Abstract
We investigate the effects of reflecting connectivity in a network composed of FitzHugh–Nagumo (FHN) elements linked in a ring topology. Reflecting connectivity is inspired by the linking between the opposite hemispheres in the mammalian brain. To study synchronization phenomena and coexistence of synchronous and asynchronous domains (chimera states) under the influence of this connectivity, we use two versions of the FHN model: Version I where membrane and recovery potentials are interlinked via a rotational matrix (Omelchenko et al. PRL 110:224101, 2013), and version II where only the membrane potentials are linked and not the recovery ones (Shepelev et al., Phys. Lett. A 381:1398, 2017). In both realizations, the reflecting connectivity forces the activity to organise in two connected semirings. Our numerical results give evidence that, for FHN-I and positive (negative) coupling strength, coherent (incoherent) oscillatory regions develop and reside in the junctions between the two semirings. For the FHN-II model which supports multistability, our simulations indicate that for appropriate values of the parameters the two semirings are separated via oscillating elements. The center parts of the two semirings are composed either by elements which are frozen at the FHN fixed points (coexistence of fixed points and oscillatory states), or by coherent oscillatory elements (coexistence of coherent and incoherent nodes). Notable, here, is the case of oscillation death, which has also been reported for other oscillators coupled via one of their variables. These results can be useful for interpreting inhomogeneous and localised coordinated activity observed in the mammalian brain.
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Y. Kuramoto, D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena Complex Syst. 5, 380 (2002)
Y. Kuramoto, Reduction methods applied to nonlocally coupled oscillator systems, in Nonlinear Dynamics and Chaos: Where do we go from here?, ed. by S.J. Hogan, A.R. Champneys, A.R. Krauskopf, M. di Bernado, R. Eddie Wilson, H.M. Osinga, M.E. Homer (CRC Press, New York, 2002), pp. 209–227
D.M. Abrams, S.H. Strogatz, Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004)
I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, When nonlocal coupling between oscillators becomes stronger: patched synchrony or multi-chimera states. Phys. Rev. Lett. 110, 224101 (2013)
N.D. Tsigkri-DeSmedt, J. Hizanidis, P. Hövel, A. Provata, Multi-chimera states and transitions in the Leaky Integrate-and-Fire model with nonlocal and hierarchical connectivity. Eur. Phys. J. Spec. Top. 225, 1149–1164 (2016)
S. Luccioli, A. Politi, Irregular collective behavior of heterogeneous neural networks. Phys. Rev. Lett. 105, 158104 (2010)
S. Olmi, A. Politi, A. Torcini, Collective chaos in pulse-coupled neural networks. Europhys. Lett. 92, 60007 (2010)
N.D. Tsigkri-DeSmedt, J. Hizanidis, E. Schöll, P. Hövel, A. Provata, Chimeras in Leaky Integrate-and-Fire neural networks: effects of reflecting connectivities. Eur. Phys. J. B 90, 139 (2017)
J. Hizanidis, V. Kanas, A. Bezerianos, T. Bountis, Chimera states in networks of nonlocally coupled Hindmarsh–Rose neuron models. Int. J. Bifur. Chaos 24, 1450030 (2014)
Z. Wei, F. Parastesh, H. Azarnoush, S. Jafari, D. Ghosh, M. Perc, M. Slavinec, Nonstationary chimeras in a neuronal network. Europhys. Lett. 123, 48003 (2018)
I.A. Shepelev, T.E. Vadivasova, A.V. Bukh, G.I. Strelkova, V.S. Anishchenko, New type of chimera structures in a ring of bistable FitzHugh–Nagumo oscillators with nonlocal interaction. Phys. Lett. A 381, 1398–1404 (2017)
J. Hizanidis, E. Panagakou, I. Omelchenko, E. Schöll, P. Hövel, A. Provata, Chimera states in population dynamics: networks with fragmented and hierarchical connectivities. Phys. Rev. E 92, 012915 (2015)
