Effect of diluted connectivities on cluster synchronization of adaptively coupled oscillator networks

Document Type : Article


1 Institute of Theoretical Physics, Technische Universitat Berlin, Hardenbergstrabe 36, 10623 Berlin, Germany

2 - Institute of Theoretical Physics, Technische Universitat Berlin, Hardenbergstrabe 36, 10623 Berlin, Germany - Institute of Mathematics, Technische Universitat Berlin, Strabe des 17. Juni 136, 10623 Berlin, Germany

3 Institute of Mathematics, Technische Universitat Berlin, Strabe des 17. Juni 136, 10623 Berlin, Germany

4 -Institute of Theoretical Physics, Technische Universitat Berlin, Hardenbergstrabe 36, 10623 Berlin, Germany -Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universitat, Philippstrabe 13, 10115 Berlin, Germany -Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473 Potsdam, Germany


Synchronization in networks of oscillatory units is an emergent phenomenon that has been observed in various systems, from power grids to ensembles of nerve cells. Many real-world networks have adaptive properties, meaning that their connectivities change with time, depending on the dynamical state of the system. Networks of adaptively coupled oscillators show various synchronization phenomena, such as hierarchical multifrequency clusters, traveling waves, or chimera states. While these self-organized patterns have been previously studied on all-to-all coupled networks, this work extends the investigations towards more complex networks, analyzing the influence of random network topologies for various degrees of dilution of the connectivities. Using numerical and analytical approaches, we investigate the robustness of multicluster states on networks of adaptively coupled Kuramoto-Sakaguchi oscillators against the random dilution of the underlying network topology. We utilize the master stability approach for adaptive networks in order to highlight the interplay between adaptivity and topology. With this, we show the robustness of multifrequency cluster states to diluted connectivities.


