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Crowd synchrony in chaotic oscillators

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Abstract

Quorum sensing is a phenomenon wherein the size of an ensemble (population density) decides its dynamical state. This happens when its constituting elements alter their dynamics coherently with the change in their population. In case of chaotic ensemble, as a precursor to the emergence of global chaotic synchronization, the chaotic elements undergo various dynamical transitions (chaotic state \(\rightarrow \) silent/periodic state \(\rightarrow \) periodic state/ intermittent chaos \(\rightarrow \) chaotic state) sequentially with the increase in their population. Among these sequence of quorum transitions, we mark the transition periodic state/intermittent chaos \(\rightarrow \) chaotic state (i.e the final quorum transition between the unsynchronized and synchronized chaotic states) as a ‘crowd synchrony’ transition for convenience. In contrast to the conventional quorum sensing mechanism, we study this population-based phenomenon by exploring the scenario wherein the chaotic elements interact via an element (of same kind) resting in a steady state, i.e the surrounding of the elements is not only dynamic but also incorporates the underlying features of the elements. Apart from the other advantages (discussed in the text), considering the surrounding of the elements as an element, resolves the issue regarding the dimensionality of the surround (which has not been addressed before) when more than one interacting species are involved. The proposed mechanism has been tested on two different class of chaotic oscillators: spiking neuron (i.e a relaxed biological oscillator) and Chua’s circuit, i.e a sinusoidal-type electrical oscillator (for the experimental verification).

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References

  1. Konigsberg, I.R.: Diffusion-mediated control of myoblast fusion. Dev. Biol. 26, 133–152 (1971)

    Article  Google Scholar 

  2. Martz, E., Steinberg, M.S.: The role of cell–cell contact in contact inhibition of cell division: a review and new evidence. J. Cell. Physiol. 79, 189–210 (1972)

    Article  Google Scholar 

  3. DeHaan, R.L., Sachs, H.G.: Cell coupling in developing systems: the heart cell paradigm. Curr. Top. Dev. Biol. 7, 193–228 (1972)

    Article  Google Scholar 

  4. Aldridge, J., Pye, E.K.: Cell density dependence of oscillatory metabolism. Nature 259, 670–671 (1976)

    Article  Google Scholar 

  5. Dano, S., Sorensen, P.G., Hynne, F.: Sustained oscillations in living cells. Nature 402, 320–322 (1999)

    Article  Google Scholar 

  6. Miller, M.B., Bassler, B.L.: Quorum sensing in bacteria. Ann. Rev. Microbiol. 55, 165–199 (2001)

    Article  Google Scholar 

  7. Garcia-Ojalvo, J., Elowitz, M.B., Strogatz, S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101(30), 10,955–10,960 (2004)

    Article  MathSciNet  Google Scholar 

  8. Waters, C.M., Bassler, B.L.: Quorum sensing: cell-to-cell communication in bacteria. Ann. Rev. Cell Dev. Biol. 21(1), 319–346 (2005)

    Article  Google Scholar 

  9. Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B., Ott, E.: Theoretical mechanics: crowd synchrony on the millennium bridge. Nature 438, 43–44 (2005)

    Article  Google Scholar 

  10. Camilli, A., Bassler, B.L.: Bacterial small-molecule signaling pathways. Science 311(5764), 1113–1116 (2006)

    Article  Google Scholar 

  11. De Monte, S., d’Ovidio, F., Dano, S., Sorensen, P.G.: Dynamical quorum sensing: population density encoded in cellular dynamics. Proc. Natl. Acad. Sci. 104(47), 18,377–18,381 (2007)

    Article  Google Scholar 

  12. Danino, T., Mondragon-Palomino, O., Tsimring, L., Hasty, J.: A synchronized quorum of genetic clocks. Nature 463, 326–330 (2010)

    Article  Google Scholar 

  13. Gregor, T., Fujimoto, K., Masaki, N., Sawai, S.: The onset of collective behavior in social amoebae. Science 328(5981), 1021–1025 (2010)

