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Adaptive coupling induced multi-stable states in complex networks. (English) Zbl 1285.34050

Summary: Adaptive coupling, where the coupling is dynamical and depends on the behaviour of the oscillators in a complex system, is one of the most crucial factors to control the dynamics and streamline various processes in complex networks. In this paper, we have demonstrated the occurrence of multi-stable states in a system of identical phase oscillators that are dynamically coupled. We find that the multi-stable state is comprised of a two cluster synchronization state where the clusters are in anti-phase relationship with each other and a desynchronization state. We also find that the phase relationship between the oscillators is asymptotically stable irrespective of whether there is synchronization or desynchronization in the system. The time scale of the coupling affects the size of the clusters in the two cluster state. We also investigate the effect of both the coupling asymmetry and plasticity asymmetry on the multi-stable states. In the absence of coupling asymmetry, increasing the plasticity asymmetry causes the system to go from a two clustered state to a desynchronization state and then to a two clustered state. Further, the coupling asymmetry, if present, also affects this transition. We also analytically find the occurrence of the above mentioned multi-stable-desynchronization-multi-stable state transition. A brief discussion on the phase evolution of nonidentical oscillators is also provided. Our analytical results are in good agreement with our numerical observations.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
93C40 Adaptive control/observation systems
05C82 Small world graphs, complex networks (graph-theoretic aspects)
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[1] Winfree, A. T., Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16, 15 (1967)
[2] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0558.76051
[3] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization — A Universal Concept in Nonlinear Sciences (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0993.37002
[4] Strogatz, S. H., Exploring complex networks, Nature, 410, 268 (2001) · Zbl 1370.90052
[5] Singer, W., Neuronal synchrony: a versatile code for the definition of relations?, Neuron, 24, 49 (1999); Fries, P., A mechanism for cognitive dynamics: neuronal communication through neuronal coherence, Trends. Cogn. Sci., 9, 474 (2005); Yamaguchi, Y.; Sato, N.; Wagatsuma, H.; Wu, Z.; Molter, C.; Aota, Y., A unified view of theta-phase coding in the entorhinal-hippocampal system, Curr. Opin. Neurobiol., 17, 197 (2007)
[6] Sheeba, Jane H.; Stefanovska, A.; McClintock, P. V.E., Neuronal synchrony during anesthesia: a thalamocortical model, Biophys. J., 95, 2722 (2008)
[7] Timmermann, L.; Gross, J.; Dirks, M.; Volkmann, J.; Freund, H.; Schnitzler, A., The cerebral oscillatory network of parkinsonian resting tremor, Brain, 126, 199 (2003); Goldberg, J. A.; Boraud, T.; Maraton, S.; Haber, S. N.; Vaadia, E.; Bergman, H., Enhanced synchrony among primary motor cortex neurons in the 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine primate model of Parkinson’s disease, J. Neurosci., 22, 4639 (2002)
[8] Percha, B.; Dzakpasu, R.; Zochowski, M., Transition from local to global phase synchrony in small world neural network and its possible implications for epilepsy, Phys. Rev. E., 72, 031909 (2005); Zucconi, M.; Manconi, M.; Bizzozero, D.; Rundo, F.; Stam, C. J.; Ferini-Strambi, L.; Ferri, R., EEG synchronisation during sleep-related epileptic seizures as a new tool to discriminate confusional arousals from paroxysmal arousals: preliminary findings, Neurol. Sci., 26, 199 (2005)
