Chimera states occur when identically coupled groups of nonlinear oscillators exhibit radically different dynamics, with one group exhibiting synchronized oscillations and the other desynchronized behavior. This dynamical phenomenon has recently been studied in computational models and demonstrated experimentally in mechanical, optical, and chemical systems. The theoretical basis of these states is currently under active investigation. Chimera behavior is of particular relevance in the context of neural synchronization, given the phenomenon of unihemispheric sleep and the recent observation of asymmetric sleep in human patients with sleep apnea. The similarity of neural chimera states to neural “bump” states, which have been suggested as a model for working memory and visual orientation tuning in the cortex, adds to their interest as objects of study. Chimera states have been demonstrated in the FitzHugh-Nagumo model of excitable cells and in the Hindmarsh-Rose neural model. Here, we demonstrate chimera states and chimera-like behaviors in a Hodgkin-Huxley-type model of thermally sensitive neurons both in a system with Abrams-Strogatz (mean field) coupling and in a system with Kuramoto (distance-dependent) coupling. We map the regions of parameter space for which chimera behavior occurs in each of the two coupling schemes.
References
1.
Abeyratne
, U. R.
, Swarnkar
, V.
, Hukins
, C.
, and Duce
, B.
, “Interhemispheric asynchrony correlates with severity of respiratory disturbance index in patients with sleep apnea
,” IEEE Trans. Biomed. Eng.
57
, 2947
–2955
(2010
).2.
Abrams
, D. M.
, Mirollo
, R.
, Strogatz
, S. H.
, and Wiley
, D. A.
, “Solvable model for Chimera states of coupled oscillators
,” Phys. Rev. Lett.
101
, 084103
(2008
).3.
Abrams
, D. M.
and Strogatz
, S. H.
, “Chimera states for coupled oscillators
,” Phys. Rev. Lett.
93
(17
), 174102
(2004
).4.
Adamchic
, I.
, Hauptmann
, C.
, Barnikol
, U. B.
, Pawelczyk
, N.
, Popovych
, O.
, Barnikol
, T. T.
, Silchenko
, A.
, Volkmann
, J.
, Deuschl
, G.
, Meissner
, W. G.
, Maarouf
, M.
, Sturm
, V.
, Freund
, H. J.
, and Tass
, P. A.
, “Coordinated reset neuromodulation for Parkinson's disease: Proof-of-concept study
,” Mov. Disord.
29
(13
), 1679
–1684
(2014
).5.
Bahar
, S.
, “Burst-enhanced synchronization in an array of noisy coupled neurons
,” Fluctuation Noise Lett.
4
(1
), L87
–L96
(2004
).6.
Bartolomei
, F.
, Guye
, M. F.
, and Wendling
, F.
, “Abnormal binding and disruption in large scale networks involved in human partial seizures
,” EPJ Nonlinear Biomed. Phys.
1
, 4
(2013
).7.
Battaglia
, D.
, Witt
, A.
, Wolf
, F.
, and Geisel
, T.
, “Dynamic effective connectivity of inter-areal brain circuits
,” PLoS Comput. Biol.
8
(3
), e1002438
(2012
).8.
Bola
, M.
and Sabel
, B. A.
, “Dynamic reorganization of brain functional networks during cognition
,” Neuroimage
114
, 398
–413
(2015
).9.
Braun
, W.
, Eckhardt
, B.
, Braun
, H. A.
, and Huber
, M. T.
, “Phase-space structure of a thermoreceptor
,” Phys. Rev. E
62
, 6352
–6360
(2000
).10.
Braun
, H. A.
, Huber
, M. T.
, Dewald
, M.
, Schäfer
, K.
, and Voigt
, K.
, “Computer simulations of neuronal signal transduction: The role of nonlinear dynamics and noise
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
8
(5
), 881
–889
(1998
).11.
Braun
, U.
, Schäfer
, A.
, Walter
, H.
, Erk
, S.
, Romanczuk-Seiferth
, N.
, Haddad
, L.
, Schweiger
, J. I.
, Grimm
, O.
, Heinz
, A.
, Tost
, H.
, Meyer-Lindenberg
, A.
, and Bassett
, D. S.
, “Dynamic reconfiguration of frontal brain networks during executive cognition in humans
,” Proc. Natl. Acad. Sci. U. S. A.
112
(37
), 11678
–11683
(2015
).12.
Chen
, X.
and Dzakpasu
, R.
, “Observed network dynamics from altering the balance between excitatory and inhibitory neurons in cultured networks
,” Phys. Rev. E
82
(3 Pt 1
), 031907
(2010
).13.
Chow
, C. C.
and Coombes
, S.
“Existence and wandering of bumps in a spiking neural network model
,” SIAM J. Appl. Dyn. Syst.