[1] Newman, M.E.J. The structure and function of complex networks". SIAM Review, 45(2), pp. 167{256 (2003).
[2] Boccaletti, S., Pisarchik, A.N., del Genio, C.I., and Amann, A. Synchronization: From Coupled Systems to Complex Networks. Cambridge: Cambridge University Press (2018).
[3] Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J., and Heagy, J.F. Fundamentals of synchronization in chaotic systems, concepts, and applications". Chaos, 7(4), pp. 520{543 (1997).
[4] Pikovsky, A., Rosenblum, M., and Kurths, J. Synchronization: a universal concept in nonlinear sciences. Cambridge: Cambridge University Press, 1st edition (2001).
[5] Strogatz, S.H. Exploring complex networks". Nature, 410, pp. 268{276 (2001).
[6] Yanchuk, S., Maistrenko, Y., and Mosekilde, E. Synchronization of time-continuous chaotic oscillators". Chaos, 13(1), pp. 388{400 (2003).
[7] Arenas, A., Daz-Guilera, A., Kurths, J., Moreno, Y., and Zhou, C. Synchronization in complex networks". Phys. Rep., 469(3), pp. 93{153 (2008).
[8] Huygens, C. Horologium Oscillatorium: sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Christiaan Huygens, 1st edition (1673).
[9] Kuramoto, Y. Chemical Oscillations, Waves and Turbulence. Berlin: Springer-Verlag (1984).
[10] Dai, X., Li, X., Guo, H., Jia, D., Perc, M., Manshour, P., Wang, Z., and Boccaletti, S. Discontinuous transitions and rhythmic states in the d-dimensional kuramoto model induced by a positive feedback with the global order parameter". Phys. Rev. Lett., 125(19), p. 194101 (2020).
[11] Yanchuk, S., Maistrenko, Y., and Mosekilde, E. Partial synchronization and clustering in a system of di usively coupled chaotic oscillators". Math. Comp. Simul., 54, pp. 491{508 (2001).
[12] Sorrentino, F. and Ott, E. Network synchronization of groups". Phys. Rev. E, 76(5), 056114 (2007).
[13] Dahms, T., Lehnert, J., and Scholl, E. Cluster and group synchronization in delay-coupled networks". Phys. Rev. E, 86(1), p. 016202 (2012).
[14] Miyakawa, K., Okano, T., and Yamazaki, T. Cluster synchronization in a chemical oscillator network with adaptive coupling". Journal of the Physical Society of Japan, 82(3), p. 034005 (2013).
[15] Zhang, Y. and Motter, A.E. Symmetry-independent stability analysis of synchronization patterns". SIAM Rev., 62(4), pp. 817{836 (2020).
[16] Kuramoto, Y. and Battogtokh, D. Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators." Nonlin. Phen. in Complex Sys., 5(4), pp. 380{385 (2002).
[17] Abrams, D.M. and Strogatz, S.H. Chimera states for coupled oscillators". Phys. Rev. Lett., 93(17), p. 174102 (2004).
[18] Motter, A.E. Nonlinear dynamics: Spontaneous synchrony breaking". Nat. Phys., 6(3), pp. 164{165 (2010).
[19] Panaggio, M.J. and Abrams, D.M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators". Nonlinearity, 28, p. R67 (2015).
[20] Scholl, E. Synchronization patterns and chimera states in complex networks: interplay of topology and dynamics". Eur. Phys. J. Spec. Top., 225, pp. 891{919 (2016).
[21] Sawicki, J., Omelchenko, I., Zakharova, A., and Scholl, E. Delay controls chimera relay synchronization in multiplex networks". Phys. Rev. E, 98, p. 062224 (2018).
[22] Omel'chenko, O.E. and Knobloch, E. Chimerapedia: coherence{incoherence patterns in one, two and three dimensions". New J. Phys., 21(9), p. 093034 (2019).
[23] Andrzejak, R.G., Ruzzene, G., Scholl, E., and Omelchenko, I. Two populations of coupled quadratic maps exhibit a plentitude of symmetric and symmetry broken dynamics". Chaos, 30(3), p. 033125 (2020).
[24] Drauschke, F., Sawicki, J., Berner, R., Omelchenko, I., and Scholl, E. E ect of topology upon relay synchronization in triplex neuronal networks". Chaos, 30, p. 051104 (2020).
[25] Scholl, E., Zakharova, A., and Andrzejak, R.G. Chimera States in Complex Networks. Research Topics, Front. Appl. Math. Stat. Lausanne: Frontiers Media SA (2020). Ebook.
[26] Zakharova, A. Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay. Understanding Complex Systems. Springer (2020).
[27] Scholl, E. Chimeras in physics and biology: Synchronization and desynchronization of rhythms". Nova Acta Leopoldina, 425 (2020). Invited contribution.
[28] Parastesh, F., Jafari, S., Azarnoush, H., Shahriari, Z., Wang, Z., Boccaletti, S., and Perc, M. Chimeras". Phys. Rep., 898, pp. 1{114 (2021).
[29] Singer, W. Neuronal Synchrony: A Versatile Code Review for the De nition of Relations?" Neuron, 24, pp. 49{65 (1999).
[30] Wang, X.J. Neurophysiological and computational principles of cortical rhythms in cognition". Phys. Rev., 90(3), pp. 1195{1268 (2010).
[31] Fell, J. and Axmacher, N. The role of phase synchronization in memory processes". Nat. Rev. Neurosci., 12(2), pp. 105{118 (2011).
[32] Hammond, C., Bergman, H., and Brown, P. Pathological synchronization in Parkinson's disease: networks, models and treatments". Trends Neurosci., 30, pp. 357{364 (2007).