[9] Pfurtscheller, G.; Neuper, C., Event-related synchronization of mu rhythm in the EEG over the cortical hand area in man, Neurosci. Lett., 174, 93 (1994); Krause, C. M.; Lang, H.; Laine, M.; Pörn, B., Event-related. EEG desynchronization and synchronization during an auditory memory task, Electroencephalogr. Clin. Neurophysiol., 98, 319 (1996); Leocani, L.; Toro, C.; Manganotti, P.; Zhuang, P.; Hallet, M., Event-related coherence and event-related desynchronization/synchronization in the 10 Hz and 20 Hz EEG during self-paced movements, Electroencephalogr. Clin. Neurophysiol., 104, 199-206 (1997); Pfurtschelle, G.; Lopes da Silva, F. H., Event-related EEG/MEG synchronization and desynchronization: basic principles, Clin. Neurophysiol., 110, 1842 (1999)
[10] Sheeba, Jane H.; Chandrasekar, V. K.; Lakshmanan, M., Event-related desynchronization in diffusively coupled oscillator models, Phys. Rev. Lett., 103, 074101 (2009)
[11] Acebron, J.; Bonilla, L.; Perez Vicente, C.; Ritort, F.; Spigler, R., The Kuramoto model: a simple paradigm for synchronization phenomena, Rev. Modern Phys., 77, 137 (2005)
[12] Albert, R.; Barabasi, A.-L., Statistical mechanics of complex networks, Rev. Modern Phys., 74, 47 (2002) · Zbl 1205.82086
[13] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: structure and dynamics, Phys. Rep., 424, 175 (2006) · Zbl 1371.82002
[14] Wu, L.; Zhu, S.; Luo, X., Diversity-induced resonance on weighted scale-free networks, Chaos, 20, 033113 (2010) · Zbl 1311.34089
[15] Perc, M., Stochastic resonance on excitable small-world networks via a pacemaker, Phys. Rev. E, 76, 066203 (2007)
[16] Seliger, P.; Young, S. C.; Tsimring, L. S., Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E, 65, 041906 (2002) · Zbl 1244.34078
[17] Zanette, D. H.; Mikhailov, A. S., Dynamical clustering in oscillator ensembles with time-dependent interactions, Europhys. Lett., 65, 465 (2004)
[18] Zimmermann, M. G.; Eguiluz, V. M.; San Miguel, M., Coevolution of dynamical states and interactions in dynamic networks, Phys. Rev. E, 69, 065102 (2004)
[19] Pacheco, J. M.; Traulsen, A.; Nowak, M. A., Co-evolution of strategy and structure in complex networks with dynamical linking, Phys. Rev. Lett., 97, 258103 (2006)
[20] Masuda, N.; Kori, H., Formation of feedforward networks and frequency synchrony by spike-timing-dependent plasticity, J. Comput. Neurosci., 22, 327 (2007)
[21] Maistrenko, Y. L.; Lysyansky, B.; Hauptmann, C.; Burylko, O.; Tass, P. A., Multistability in the Kuramoto model with synaptic plasticity, Phys. Rev. E, 75, 066207 (2007)
[22] Cateau, H.; Kitano, K.; Fukai, T., Interplay between a phase response curve and spike-timing-dependent plasticity leading to wireless clustering, Phys. Rev. E, 77, 051909 (2008)
[23] Chen, M.; Shang, Y.; Zou, Y.; Kurths, J., Synchronization in the Kuramoto model: a dynamical gradient network approach, Phys. Rev. E, 77, 027101 (2008)
[24] Gilson, M.; Burkitt, A. N.; Grayden, D. B.; Thomas, D. A.; van Hemmen, J. L., Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks III: partially connected neurons driven by spontaneous activity, Biol. Cybernet., 101, 411 (2009) · Zbl 1344.92019
[25] Szolnoki, A.; Perc, M., Resolving social dilemmas on evolving random networks, Europhys. Lett., 86, 30007 (2009)
[26] Szolnoki, A.; Perc, M., Emergence of multilevel selection in the prisoner’s dilemma game on coevolving random networks, New J. Phys., 11, 093033 (2009)
[27] Poncela, J.; Gomez-Gardenes, J.; Traulsen, A.; Moreno, Y., Evolutionary game dynamics in a growing structured population, New J. Phys., 11, 083031 (2009)
[28] Perc, M.; Szolnoki, A., Coevolutionary games—a mini review, Biosystems, 99, 109 (2010)
[29] Iwasa, M.; Iida, K.; Tanaka, D., Hierarchical cluster structures in a one-dimensional swarm oscillator model, Phys. Rev. E, 81, 046220 (2010)
[30] Iwasa, M.; Tanaka, D., Dimensionality of clusters in a swarm oscillator model, Phys. Rev. E, 81, 066214 (2010)
[31] Bi, G. Q.; Poo, M. M., Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type, J. Neurosci., 18, 10464 (1998)
[32] Markram, H.; Lubke, J.; Frotscher, M.; Sakmann, B., Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs, Science, 275, 213 (1997)
[33] Caporale, N.; Dan, Y., Spike timing-dependent plasticity: a Hebbian learning rule, Annu. Rev. Neurosci., 31, 25 (2008)
