Adaptive dynamical networks are network systems in which the structure co-evolves and interacts with the dynamical state of the nodes. We study an adaptive dynamical network in which the structure changes on a slower time scale relative to the fast dynamics of the nodes. We identify a phenomenon we refer to as recurrent adaptive chaotic clustering (RACC), in which chaos is observed on a slow time scale, while the fast time scale exhibits regular dynamics. Such slow chaos is further characterized by long (relative to the fast time scale) regimes of frequency clusters or frequency-synchronized dynamics, interrupted by fast jumps between these regimes. We also determine parameter values where the time intervals between jumps are chaotic and show that such a state is robust to changes in parameters and initial conditions.
REFERENCES
1.
L. F.
Abbott
and S. B.
Nelson
, “Synaptic plasticity: Taming the beast
,” Nat. Neurosci.
3
, 1178
–1183
(2000
). 2.
W.
Gerstner
, W. M.
Kistler
, R.
Naud
, and L.
Paninski
, Neuronal Dynamics
(Cambridge University Press
, Cambridge
, 2014
).3.
O.
Popovych
, S.
Yanchuk
, and P.
Tass
, “Self-organized noise resistance of oscillatory neural networks with spike timing-dependent plasticity
,” Sci. Rep.
3
, 2926
(2013
). 4.
T.
Gross
, C. J. D.
D’Lima
, and B.
Blasius
, “Epidemic dynamics on an adaptive network
,” Phys. Rev. Lett.
96
, 208701
(2006
). 5.
T.
Gross
and H.
Sayama
, “Adaptive networks,” in Adaptive Networks: Theory, Models and Applications, edited by T. Gross and H. Sayama (Springer Berlin Heidelberg, Berlin, 2009), pp. 1–8.6.
L.
Horstmeyer
and C.
Kuehn
, “Adaptive voter model on simplicial complexes
,” Phys. Rev. E
101
, 022305
(2020
). 7.
F.
Schweitzer
, “Social percolation revisited: From 2d lattices to adaptive networks
,” Physica A
570
, 125687
(2021
). 8.
E. A.
Martens
and K.
Klemm
, “Cyclic structure induced by load fluctuations in adaptive transportation networks,” in Progress in Industrial Mathematics at ECMI 2018, edited by I. Faragó, F. Izsák, and P. L. Simon (Springer International Publishing, Cham, 2019), pp. 147–155.9.
A.
Pikovsky
and Y.
Maistrenko
, Synchronization: Theory and Application
(Kluwer Academic Publishers
, 2003
), Vol. 109.10.
A.
Pikovsky
, M.
Rosenblum
, and J.
Kurths
, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series (Cambridge University Press, 2001).11.
R.
Berner
, T.
Gross
, C.
Kuehn
, J.
Kurths
, and S.
Yanchuk
, “Adaptive dynamical networks
,” Phys. Rep.
1031
, 1
–59
(2023
). 12.
T.
Gross
and B.
Blasius
, “Adaptive coevolutionary networks: A review
,” J. R. Soc. Interface
5
, 259
–271
(2008
). 13.
N.
Caporale
and Y.
Dan
, “Spike timing-dependent plasticity: A Hebbian learning rule
,” Annu. Rev. Neurosci.
31
, 25
–46
(2008
). 14.
D. V.
Kasatkin
, S.
Yanchuk
, E.
Schöll
, and V. I.
Nekorkin
, “Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings
,” Phys. Rev. E
96
, 62211
(2017
). 15.
R.
Berner
, E.
Schöll
, and S.
Yanchuk
, “Multiclusters in networks of adaptively coupled phase oscillators
,” SIAM J. Appl. Dyn. Syst.
18
, 2227
–2266
(2019
). 16.
P.
Feketa
, A.
Schaum
, and T.
Meurer
, “Synchronization and multicluster capabilities of oscillatory networks with adaptive coupling
,” IEEE Trans. Automat. Control
66
, 3084
–3096
(2021
). 17.