5
, 552
–574
(2006
).14.
Feldt
, S.
, Wang
, J. X.
, Shtrahman
, E.
, Dzakpasu
, R.
, Olariu
, E.
, and Zochowski
, M.
, “Functional clustering in hippocampal cultures: Relating network structure and dynamics
,” Phys. Biol.
7
(4
), 046004
(2010
).15.
Feudel
, U.
, Neiman
, A.
, Pei
, X.
, Wojtenek
, W.
, Braun
, H.
, Huber
, M.
, and Moss
, F.
, “Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons
,” Chaos
10
(1
), 231
–239
(2000
).16.
Fox
, R. F.
, Gatland
, I. R.
, Roy
, R.
, and Vemuri
, G.
, “Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise
,” Phys. Rev. A
38
, 5938
–5940
(1988
).17.
Fries
, P.
, Reynolds
, J. H.
, Rorie
, A. E.
, and Desimone
, R.
, “Modulation of oscillatory neuronal synchronization by selective visual attention
,” Science
291
(5508
), 1560
–1563
(2001
).18.
Hagerstrom
, A. M.
, Murphy
, T. E.
, Roy
, R.
, Hövel
, P.
, Omelchenko
, I.
, and Schöll
, E.
, “Experimental observation of chimeras in coupled-map lattices
,” Nat. Phys.
8
, 658
–661
(2012
).19.
Hizanidis
, J.
, Kanas
, V. G.
, Bezerianos
, A.
, and Bountis
, T.
, “Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron models
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
24
(3
), 1450030
(2014
).20.
Huber
, M. T.
, Krieg
, J. C.
, Dewald
, M.
, Voigt
, K.
, and Braun
, H. A.
, “Stochastic encoding in sensory neurons: impulse patterns of mammalian cold receptors
,” Chaos, Solitons Fractals
11
, 1895
–1903
(2000
).21.
Isomura
, Y.
, Fujiwara-Tsukamoto
, Y.
, and Takada
, M.
, “A network mechanism underlying hippocampal seizure-like synchronous oscillations
,” Neurosci. Res.
61
(3
), 227
–33
(2008
).22.
Kedziora
, D. J.
, Abeysuriya
, R. G.
, Phillips
, A. J. K.
, and Robinson
, P. A.
, “Physiologically based quantitative modeling of unihemispheric sleep
,” J. Theor. Biol.
314
, 109
–119
(2012
).23.
Kuramoto
, Y.
and Battogtokh
, D.
, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,” Nonlinear Phenom. Complex Syst.
5
, 380
–385
(2002
); available at http://www.j-npcs.org/online/vol2002/v5no4/v5no4p380.pdf.24.
Laing
, C. R.
, “Chimera states in heterogeneous networks
,” Chaos
19
, 013113
(2009a
).25.
Laing
, C. R.
, “The dynamics of chimera states in heterogeneous Kuramoto networks
,” Physica D
238
, 1569
–1588
(2009b
).26.
Laing
, C. R.
, “Fronts and bumps in spatially extended Kuramoto networks
,” Physica D
240
, 1960
–1971
(2011
).27.
Laing
, C. R.
, “Exact neural fields incorporating gap junctions
,” SIAM J. Appl. Dyn. Syst.
14
, 1899
–1929
(2015
).28.
Laing
, C. R.
and Chow
, C. C.
, “Stationary bumps in networks of spiking neurons
,” Neural Comput.
13
, 1473
–1494
(2001
).29.
Lyamin
, O. I.
, Manger
, P. R.
, Ridgway
, S. H.
, Mukhametov
, L. M.
, and Siegel
, J. M.
, “Cetacean sleep: An unusual form of mammalian sleep
,” Neurosci. Biobehav. Rev.
32
, 1451
–1484
(2008
).30.
Martens
, E. A.
, Thutupalli
, S.
, Fourrière
, A.
, and Hallatschek
, O.
, “Chimera states in mechanical oscillator networks
,” Proc. Natl. Acad. Sci. U. S. A.
110
(26
), 10563
–10567
(2013
).31.
Mathews
, C. G.
, Lesku
, J. A.
, Lima
, S. L.
, and Amlaner
, C. J.
, “Asynchronous eye closure as an anti-predator behavior in the western fence lizard (Sceloporus occidentalis)
,” Ethology
112
, 286
–292
(2006
).32.
Mukhametov
, L. M.
, in Sleep Mechanisms
, edited by Borbely
, A.
and Valatx
, J. L.
(Springer Verlag
, 1984
), pp. 227
–238
.33.
Niedringhaus
, M.
, Chen
, X.
, and Dzakpasu
, R.
, “Long-term dynamical constraints on pharmacologically evoked potentiation imply activity conservation within in vitro hippocampal networks
,” PLoS One
10
(6
), e0129324
(2015
).34.
