phys. Complex
2
,
035019
(
2021
).
40
G.
St-Onge
,
H.
Sun
,
A.
Allard
,
L.
Hébert-Dufresne
, and
G.
Bianconi
, “
Universal nonlinear infection kernel from heterogeneous exposure on higher-order networks
,”
Phys. Rev. Lett.
127
,
158301
(
2021
).
41
G.
St-Onge
,
V.
Thibeault
,
A.
Allard
,
L. J.
Dubé
, and
L.
Hébert-Dufresne
, “
Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks
,”
Phys. Rev. E
103
,
032301
(
2021
).
42
G.
St-Onge
,
I.
Iacopini
,
V.
Latora
,
A.
Barrat
,
G.
Petri
,
A.
Allard
, and
L.
Hébert-Dufresne
, “
Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs
,”
Commun. Phys.
5
,
25
(
2022
).
43
H.
Sun
,
F.
Radicchi
,
J.
Kurths
, and
G.
Bianconi
, “
The dynamic nature of percolation on networks with triadic interactions
,”
Nat. Commun.
14
,
1308
(
2023
).
44
G.
Bianconi
,
I.
Kryven
, and
R. M.
Ziff
, “
Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
,”
Phys. Rev. E
100
,
062311
(
2019
).
45
H.
Sun
,
R. M.
Ziff
, and
G.
Bianconi
, “
Renormalization group theory of percolation on pseudofractal simplicial and cell complexes
,”
Phys. Rev. E
102
,
012308
(
2020
).
46
U.
Alvarez-Rodriguez
,
F.
Battiston
,
G. F.
de Arruda
,
Y.
Moreno
,
M.
Perc
, and
V.
Latora
, “
Evolutionary dynamics of higher-order interactions in social networks
,”
Nat. Hum. Behav.
5
,
586
595
(
2021
).
47
P. S.
Skardal
and
A.
Arenas
, “
Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes
,”
Phys. Rev. Lett.
122
,
248301
(
2019
).
48
P. S.
Skardal
,
L.
Arola-Fernández
,
D.
Taylor
, and
A.
Arenas
, “
Higher-order interactions can better optimize network synchronization
,”
Phys. Rev. Res.
3
,
043193
(
2021
).
49
M.
Lucas
,
G.
Cencetti
, and
F.
Battiston
, “
Multiorder Laplacian for synchronization in higher-order networks
,”
Phys. Rev. Res.
2
,
033410
(
2020
).
50
E.
Gengel
,
E.
Teichmann
,
M.
Rosenblum
, and
A.
Pikovsky
, “
High-order phase reduction for coupled oscillators
,”
J. Phys. Complex
2
,
015005
(
2020
).
51
M. H.
Matheny
,
J.
Emenheiser
,
W.
Fon
,
A.
Chapman
,
A.
Salova
,
M.
Rohden
,
J.
Li
,
M.
Hudoba de Badyn
,
M.
Pósfai
,
L.
Duenas-Osorio
, and
M.
Mesbahi
, “
Exotic states in a simple network of nanoelectromechanical oscillators
,”
Science
363
,
eaav7932
(
2019
).
52
Y.
Zhang
,
M.
Lucas
, and
F.
Battiston
, “
Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
,”
Nat. Commun.
14
,
1605
(
2023
).
53
L. V.
Gambuzza
,
F.
Di Patti
,
L.
Gallo
,
S.
Lepri
,
M.
Romance
,
R.
Criado
,
M.
Frasca
,
V.
Latora
, and
S.
Boccaletti
, “
Stability of synchronization in simplicial complexes
,”
Nat. Commun.
12
,
1255
(
2021
).
54
S.
Kundu
and
D.
Ghosh
, “
Higher-order interactions promote chimera states
,”
Phys. Rev. E
105
,
L042202
(
2022
).
55
C.
Bick
,
P.
Ashwin
, and
A.
Rodrigues
, “
Chaos in generically coupled phase oscillator networks with nonpairwise interactions
,”
Chaos
26
,
094814
(
2016
).
56
C.
Bick
, “
Heteroclinic dynamics of localized frequency synchrony: Heteroclinic cycles for small populations
,”
J. Nonlinear Sci.
29
,
2571
2600
(
2019
).
57
C.
Bick
and
A.