O. V.
Popovych
, M. N.
Xenakis
, and P. A.
Tass
, “The spacing principle for unlearning abnormal neuronal synchrony
,” PLoS One
10
, e0117205
(2015
). 18.
T.
Aoki
, “Self-organization of a recurrent network under ongoing synaptic plasticity
,” Neural Netw.
62
, 11
–19
(2015
). 19.
V.
Röhr
, R.
Berner
, E. L.
Lameu
, O. V.
Popovych
, and S.
Yanchuk
, “Frequency cluster formation and slow oscillations in neural populations with plasticity
,” PLoS One
14
, e0225094
(2019
). 20.
M.
Thiele
, R.
Berner
, P. A.
Tass
, E.
Schöll
, and S.
Yanchuk
, “Asymmetric adaptivity induces recurrent synchronization in complex networks
,” Chaos
33
, 023123
(2023
). 21.
S.
Bornholdt
and T.
Röhl
, “Self-organized critical neural networks
,” Phys. Rev. E
67
, 066118
(2003
). 22.
J.
Fialkowski
, S.
Yanchuk
, I. M.
Sokolov
, E.
Schöll
, G. A.
Gottwald
, and R.
Berner
, “Heterogeneous nucleation in finite-size adaptive dynamical networks
,” Phys. Rev. Lett.
130
, 067402
(2023
). 23.
C.
Kuehn
, “Multiscale dynamics of an adaptive catalytic network
,” Math. Model. Nat. Phenom.
14
, 402
(2019
). 24.
C.
Kuehn
, Multiple Time Scale Dynamics
(Springer-Verlag GmbH
, 2015
), Vol. 191.25.
S. H.
Strogatz
, “Exploring complex networks
,” Nature
410
, 268
–276
(2001
). 26.
S.
Boccaletti
, V.
Latora
, Y.
Moreno
, M.
Chavez
, and D.-U.
Hwang
, “Complex networks: Structure and dynamics
,” Phys. Rep.
424
, 175
–308
(2006
). 27.
A. S. d.
Mata
, “Complex networks: A mini-review
,” Braz. J. Phys.
50
, 658
–672
(2020
). 28.
Y.
Zou
, R. V.
Donner
, N.
Marwan
, J. F.
Donges
, and J.
Kurths
, “Complex network approaches to nonlinear time series analysis
,” Phys. Rep.
787
, 1
–97
(2019
). 29.
O. V.
Maslennikov
and V. I.
Nekorkin
, “Adaptive dynamical networks
,” Phys. Usp.
60
, 694
(2017
). 30.
J.
Sawicki
, R.
Berner
, S. A. M.
Loos
, M.
Anvari
, R.
Bader
, W.
Barfuss
, N.
Botta
, N.
Brede
, I.
Franovi
, D. J.
Gauthier
, S.
Goldt
, A.
Hajizadeh
, P.
Hövel
, O.
Karin
, P.
Lorenz-Spreen
, C.
Miehl
, J.
Mölter
, S.
Olmi
, E.
Schöll
, A.
Seif
, P. A.
Tass
, G.
Volpe
, S.
Yanchuk
, and J.
Kurths
, “Perspectives on adaptive dynamical systems
,” Chaos
33
, 071501
(2023
). 31.
C.
Zhou
and J.
Kurths
, “Dynamical weights and enhanced synchronization in adaptive complex networks
,” Phys. Rev. Lett.
96
, 164102
(2006
). 32.
M.
Wiedermann
, J. F.
Donges
, J.
Heitzig
, W.
Lucht
, and J.
Kurths
, “Macroscopic description of complex adaptive networks coevolving with dynamic node states
,” Phys. Rev. E
91
, 052801
(2015
). 33.
E. A.
Martens
and K.
Klemm
, “Transitions from trees to cycles in adaptive flow networks
,” Front. Phys.
5
, 62
(2017
). 34.
B.
Duchet
, C.
Bick
, and A.