    Article  Google Scholar 

  14. Chen, A.: Modelling a synthetic biological chaotic system: relaxation oscillators coupled by quorum sensing. Nonlinear Dyn. 63, 711–718 (2011)

    Article  Google Scholar 

  15. Rubin, J.J., Rubin, J.E., Ermentrout, G.B.: Analysis of synchronization in a slowly changing environment: how slow coupling becomes fast weak coupling. Phys. Rev. Lett. 110, 204,101 (2013)

    Article  Google Scholar 

  16. Singh, H., Parmananda, P.: Alternate coupling mechanism for dynamical quorum sensing. J. Phys. Chem. A 116(42), 10,269–10,275 (2012)

    Article  Google Scholar 

  17. Toth, R., Taylor, A.F., Tinsley, M.R.: Collective behavior of a population of chemically coupled oscillators. J. Phys. Chem. B 110(20), 10,170–10,176 (2006)

    Article  Google Scholar 

  18. Taylor, A.F., Tinsley, M.R., Wang, F., Huang, Z., Showalter, K.: Dynamical quorum sensing and synchronization in large populations of chemical oscillators. Science 323(5914), 614–617 (2009)

    Article  Google Scholar 

  19. Tinsley, M.R., Taylor, A.F., Huang, Z., Showalter, K.: Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing. Phys. Rev. Lett. 102, 158,301 (2009)

    Article  Google Scholar 

  20. Tinsley, M., Taylor, A., Huang, Z., Wang, F., Showalter, K.: Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles. Phys. D Nonlinear Phenom. 239(11), 785–790 (2010)

    Article  Google Scholar 

  21. Li, B.W., Fu, C., Zhang, H., Wang, X.: Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators. Phys. Rev. E 86, 046,207 (2012)

  22. Zamora-Munt, J., Masoller, C., Garcia-Ojalvo, J., Roy, R.: Crowd synchrony and quorum sensing in delay-coupled lasers. Phys. Rev. Lett. 105, 264,101 (2010)

  23. Cohen, E., Rosenbluh, M., Kanter, I.: Phase transition in crowd synchrony of delay-coupled multilayer laser networks. Opt. Express 20, 19,683–19,689 (2012)

    Article  Google Scholar 

  24. Hellen, E.H., Dana, S.K., Zhurov, B., Volkov, E.: Electronic implementation of a repressilator with quorum sensing feedback. PLoS One 8, e62,997 (2013)

    Article  Google Scholar 

  25. Singh, H., Parmananda, P.: Quorum sensing via static coupling demonstrated by Chua’s circuits. Phys. Rev. E 88, 040,903 (2013)

    Article  Google Scholar 

  26. Schwab, D.J., Baetica, A., Mehta, P.: Dynamical quorum-sensing in oscillators coupled through an external medium. Phys. D Nonlinear Phenom. 241(21), 1782–1788 (2012)

    Article  MATH  Google Scholar 

  27. Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B Biol. Sci. 221(1222), 87–102 (1984)

    Article  Google Scholar 

  28. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. part 1: Theory. Meccanica 15(1), 9–20 (1980)

    Article  MATH  Google Scholar 

  29. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Prog. Theor. Phys. 75(5), 1105–1110 (1986)

    Article  Google Scholar 

  31. Pikovsky, A.S., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  32. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)

    MATH  Google Scholar 

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Acknowledgments

Financial support from IIT Bombay and DST, India is acknowledged.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Bombay, Powai, 400 076, Mumbai, India

    Harpartap Singh & P. Parmananda

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  1. Harpartap Singh
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  2. P. Parmananda
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Correspondence to Harpartap Singh.

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Singh, H., Parmananda, P. Crowd synchrony in chaotic oscillators. Nonlinear Dyn 80, 767–776 (2015). https://doi.org/10.1007/s11071-015-1904-9

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