Lohse
, “
Heteroclinic dynamics of localized frequency synchrony: Stability of heteroclinic cycles and networks
,”
J. Nonlinear Sci.
29
,
2547
2570
(
2019
).
58
S.
Adhikari
,
J. G.
Restrepo
, and
P. S.
Skardal
, “
Synchronization of phase oscillators on complex hypergraphs
,”
Chaos
33
,
033116
(
2023
).
59
K.
Kovalenko
,
X.
Dai
,
K.
Alfaro-Bittner
,
A.
Raigorodskii
,
M.
Perc
, and
S.
Boccaletti
, “
Contrarians synchronize beyond the limit of pairwise interactions
,”
Phys. Rev. Lett.
127
,
258301
(
2021
).
60
I.
León
and
D.
Pazó
, “
Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation
,”
Phys. Rev. E
100
,
012211
(
2019
).
61
C. C.
Gong
and
A.
Pikovsky
, “
Low-dimensional dynamics for higher-order harmonic, globally coupled phase-oscillator ensembles
,”
Phys. Rev. E
100
,
062210
(
2019
).
62
A. P.
Millán
,
J. J.
Torres
, and
G.
Bianconi
, “
Explosive higher-order Kuramoto dynamics on simplicial complexes
,”
Phys. Rev. Lett.
124
,
218301
(
2020
).
63
L. J.
Grady
and
J. R.
Polimeni
,
Discrete Calculus: Applied Analysis on Graphs for Computational Science
(
Springer Science & Business Media
,
2010
).
64
L.
Calmon
,
J. G.
Restrepo
,
J. J.
Torres
, and
G.
Bianconi
, “
Dirac synchronization is rhythmic and explosive
,”
Commun. Phys.
5
,
1
17
(
2022
).
65
L.
Calmon
,
S.
Krishnagopal
, and
G.
Bianconi
, “
Local Dirac synchronization on networks
,”
Chaos
33
,
033117
(
2023
).
66
G.
Bianconi
, “
The topological Dirac equation of networks and simplicial complexes
,”
J. Phys. Complex
2
,
035022
(
2021
).
67
A.
Arnaudon
,
R. L.
Peach
,
G.
Petri
, and
P.
Expert
, “
Connecting Hodge and Sakaguchi–Kuramoto through a mathematical framework for coupled oscillators on simplicial complexes
,”
Commun. Phys.
5
,
1
12
(
2022
).
68
J. A.
Acebrón
,
L. L.
Bonilla
,
C. J.
Pérez Vicente
,
F.
Ritort
, and
R.
Spigler
, “
The Kuramoto model: A simple paradigm for synchronization phenomena
,”
Rev. Mod. Phys.
77
,
137
185
(
2005
).
69
A.
Pikovsky
and
M.
Rosenblum
, “
Dynamics of globally coupled oscillators: Progress and perspectives
,”
Chaos
25
,
097616
(
2015
).
70
F. A.
Rodrigues
,
T. K. D.
Peron
,
P.
Ji
, and
J.
Kurths
, “
The Kuramoto model in complex networks
,”
Phys. Rep.
610
,
1
98
(
2016
).
71
A. N.
Hirani
,
Discrete Exterior Calculus
(
California Institute of Technology
,
2003
).
72
L.-H.
Lim
, “
Hodge Laplacians on graphs
,”
SIAM Rev.
62
,
685
715
(
2020
).
73
R. W.
Ghrist
,
Elementary Applied Topology
(
Createspace Seattle
,
2014
), Vol. 1.
74
E.
Nijholt
and
L.
DeVille
, “
Dynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces
,”
Chaos
32
,
093131
(
2022
).
75
T.
Carletti
,
L.
Giambagli
, and
G.
Bianconi
, “
Global topological synchronization on simplicial and cell complexes
,”
Phys. Rev. Lett.
130
,
187401
(
2023
).
76
H.
Daido
, “
Generic scaling at the onset of macroscopic mutual entrainment in limit-cycle oscillators with uniform all-to-all coupling
,”
Phys. Rev. Lett.
73
,
760
(
1994
).
77
D.
Hansel
,
G.
Mato
, and
C.
Meunier
, “
Clustering and slow switching in globally coupled phase oscillators
,”
Phys. Rev. E
48
,
3470
(
1993
).