[33] Schi , S.J. Towards model-based control of Parkinson's disease". Phil. Trans. R. Soc. A, 368, pp. 2269{2308 (2010).
[34] Kromer, J.A. and Tass, P.A. Long-lasting desynchronization by decoupling stimulation". Phys. Rev. Research, 2(3), p. 033101 (2020).
[35] Kromer, J.A., Khaledi-Nasab, A., and Tass, P.A. Impact of number of stimulation sites on long-lasting desynchronization e ects of coordinated reset stimulation". Chaos, 30(8), p. 083134 (2020).
[36] Lehnertz, K., Bialonski, S., Horstmann, M.T., Krug, D., Rothkegel, A., Staniek, M., and Wagner, T. Synchronization phenomena in human epileptic brain networks". J. Neurosci. Methods, 183(1), pp. 42{48 (2009).
[37] Mormann, F., Lehnertz, K., David, P., and Elger, C.E. Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients". Physica D, 144(3-4), pp. 358{369 (2000).
[38] Jiruska, P., de Curtis, M., Je erys, J.G.R., Schevon, C.A., Schi , S.J., and Schindler, K. Synchronization and desynchronization in epilepsy: controversies and hypotheses". J. Physiol., 591.4, pp. 787{797 (2013).
[39] Jirsa, V.K., Stacey, W.C., Quilichini, P.P., Ivanov, A.I., and Bernard, C. On the nature of seizure dynamics". Brain, 137, p. 2210 (2014).
[40] Rothkegel, A. and Lehnertz, K. Irregular macroscopic dynamics due to chimera states in small-world networks of pulse-coupled oscillators". New J. Phys., 16, p. 055006 (2014).
[41] Andrzejak, R.G., Rummel, C., Mormann, F., and Schindler, K. All together now: Analogies between chimera state collapses and epileptic seizures". Sci. Rep., 6, p. 23000 (2016).
[42] Chouzouris, T., Omelchenko, I., Zakharova, A., Hlinka, J., Jiruska, P., and Scholl, E. Chimera states in brain networks: empirical neural vs. modular fractal connectivity". Chaos, 28(4), p. 045112 (2018).
[43] Gerster, M., Berner, R., Sawicki, J., Zakharova, A., Skoch, A., Hlinka, J., Lehnertz, K., and Scholl, E. FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena". Chaos, 30, p. 123130 (2020).
[44] Pecora, L.M. and Carroll, T.L. Master Stability Functions for Synchronized Coupled Systems". Phys. Rev. Lett., 80(10), pp. 2109{2112 (1998).
[45] Brechtel, A., Gramlich, P., Ritterskamp, D., Drossel, B., and Gross, T. Master stability functions reveal di usion-driven pattern formation in networks". Phys. Rev. E, 97(3), p. 032307 (2018).
[46] Berner, R., Sawicki, J., and Scholl, E. Birth and stabilization of phase clusters by multiplexing of adaptive networks". Phys. Rev. Lett., 124(8), p. 088301 (2020).
[47] Choe, C.U., Dahms, T., Hovel, P., and Scholl, E. Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states". Phys. Rev. E, 81(2), p. 025205(R) (2010).
[48] Flunkert, V., Yanchuk, S., Dahms, T., and Scholl, E. Synchronizing distant nodes: a universal classi cation of networks".  Phys. Rev. Lett., 105, p. 254101 (2010).
[49] Greenshields, C. Master stability function for systems with two coupling delays" (2010).
Private communication.
[50] Lehnert, J., Dahms, T., Hovel, P., and Scholl, E. Loss of synchronization in complex neural networks with delay". Europhys. Lett., 96, p. 60013 (2011).
[51] Flunkert, V., Yanchuk, S., Dahms, T., and Scholl, E. Synchronizability of networks with strongly delayed links: a universal classi cation". Contemp. Math., 48, pp. 134{148 (2013).
English version: J. of Math. Sciences (Springer), 2014.
[52] Keane, A., Dahms, T., Lehnert, J., Suryanarayana, S.A., Hovel, P., and Scholl, E. Synchronisation in networks of delay-coupled type-I excitable systems". Eur. Phys. J. B, 85(12), p. 407 (2012).
[53] Wille, C., Lehnert, J., and Scholl, E. Synchronization-desynchronization transitions in complex  networks: An interplay of distributed time delay and inhibitory nodes". Phys. Rev. E, 90, p. 032908 (2014).
[54] Kyrychko, Y.N., Blyuss, K.B., and Scholl, E. Synchronization of networks of oscillators with distributed-delay coupling". Chaos, 24, p. 043117 (2014).
[55] Lehnert, J. Controlling synchronization patterns in complex networks. Springer Theses. Heidelberg: Springer (2016).
[56] Huddy, S.R. and Sun, J. Master stability islands for amplitude death in networks of delaycoupled oscillators". Phys. Rev. E, 93, p. 052209 (2016).
[57] Borner, R., Schultz, P., Unzelmann, B., Wang, D., Hellmann, F., and Kurths, J. Delay master stability of inertial oscillator networks". Phys. Rev. Research, 2(2), p. 023409 (2020).
[58] Sorrentino, F. Synchronization of hypernetworks of coupled dynamical systems". New J. Phys., 14, p. 033035 (2012).
[59] Mulas, R., Kuehn, C., and Jost, J. Coupled dynamics on hypergraphs: Master stability of steady states and synchronization". Phys. Rev. E, 101(6), p. 062313 (2020).
[60] Berner, R., Vock, S., Scholl, E., and Yanchuk, S. Desynchronization transitions in adaptive networks". Phys. Rev. Lett., 126(2), p. 028301 (2021).