[34] Zhou, C. S.; Kurths, J., Dynamical weights and enhanced synchronization in adaptive complex networks, Phys. Rev. Lett., 96, 164102 (2006); Ren, Q.; Zhao, J., Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Rev. E, 76, 016207 (2007)
[35] Hou, J. L.; Zhao, J., The order-oscillation induced by negative feedback in the adaptive scheme, Phys. Lett. A, 374, 929-932 (2010) · Zbl 1235.34153
[36] Chandrasekar, V. K.; Sheeba, Jane H.; Lakshmanan, M., Mass synchronization: occurrence and its control with possible applications to brain dynamics, Chaos, 20, 045106 (2010) · Zbl 1311.92016
[37] Aoki, T.; Aoyagi, T., Co-evolution of phases and connection strengths in a network of phase oscillators, Phys. Rev. Lett., 102, 034101 (2009)
[38] Aoki, T.; Aoyagi, T., Self-organized network of phase oscillators coupled by activity-dependent interactions, Phys. Rev. E, 84, 066109 (2011)
[39] Yuan, W.; Zhou, C., Interplay between structure and dynamics in adaptive complex networks: emergence and amplification of modularity by adaptive dynamics, Phys. Rev. E, 84, 016116 (2011)
[40] Zhu, J.; Zhao, M.; Yu, W.; Zhou, C.; Wang, B., Better synchronizability in generalized adaptive networks, Phys. Rev. E, 81, 026201 (2010)
[41] Lama, M. S.; Lopez, J. M.; Wio, H. S., Spontaneous emergence of contrarian-like behaviour in an opinion spreading model, Europhys. Lett., 72, 851 (2005)
[42] Demirel, Guven; Prizak, Roshan; Nitish Reddy, P.; Gross, Thilo, Opinion formation and cyclic dominance in adaptive networks
[43] Pluchino, Alessandro; Boccaletti, Stefano; Latora, Vito; Rapisarda, Andrea, Opinion dynamics and synchronization in a network of scientific collaborations, Physica A, 372, 316 (2006)
[44] Furusawa, C.; Kaneko, K., Zipf’s law in gene expression, Phys. Rev. Lett., 90, 088102 (2003)
[45] Jain, S.; Krishna, S., A model for the emergence of cooperation, interdependence, and structure in evolving networks, Proc. Natl. Acad. Sci. USA, 98, 543 (2001)
[46] Tero, A.; Takagi, S.; Saigusa, T.; Ito, K.; Bebber, D. P.; Fricker, M. D.; Yumiki, K.; Kobayashi, R.; Nakagaki, T., Rules for biologically inspired adaptive network design, Science, 327, 439 (2010) · Zbl 1226.90021
[47] Harris, K. D.; Csicsvari, J.; Hirase, H.; Dragoi, G.; Buzsaki, G., Organization of cell assemblies in the hippocampus, Nature (London), 424, 552 (2003)
[48] Gross, T.; Blasius, B., Adaptive coevolutionary networks: a review, J. R. Soc. Interface, 5, 259 (2008)
[49] Abrams, Daniel M.; Mirollo, Rennie; Strogatz, Steven H.; Wiley, Daniel A., Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101, 084103 (2008)
[50] Hong, H.; Strogatz, S. H., Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators, Phys. Rev. Lett., 106, 054102 (2011)
[51] Hong, H.; Strogatz, S. H., Conformists and contrarians in a Kuramoto model with identical natural frequencies, Phys. Rev. E, 84, 046202 (2011)
[52] Wu, D.; Zhu, S.; Luo, X.; Wu, L., Effects of adaptive coupling on stochastic resonance of small-world networks, Phys. Rev. E, 84, 021102 (2011)
[53] Huang, D., Synchronization in adaptive weighted networks, Phys. Rev. E, 74, 046208 (2006)
[54] Li, M.; Wang, X.; Fan, Y.; Di, Z.; Lai, C., Onset of synchronization in weighted complex networks: the effect of weight-degree correlation, Chaos, 21, 025108 (2011) · Zbl 1317.34108
[55] Ko, Tae-Wook; Bard Ermentrout, G., Partially locked states in coupled oscillators due to inhomogeneous coupling, Phys. Rev. E, 78, 016203 (2008)
[56] Tanaka, T.; Aoyagi, T., Multistable attractors in a network of phase oscillators with three-body interactions, Phys. Rev. Lett., 106, 224101 (2011)
[57] Izhikevich, E. M., Phase models with explicit time delays, Phys. Rev. E, 58, 905 (1998)
[58] Strogatz, Steven H., From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143, 1-20 (2000) · Zbl 0983.34022
[59] Paissan, Gabriel H.; Zanette, Damian H., Synchronization of phase oscillators with heterogeneous coupling: a solvable case, Physica D, 237, 818-828 (2008) · Zbl 1163.34032
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