78
H.
Daido
, “
Order function and macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators
,”
Prog. Theor. Phys.
88
,
1213
1218
(
1992
).
79
H.
Daido
, “
Critical conditions of macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators
,”
Prog. Theor. Phys.
89
,
929
934
(
1993
).
80
H.
Daido
, “
A solvable model of coupled limit-cycle oscillators exhibiting partial perfect synchrony and novel frequency spectra
,”
Physica D
69
,
394
403
(
1993
).
81
H.
Daido
, “
Multi-branch entrainment and multi-peaked order-functions in a phase model of limit-cycle oscillators with uniform all-to-all coupling
,”
J. Phys. A: Math. Gen.
28
,
L151
(
1995
).
82
H.
Daido
, “
Multibranch entrainment and scaling in large populations of coupled oscillators
,”
Phys. Rev. Lett.
77
,
1406
(
1996
).
83
H.
Daido
, “
Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function
,”
Physica D
91
,
24
66
(
1996
).
84
P.
Ashwin
,
G.
Orosz
,
J.
Wordsworth
, and
S.
Townley
, “
Dynamics on networks of cluster states for globally coupled phase oscillators
,”
SIAM J. Appl. Dyn. Syst.
6
,
728
758
(
2007
).
85
M.
Komarov
and
A.
Pikovsky
, “
The Kuramoto model of coupled oscillators with a bi-harmonic coupling function
,”
Physica D
289
,
18
31
(
2014
).
86
M.
Komarov
and
A.
Pikovsky
, “
Dynamics of multifrequency oscillator communities
,”
Phys. Rev. Lett.
110
,
134101
(
2013
).
87
V.
Chandrasekar
,
M.
Manoranjani
, and
S.
Gupta
, “
Kuramoto model in the presence of additional interactions that break rotational symmetry
,”
Phys. Rev. E
102
,
012206
(
2020
).
88
X.
Jiang
,
L.-H.
Lim
,
Y.
Yao
, and
Y.
Ye
, “
Statistical ranking and combinatorial Hodge theory
,”
Math. Program.
127
,
203
244
(
2011
).
89
L.
Eldén
, “
A weighted pseudoinverse, generalized singular values, and constrained least squares problems
,”
BIT
22
,
487
502
(
1982
).
90
D.
Manik
,
M.
Timme
, and
D.
Witthaut
, “
Cycle flows and multistability in oscillatory networks
,”
Chaos
27
,
083123
(
2017
).
91
F.
Dörfler
and
F.
Bullo
, “
On the critical coupling for Kuramoto oscillators
,”
SIAM J. Appl. Dyn. Syst.
10
,
1070
1099
(
2011
).
92
F.
Dörfler
and
F.
Bullo
, “
Synchronization in complex networks of phase oscillators: A survey
,”
Automatica
50
,
1539
1564
(
2014
).
93
F.
Dörfler
and
F.
Bullo
, “Exploring synchronization in complex oscillator networks,” in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC) (IEEE, 2012), pp. 7157–7170.
94
M.
Black
and
W.
Maxwell
, “Effective resistance and capacitance in simplicial complexes and a quantum algorithm,” in 32nd International Symposium on Algorithms and Computation (ISAAC 2021), Leibniz International Proceedings in Informatics (LIPIcs), edited by H.-K. Ahn and K. Sadakane (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, 2021), Vol. 212, pp. 31:1–31:27.
95
R.
Ghorbanchian
,
J. G.
Restrepo
,
J. J.
Torres
, and
G.
Bianconi
, “
Higher-order simplicial synchronization of coupled topological signals
,”
Commun. Phys.
4
,
120
(
2021
).
96
H.
Hong
and
S. H.
Strogatz
, “
Conformists and contrarians in a Kuramoto model with identical natural frequencies
,”
Phys. Rev. E
84
,
046202
(
2011
).
97
H.
Sakaguchi
and
Y.
Kuramoto
, “
A soluble active rotater model showing phase transitions via mutual entertainment
,”
Prog. Theor. Phys.
76
,
576
581
(
1986
).
98
C.
Anné
and
N.
Torki-Hamza
, “
The Gauss-Bonnet operator of an infinite graph
,”
Anal. Math. Phys.