B.W. Li, H. Dierckx, Spiral wave chimeras in locally coupled oscillator systems. Phys. Rev. E 93, 020202 (2016)
S. Nkomo, M.R. Tinsley, K. Showalter, Chimera and chimera-like states in populations of nonlocally coupled homogeneous and heterogeneous chemical oscillators. Chaos 26, 094826 (2016)
L. Bauer, J. Bassett, P. Hövel, Y.N. Kyrychko, K.B. Blyuss, Chimera states in multi-strain epidemic models with temporary immunity. Chaos 27, 114317 (2017)
P. Jaros, Y. Maistrenko, T. Kapitaniak, Chimera states on the route from coherence to rotating waves. Phys. Rev. E 91, 022907 (2015)
T. Kapitaniak, J. Kurths, Synchronized pendula: from Huygens’ clocks to chimera states. Eur. Phys. J. Spec. Top. 223, 609–612 (2014)
D. Dudkowski, J. Grabski, J. Wojewoda, P. Perlikowski, Y. Maistrenko, T. Kapitaniak, Experimental multistable states for small network of coupled pendula. Sci. Rep. 6, 29833 (2016)
N. Lazarides, G. Neofotistos, G.P. Tsironis, Chimeras in SQUID metamaterials. Phys. Rev. B 91, 054303 (2015)
J. Hizanidis, N. Lazarides, G.P. Tsironis, Robust chimera states in SQUID metamaterials with local interactions. Phys. Rev. E 94 (2016)
J. Hizanidis, N.E. Kouvaris, G. Zamora-López, A. Díaz-Guilera, C.G. Antonopoulos, Chimera-like states in modular neural networks. Sci. Rep. 6, 19845 (2016)
N.C. Rattenborg, C.J. Amlaner, S.L. Lima, Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep. Neurosci. Biobehav. Rev. 24, 817–842 (2000)
N.C. Rattenborg, Do birds sleep in flight? Naturwissenschaften 93, 413–425 (2006)
F. Mormann, K. Lehnertz, P. David, C.E. Elger, Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Physica D 144, 358 (2000)
F. Mormann, T. Kreuz, R.G. Andrzejak, P. David, K. Lehnertz, C.E. Elger, Epileptic seizures are preceded by a decrease in synchronization. Epilepsy Res. 53, 173 (2003)
R.G. Andrzejak, C. Rummel, F. Mormann, K. Schindler, All together now: analogies between chimera state collapses and epileptic seizures. Sci. Rep. 6, 23000 (2016)
H.M. Mitchell, P.S. Dodds, J.M. Mahoney, C.M. Danforth, Chimera states and seizures in a mouse neuronal model. Int. J. Bifurcat. Chaos. 30, 2050256 (2020)
M. Gerster, R. Berner, J. Sawicki, A. Zakharova, A. Skoch, J. Hlinka, K. Lehnertz, E. Schöll, FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena. Chaos 30, 123130 (2020)
E.A. Martens, S. Thutupalli, A. Fourrière, O. Hallatschek, Chimera states in mechanical oscillator networks. Proc. Natl. Acad. Sci. 110(26), 10563–10567 (2013)
S. Olmi, E.A. Martens, S. Thutupalli, A. Torcini, Intermittent chaotic chimeras for coupled rotators. Phys. Rev. E 92, 030901 (2015)
M.R. Tinsley, S. Nkomo, K. Showalter, Chimera and phase-cluster states in populations of coupled chemical oscillators. Nat. Phys. 8, 662 (2012)
J.F. Totz, J. Rode, M.R. Tinsley, K. Showalter, H. Engel, Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys. 14, 282–285 (2018)
J.F. Totz, M.R. Tinsley, H. Engel, K. Showalter, Transition from spiral wave chimeras to phase cluster states. Sci. Rep. 10, 7821 (2020)
A.M. Hagerstrom, T.E. Murphy, R. Roy, P. Hövel, I. Omelchenko, E. Schöll, Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658 (2012)
M.J. Panaggio, D. Abrams, Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28, R67–R87 (2015)
E. Schöll, Synchronization patterns and chimera states in complex networks: interplay of topology and dynamics. Eur. Phys. J. Spec. Top. 225, 891–919 (2016)
O.E. Omel’Chenko, The mathematics behind chimera states. Nonlinearity 31, R121 (2018)
S. Majhi, B.K. Bera, D. Ghosh, M. Perc, Chimera states in neuronal networks: a review. Phys. Life Rev. 28, 100–121 (2019)
F. Parastesh, S. Jafari, H. Azarnoush, Z. Shahriari, Z. Wang, S. Boccaletti, M. Perc, Chimeras. Phys. Rep. 898, 1–114 (2021)