Byrne
, “Mean-field approximations with adaptive coupling for networks with spike-timing-Dependent plasticity
,” Neural Comput.
35
, 1481
–1528
(2023
). 35.
T.
Aoki
and T.
Aoyagi
, “Co-evolution of phases and connection strengths in a network of phase oscillators
,” Phys. Rev. Lett.
102
, 034101
(2009
). 36.
R.
Berner
, S.
Vock
, E.
Schöll
, and S.
Yanchuk
, “Desynchronization transitions in adaptive networks
,” Phys. Rev. Lett.
126
, 028301
(2021
). 37.
B.
Jüttner
and E. A.
Martens
, “Complex dynamics in adaptive phase oscillator networks
,” Chaos
33
, 053106
(2023
). 38.
H.
Markram
, W.
Gerstner
, and P. J.
Sjöström
, “Spike-timing-dependent plasticity: A comprehensive overview
,” Front. Synaptic Neurosci.
4
, 2
(2012
). 39.
N.
Fenichel
, “Geometric singular perturbation theory for ordinary differential equations
,” J. Differ. Equ.
31
, 53
–98
(1979
). 40.
M.
Krupa
and P.
Szmolyan
, “Relaxation oscillation and canard explosion
,” J. Differ. Equ.
174
, 312
–368
(2001
). 41.
P.
Szmolyan
and M.
Wechselberger
, “Canards in R3
,” J. Differ. Equ.
177
, 419
–453
(2001
). 42.
S.
Wieczorek
, P.
Ashwin
, C. M.
Luke
, and P. M.
Cox
, “Excitability in ramped systems: The compost-bomb instability
,” Proc. R. Soc. A
467
, 1243
–1269
(2011
). 43.
M.
Ciszak
, F.
Marino
, A.
Torcini
, and S.
Olmi
, “Emergent excitability in populations of nonexcitable units
,” Phys. Rev. E
102
, 050201
(2020
). 44.
Y.
Kuramoto
, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics Vol. 19 (Springer Berlin Heidelberg, Berlin, 1984).45.
H.
Sakaguchi
and Y.
Kuramoto
, “A soluble active rotater model showing phase transitions via mutual entertainment
,” Prog. Theor. Phys.
76
, 576
–581
(1986
). 46.
F. C.
Hoppensteadt
and E. M.
Izhikevich
, Neuron, Applied Mathematical Sciences Vol. 126 (Springer New York, New York, 1997).47.
B.
Pietras
and A.
Daffertshofer
, “Network dynamics of coupled oscillators and phase reduction techniques
,” Phys. Rep.
819
, 1
–105
(2019
). 48.
A. L.
Hodgkin
and A. F.
Huxley
, “A quantitative description of membrane current and its application to conduction and excitation in nerve
,” J. Phys.
117
, 500
–544
(1952
). 49.
A.
Bergner
, M.
Frasca
, G.
Sciuto
, A.
Buscarino
, E. J.
Ngamga
, L.
Fortuna
, and J.
Kurths
, “Remote synchronization in star networks
,” Phys. Rev. E
85
, 026208
(2012
). 50.
I.
Leyva
, I.
Sendiña-Nadal
, R.
Sevilla-Escoboza
, V. P.
Vera-Avila
, P.
Chholak
, and S.
Boccaletti
, “Relay synchronization in multiplex networks
,” Sci. Rep.
8
, 8629
(2018
). 51.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory
,” Meccanica
15
, 9
–20
(1980
). 52.
A.
Wolf
, J. B.
Swift
, H. L.
Swinney
, and J. A.
Vastano
, “Determining Lyapunov exponents from a time series
,” Physica D
16
, 285
–317
(1985
). 53.
A.
Vanselow
, S.
Wieczorek
, and U.
Feudel
, “When very slow is too fast—Collapse of a predator-prey system
,” J. Theor. Biol.
479
, 64
–72
(2019
). © 2024 Author(s). Published under an exclusive license by AIP Publishing.
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