5
,
137
159
(
2015
).
99
S.
Lloyd
,
S.
Garnerone
, and
P.
Zanardi
, “
Quantum algorithms for topological and geometric analysis of data
,”
Nat. Commun.
7
,
10138
(
2016
).
100
L.
Calmon
,
M. T.
Schaub
, and
G.
Bianconi
, “
Dirac signal processing of higher-order topological signals
,”
New J. Phys.
25
,
093013
(
2023
).
101
J.
Wee
,
G.
Bianconi
, and
K.
Xia
, “
Persistent Dirac for molecular representation
,”
Sci. Rep.
13
,
11183
(
2023
).
102
M. L.
Kringelbach
and
G.
Deco
, “
Brain states and transitions: Insights from computational neuroscience
,”
Cell Rep.
32
,
108128
(
2020
).
103
G.
Buzsaki
,
Rhythms of the Brain
(
Oxford University Press
,
2006
).
104
L.-D.
Lord
,
T.
Carletti
,
H.
Fernandes
,
F. E.
Turkheimer
, and
P.
Expert
, “
Altered dynamical integration/segregation balance during anesthesia-induced loss of consciousness
,”
Front. Netw. Physiol.
3
,
1279646
(
2023
).
105
J.
Cabral
,
E.
Hugues
,
O.
Sporns
, and
G.
Deco
, “
Role of local network oscillations in resting-state functional connectivity
,”
NeuroImage
57
,
130
139
(
2011
).
106
L.-D.
Lord
,
P.
Expert
,
J. F.
Huckins
, and
F. E.
Turkheimer
, “
Cerebral energy metabolism and the brain’s functional network architecture: An integrative review
,”
J. Cereb. Blood Flow Metab.
33
,
1347
1354
(
2013
).
107
D.
Cumin
and
C.
Unsworth
, “
Generalising the Kuramoto model for the study of neuronal synchronisation in the brain
,”
Physica D
226
,
181
196
(
2007
).
108
R.
Schmidt
,
K. J.
LaFleur
,
M. A.
de Reus
,
L. H.
van den Berg
, and
M. P.
van den Heuvel
, “
Kuramoto model simulation of neural hubs and dynamic synchrony in the human cerebral connectome
,”
BMC Neurosci.
16
,
1
13
(
2015
).
109
M.
Pope
,
M.
Fukushima
,
R. F.
Betzel
, and
O.
Sporns
, “
Modular origins of high-amplitude cofluctuations in fine-scale functional connectivity dynamics
,”
Proc. Natl. Acad. Sci. U.S.A.
118
,
e2109380118
(
2021
).
110
M.
Pope
,
C.
Seguin
,
T. F.
Varley
,
J.
Faskowitz
, and
O.
Sporns
, “Co-evolving dynamics and topology in a coupled oscillator model of resting brain function,” bioRxiv, 2023-01 (2023).
111
F.
Váša
,
M.
Shanahan
,
P. J.
Hellyer
,
G.
Scott
,
J.
Cabral
, and
R.
Leech
, “
Effects of lesions on synchrony and metastability in cortical networks
,”
NeuroImage
118
,
456
467
(
2015
).
112
E.
Schneidman
,
M. J.
Berry
,
R.
Segev
, and
W.
Bialek
, “
Weak pairwise correlations imply strongly correlated network states in a neural population
,”
Nature
440
,
1007
1012
(
2006
).
113
S.
Yu
,
H.
Yang
,
H.
Nakahara
,
G. S.
Santos
,
D.
Nikolić
, and
D.
Plenz
, “
Higher-order interactions characterized in cortical activity
,”
J. Neurosci.
31
,
17514
17526
(
2011
).
114
J.
Faskowitz
,
R. F.
Betzel
, and
O.
Sporns
, “
Edges in brain networks: Contributions to models of structure and function
,”
Netw. Neurosci.
6
,
1
28
(
2022
).
115
A.
Gidon
,
T. A.
Zolnik
,
P.
Fidzinski
,
F.
Bolduan
,
A.
Papoutsi
,
P.
Poirazi
,
M.
Holtkamp
,
I.
Vida
, and
M. E.
Larkum
, “
Dendritic action potentials and computation in human layer 2/3 cortical neurons
,”
Science
367
,
83
87
(
2020
).