A. Schmidt, T. Kasimatis, J. Hizanidis, A. Provata, P. Hövel, Chimera patterns in two-dimensional networks of coupled neurons. Phys. Rev. E 95, 032224 (2017)
S. Kundu, S. Majhi, B.K. Bera, D. Ghosh, M. Lakshmanan, Chimera states in two-dimensional networks of locally coupled oscillators. Phys. Rev. E 97, 022201 (2018)
T. Kasimatis, J. Hizanidis, A. Provata, Three-dimensional chimera patterns in networks of spiking neuron oscillators. Phys. Rev. E 97, 052213 (2018)
Y. Maistrenko, O. Sudakov, O. Osiv, V. Maistrenko, Chimera states in three dimensions. New J. Phys. 17, 073037 (2015)
V. Maistrenko, O. Sudakov, O. Osiv, Y. Maistrenko, Multiple scroll wave chimera states. Eur. Phys. J. Spec. Top. 226, 1867–1881 (2017)
S. Kundu, B.K. Bera, D. Ghosh, M. Lakshmanan, Chimera patterns in three-dimensional locally coupled systems. Phys. Rev. E 99, 022204 (2019)
I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, P. Hövel, Robustness of chimera states for coupled FitzHugh-Nagumo oscillators. Phys. Rev. E 91, 022917 (2015)
S. Ulonska, I. Omelchenko, A. Zakharova, E. Schöll, Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. Chaos 26, 094825 (2016)
J. Sawicki, I. Omelchenko, A. Zakharova, E. Schöll, Chimera states in complex networks: interplay of fractal topology and delay. Eur. Phys. J. Spec. Top. 226, 1883–1892 (2017)
J. Sawicki, I. Omelchenko, A. Zakharova, E. Schöll, Delay-induced chimeras in neural networks with fractal topology. Eur. Phys. J. B 92, 54 (2019)
I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Loss of coherence in dynamical networks: Spatial chaos and chimera states. Phys. Rev. Lett. 106, 234102 (2011)
E. Rybalova, V.S. Anishchenko, G.I. Strelkova, A. Zakharova, Solitary states and solitary state chimera in neural networks. Chaos 29, 071106 (2019)
N.D. Tsigkri-DeSmedt, I. Koulierakis, G. Karakos, A. Provata, Synchronization patterns in LIF neuron networks: merging nonlocal and diagonal connectivity. Eur. Phys. J. B 91, 305 (2018)
B.K. Bera, S. Majhi, D. Ghosh, M. Perc, Chimera states: effects of different coupling topologies. Europhys. Lett. 118, 10001 (2017)
B.K. Bera, D. Ghosh, Chimera states in purely local delay-coupled oscillators. Phys. Rev. E 93, 052223 (2016)
E.S. Finn, X. Shen, D. Scheinost, M.D. Rosenberg, J. Huang, M.M. Chun, X. Papademetris, R.T. Constable, Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity. Nat. Neurosci. 18, 1664–1671 (2016)
J.D. Murray, Mathematical Biology (Chapter 6) (Springer, Berlin, 1993)
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
S. Majhi, M. Perc, D. Ghosh, Chimera states in uncoupled neurons induced by a multilayer structure. Sci. Rep. 6, 39033 (2016)
S. Majhi, M. Perc, D. Ghosh, Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos 27, 073109 (2017)
I. A. Shepelev, T. E. Vadivasova. Variety of spatio-temporal regimes in a 2d lattice of coupled bistable Fitzhugh–Nagumo oscillators. Formation mechanisms of spiral and double-well chimeras. Commun. Nonlinear Sci. Numer. Simul. 79, 104925 (2019)
A. Zakharova, M. Kapeller, E. Sch\(\ddot{\rm o}\)ll, Chimera death: symmetry breaking in dynamical networks. Phys. Rev. Lett. 112, 154101 (2014)
Acknowledgements
A.P. would like to thank Profs.S. Lambropoulou and E. Schöll for constructive discussions. This work was supported by computational time granted from the Greek Research & Technology Network (GRNET) in the National HPC facility - ARIS - under Project ID: PR009012.
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Rontogiannis, A., Provata, A. Chimera states in FitzHugh–Nagumo networks with reflecting connectivity. Eur. Phys. J. B 94, 97 (2021). https://doi.org/10.1140/epjb/s10051-021-00097-9
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DOI: https://doi.org/10.1140/epjb/s10051-021-00097-9