Nkomo
, S.
, Tinsley
, M. R.
, and Showalter
, K.
, “Chimera states in populations of nonlocally coupled chemical oscillators
,” Phys. Rev. Lett.
110
, 244102
(2013
).35.
Olmi
, S.
, Politi
, A.
, and Torcini
, A.
, “Collective chaos in pulse-coupled neural networks
,” Europhys. Lett.
92
, 60007
(2010
).36.
Omel'chenko
, O. E.
, Maistrenko
, Y. L.
, and Tass
, P. A.
, “Chimera states: the natural link between coherence and incoherence
,” Phys. Rev. Lett.
100
, 044105
(2008
).37.
Omelchenko
, I.
, Omel'chenko
, O. E.
, Hövel
, P.
, and Schöll
, E.
, “When nonlocal coupling between oscillators becomes stronger: Patched synchrony or multichimera states
,” Phys. Rev. Lett.
110
, 224101
(2013
).38.
Omel'chenko
, O. E.
, Wolfrum
, M.
, and Maistrenko
, Y. L.
, “Chimera states as chaotic spatiotemporal patterns
,” Phys. Rev. E
81
, 065201(R)
(2010
).39.
Panaggio
, M. J.
and Abrams
, D. M.
, “Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators
,” Nonlinearity
28
, R67
–R87
(2015
).40.
Pikovsky
, A.
, Rosenblum
, M.
, and Kurths
, J.
, Synchronization: A Universal Concept in Nonlinear Sciences
(Cambridge University Press
, Cambridge
, 2001
).41.
Popovych
, O. V.
, Hauptmann
, C.
, and Tass
, P. A.
, “Effective desynchronization by nonlinear delayed feedback
,” Phys. Rev. Lett.
94
(16
), 164102
(2005
).42.
Rattenborg
, N. C.
, “Do birds sleep in flight?
,” Naturwissenschaften
93
, 413
–425
(2006
).43.
Rattenborg
, N. C.
, Lima
, S. L.
, and Amlaner
, C. J.
, “Half-awake to the risk of predation
,” Nature
397
, 397
–398
(1999
).44.
Rial
, R.
, González
, J.
, Gené
, L.
, Akaârir
, M.
, Esteban
, S.
, Gamundí
, A.
, Barceló
, P.
, and Nicolau
, C.
, “Asymmetric sleep in apneic human patients
,” Am. J. Physiol.: Regul., Integr. Comp. Physiol.
304
, R232
–R237
(2013
).45.
Sethia
, G. C.
, Sen
, A.
, and Johnston
, G. L.
, “Amplitude-mediated chimera states
,” Phys. Rev. E
88
, 042917
(2013
).46.
Tamaki
, M.
, Bang
, J. W.
, Watanabe
, T.
, and Sasaki
, Y.
, “Night watch in one brain hemisphere during sleep associated with first-night effect in humans
,” Curr. Biol.
26
, 1190
–1194
(2016
).47.
Tass
, P. A.
, Qin
, L.
, Hauptmann
, C.
, Dovero
, S.
, Bezard
, E.
, Boraud
, T.
, and Meissner
, W. G.
, “Coordinated reset has sustained aftereffects in Parkinsonian monkeys
,” Ann. Neurol.
72
(5
), 816
–820
(2012
).48.
Tinsley
, M. R.
, Nkomo
, S.
, and Showalter
, K.
, “Chimera and phase-cluster states in populations of coupled chemical oscillators
,” Nat. Phys.
8
, 662
–665
(2012
).49.
Tononi
, G.
and Koch
, C.
, “The neural correlates of consciousness: An update
,” Ann. N. Y. Acad. Sci.
1124
, 239
–261
(2008
).50.
Voytek
, B.
and Knight
, R. T.
, “Dynamic network communication as a unifying neural basis for cognition, development, aging, and disease
,” Biol. Psychiatry
77
(12
), 1089
–1097
(2015
).51.
Weihberger
, O.
and Bahar
, S.
, “Frustration, drift, and antiphase coupling in a neural array
,” Phys. Rev. E
76
, 011910
(2007
).52.
Wickramasinghe
, M.
and Kiss
, I. Z.
, “Spatially organized dynamical states in chemical oscillator networks: Synchronization, dynamical differentiation, and chimera patterns
,” PLoS One
8
(11
), e80586
(2013
).53.
Wilson
, H. R.
, “Simplified dynamics of human and mammalian neocortical neurons
,” J. Theor. Biol.
200
(4
), 375
–388
(1999
).54.
Wolfrum
, M.
and Omel'chenko
, O. E.
, “Chimera states are chaotic transients
,” Phys. Rev. E
84
, 015201(R)
(2011
).© 2016 Author(s).
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