116
D.
Haufler
and
D.
Paré
, “
Detection of multiway gamma coordination reveals how frequency mixing shapes neural dynamics
,”
Neuron
101
,
603
614
(
2019
).
117
C.
Luff
,
R. L.
Peach
,
E.-J.
Mallas
,
E.
Rhodes
,
F.
Laumann
,
E. S.
Boyden
,
D. J.
Sharp
,
M.
Barahona
, and
N.
Grossman
, “The neuron mixer and its impact on human brain dynamics,” bioRxiv, 2023-01 (2023).
118
F.
Parastesh
,
M.
Mehrabbeik
,
K.
Rajagopal
,
S.
Jafari
, and
M.
Perc
, “
Synchronization in Hindmarsh–Rose neurons subject to higher-order interactions
,”
Chaos
32
,
013125
(
2022
).
119
E.
Nijholt
,
J. L.
Ocampo-Espindola
,
D.
Eroglu
,
I. Z.
Kiss
, and
T.
Pereira
, “
Emergent hypernetworks in weakly coupled oscillators
,”
Nat. Commun.
13
,
4849
(
2022
).
120
L.
Neuhäuser
,
R.
Lambiotte
, and
M. T.
Schaub
, “
Consensus dynamics on temporal hypergraphs
,”
Phys. Rev. E
104
,
064305
(
2021
).
121
G. T.
Einevoll
,
A.
Destexhe
,
M.
Diesmann
,
S.
Grün
,
V.
Jirsa
,
M.
de Kamps
,
M.
Migliore
,
T. V.
Ness
,
H. E.
Plesser
, and
F.
Schürmann
, “
The scientific case for brain simulations
,”
Neuron
102
,
735
744
(
2019
).
122
G.
Deco
,
V. K.
Jirsa
, and
A. R.
McIntosh
, “
Emerging concepts for the dynamical organization of resting-state activity in the brain
,”
Nat. Rev. Neurosci.
12
,
43
56
(
2011
).
123
S.
Petkoski
and
V. K.
Jirsa
, “
Transmission time delays organize the brain network synchronization
,”
Philos. Trans. R. Soc. A
377
,
20180132
(
2019
).
124
E.
Montbrió
,
D.
Pazó
, and
A.
Roxin
, “
Macroscopic description for networks of spiking neurons
,”
Phys. Rev. X
5
,
021028
(
2015
).
125
P.
Clusella
,
B.
Pietras
, and
E.
Montbrió
, “
Kuramoto model for populations of quadratic integrate-and-fire neurons with chemical and electrical coupling
,”
Chaos
32
,
013105
(
2022
).
126
G.
Petri
,
P.
Expert
,
H. J.
Jensen
, and
J. W.
Polak
, “
Entangled communities and spatial synchronization lead to criticality in urban traffic
,”
Sci. Rep.
3
,
1
8
(
2013
).
127
H. K.
Lo
,
E.
Chang
, and
Y. C.
Chan
, “
Dynamic network traffic control
,”
Transp. Res. Part A
35
,
721
744
(
2001
).
128
M. W.
Levin
,
S. D.
Boyles
, and
R.
Patel
, “
Paradoxes of reservation-based intersection controls in traffic networks
,”
Transp. Res. Part A
90
,
14
25
(
2016
).
129
H.
Taher
,
S.
Olmi
, and
E.
Schöll
, “
Enhancing power grid synchronization and stability through time-delayed feedback control
,”
Phys. Rev. E
100
,
062306
(
2019
).
130
J.
He
,
C.
Lu
,
X.
Wu
,
P.
Li
, and
J.
Wu
, “
Design and experiment of wide area HVDC supplementary damping controller considering time delay in China Southern power grid
,”
IET Gener., Transm. Dis.
3
,
17
25
(
2009
).
131
M.
Pope
,
M.
Fukushima
,
R.
Betzel
, and
O.
Sporns
(2021). “KSmodel_fMRIdynamics,”
Github.
https://github.com/brain-networks/KSmodel_fMRIdynamics.
132
A.
Arnaudon
(2023). “Simplicial Kuramoto,”
Github.
https://github.com/arnaudon/simplicial-kuramoto.
You do not currently have access to this content.