[61] Gross, T. and Blasius, B. Adaptive coevolutionary networks: a review". J. R. Soc. Interface, 5(20), pp. 259{271 (2008).
[62] Aoki, T. and Aoyagi, T. Co-evolution of phases and connection strengths in a network of phase oscillators". Phys. Rev. Lett., 102, p. 034101 (2009).
[63] Kasatkin, D.V., Yanchuk, S., Scholl, E., and Nekorkin, V.I. Self-organized emergence of multi-layer structure and chimera states in dynamical networks with adaptive couplings". Phys. Rev. E, 96(6), p. 062211 (2017).
[64] Gerstner, W., Kempter, R., von Hemmen, J.L., and Wagner, H. A neuronal learning rule for sub-millisecond temporal coding". Nature, 383(6595), pp. 76{78 (1996).
[65] Abbott, L.F. and Nelson, S. Synaptic plasticity: taming the beast". Nat. Neurosci., 3(11), pp. 1178{1183 (2000).
[66] Dan, Y. and Poo, M.m. Spike timing-dependent plasticity: from synapse to perception". Physiol. Rev., 86(3), pp. 1033{1048 (2006).
[67] Clopath, C., Busing, L., Vasilaki, E., and Gerstner, W. Connectivity reflects coding: a model of voltage-based STDP with homeostasis". Nat. Neurosci., 13(3), pp. 344{352 (2010).
[68] Seliger, P., Young, S.C., and Tsimring, L.S. Plasticity and learning in a network of coupled phase oscillators". Phys. Rev. E, 65(4), p. 041906 (2002).
[69] Berner, R., Scholl, E., and Yanchuk, S. Multiclusters in networks of adaptively coupled phase oscillators". SIAM J. Appl. Dyn. Syst., 18(4), pp. 2227{2266 (2019).
[70] Sakaguchi, H. and Kuramoto, Y. A soluble active rotater model showing phase transitions via mutual entertainment". Prog. Theor. Phys, 76(3), pp. 576{581 (1986).
[71] Asl, M.M., Valizadeh, A., and Tass, P.A. Dendritic and axonal propagation delays may shape neuronal networks with plastic synapses". Front. Physiol., 9, p. 1849 (2018).
[72] Daido, H. Order function and macroscopic mutual entrainment in uniformly coupled limitcycle oscillators". Prog. Theor. Phys., 88(6), pp. 1213{1218 (1992).
[73] Rodrigues, F.A., Peron, T.K.D.M., Ji, P., and Kurths, J. The Kuramoto model in complex networks". Phys. Rep., 610, pp. 1{98 (2016).
[74] Ren, Q. and Zhao, J. Adaptive coupling and enhanced synchronization in coupled phase oscillators". Phys. Rev. E, 76(1), p. 016207 (2007).
[75] Aoki, T. and Aoyagi, T. Self-organized network of phase oscillators coupled by activitydependent interactions". Phys. Rev. E, 84, p. 066109 (2011).
[76] Picallo, C.B. and Riecke, H. Adaptive oscillator networks with conserved overall coupling: Sequential  ring and near-synchronized states". Phys. Rev. E, 83(3), p. 036206 (2011).
[77] Ha, S.Y., Noh, S.E., and Park, J. Synchronization of kuramoto oscillators with adaptive couplings". SIAM J. Appl. Dyn. Syst., 15(1), pp. 162{194 (2016).
[78] Avalos-Gaytan, V., Almendral, J.A., Leyva, I., Battiston, F., Nicosia, V., Latora, V., and Boccaletti, S. Emergent explosive synchronization in adaptive complex networks". Phys. Rev. E, 97(4), p. 042301 (2018).
[79] Maistrenko, Y., Lysyansky, B., Hauptmann, C., Burylko, O., and Tass, P.A. Multistability in the kuramoto model with synaptic plasticity". Phys. Rev. E, 75(6), p. 066207 (2007).
[80] Kasatkin, D.V. and Nekorkin, V.I. Dynamics of the phase oscillators with plastic couplings". Radiophysics and Quantum Electronics, 58(11), pp. 877{891 (2016).
[81] Nekorkin, V.I. and Kasatkin, D.V. Dynamics of a network of phase oscillators with plastic couplings". AIP Conf. Proc., 1738(1), p. 210010 (2016).
[82] Aoki, T. and Aoyagi, T. Scale-free structures emerging from co-evolution of a network and the distribution of a di usive resource on it". Phys. Rev. Lett., 109(20), p. 208702 (2012).
[83] Gushchin, A., Mallada, E., and Tang, A. Synchronization of phase-coupled oscillators with plastic coupling strength". In Information Theory and Applications Workshop ITA 2015, San Diego, CA, USA", pp. 291{300. IEEE (2015).
[84] Timms, L. and English, L.Q. Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity". Phys. Rev. E, 89(3), p. 032906 (2014).
[85] Kasatkin, D.V. and Nekorkin, V.I. Synchronization of chimera states in a multiplex system of phase oscillators with adaptive couplings". Chaos, 28, p. 093115 (2018).
[86] Kachhvah, A.D., Dai, X., Boccaletti, S., and Jalan, S. Interlayer hebbian plasticity induces  rst-order transition in multiplex networks". New J. Phys., 22, p. 122001 (2020).
[87] Berner, R., Fialkowski, J., Kasatkin, D.V., Nekorkin, V.I., Yanchuk, S., and Scholl, E. Hierarchical frequency clusters in adaptive networks of phase oscillators". Chaos, 29(10), p. 103134 (2019).
[88] Kasatkin, D.V. and Nekorkin, V.I. The e ect of topology on organization of synchronous behavior in dynamical networks with adaptive couplings". Eur. Phys. J. Spec. Top., 227, p. 1051 (2018).
[89] Berner, R., Polanska, A., Scholl, E., and Yanchuk, S. Solitary states in adaptive nonlocal oscillator networks". Eur. Phys. J. Spec. Top., 229(12-13), pp. 2183{2203 (2020).