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Emerging Asymptotic Patterns in a Winfree Ensemble with Higher-Order Couplings

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Abstract

The Winfree model is a phase-coupled synchronization model which simplifies pulse-coupled models such as the Peskin model on pacemaker cells. It is well-known that the Winfree ensemble with the first-order coupling exhibits discrete asymptotic patterns such as incoherence, locking and death depending on the coupling strength and variance of natural frequencies. In this paper, we further study higher-order couplings which makes the dynamics more close to the behaviors of the Peskin model. For this, we propose several sufficient frameworks for asymptotic patterns compared to the first-order coupling model. Our proposed conditions on the coupling strength, natural frequencies and initial data are independent of the number of oscillators so that they can be applied to the corresponding mean-field model. We also provide several numerical simulations and compare them with analytical results.

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References

  1. Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)

    Article  Google Scholar 

  2. Aeyels, D., Rogge, J.: Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators. Prog. Theor. Phys. 112, 921–941 (2004)

    Article  MATH  Google Scholar 

  3. Akhmet, M.U.: Self-synchronization of the integrate-and-fire pacemaker model with continuous couplings. Nonlinear Anal. Hybrid Syst. 6, 730–740 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Albi, G., Bellomo, N., Fermo, L., Ha, S.-Y., Kim, J., Pareschi, L., Poyato, D., Soler, J.: Vehicular traffic, crowds and swarms: from kinetic theory and multiscale methods to applications and research perspectives. Math. Models Methods Appl. Sci. 29, 1901–2005 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ariaratnam, J.T., Strogatz, S.H.: Phase diagram for the Winfree model of coupled nonlinear oscillators. Phys. Rev. Lett. 86, 4278–4281 (2001)

    Article  Google Scholar 

  6. Benedetto, D., Caglioti, E., Montemagno, U.: On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13, 1775–1786 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bronski, J., Deville, L., Park, M.J.: Fully synchronous solutions and the synchronization phase transition for the finite-\(N\) Kuramoto model. Chaos 22, 033133 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, Y.-P., Ha, S.-Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D 241, 735–754 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chopra, N., Spong, M.W.: On exponential synchronization of Kuramoto oscillators. IEEE Trans. Autom. Control 54, 353–357 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong, J.-G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11, 465–480 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dörfler, F., Bullo, F.: On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 10, 1070–1099 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dörfler, F., Bullo, F.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50, 1539–1564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ha, S.-Y., Kim, H.K., Ryoo, S.W.: Emergence of phase-locked states for the Kuramoto model in a large coupling regime. Commun. Math. Sci. 4, 1073–1091 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ha, S.-Y., Ko, D., Park, J., Ryoo, S.W.: Emergence of partial locking states from the ensemble of Winfree oscillators. Q. Appl. Math. 75, 39–68 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ha, S.-Y., Ko, D., Park, J., Zhang, X.: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3, 209–267 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ha, S.-Y., Park, J., Ryoo, S.W.: Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete Contin. Dyn. Syst. 35, 3417–3436 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ha, S.-Y., Ryoo, S.W.: Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model. Commun. Math. Phys. 377, 811–857 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jadbabaie, A., Motee, N., Barahona, M.: On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proceedings of the American Control Conference, pp. 4296–4301 (2004)

    Google Scholar 

  19. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  20. Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420 (1975)

    Article  Google Scholar 

  21. Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645–1662 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nguyen, L.H., Hong, K.-S.: Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions. Math. Comput. Simul. 82, 590–603 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oukil, W., Thieullen, Ph., Kessi, A.: Invariant cone and synchronization state stability of the mean field models. Dyn. Syst. 34, 422–433 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Oukil, W., Kessi, A., Thieullen, Ph.: Synchronization hypothesis in the Winfree model. Dyn. Syst. 32, 326–339 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peskin, C.S.: Mathematical Aspects of Heart Physiology, pp. 268–278. Courant Institute of Mathematical Sciences, New York University, New York (1975)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  27. Plotnikov, S.A., Fradkov, A.L.: On synchronization in heterogeneous FitzHugh-Nagumo networks. Chaos Solitons Fractals 121, 85–91 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Steur, E., Tyukin, I., Nijmeijer, H.: Semi-passivity and synchronization of diffusively coupled neuronal oscillators. Physica D 328, 2119–2128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Verwoerd, M., Mason, O.: A convergence result for the Kurmoto model with all-to-all couplings. SIAM J. Appl. Dyn. Syst. 10, 906–920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Verwoerd, M., Mason, O.: On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph. SIAM J. Appl. Dyn. Syst. 8, 417–453 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Verwoerd, M., Mason, O.: Global phase-locking in finite populations of phase-coupled oscillators. SIAM J. Appl. Dyn. Syst. 7, 134–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)

    Article  Google Scholar 

  34. Zong, Y., Dai, X., Gao, Z., Busawon, K., Zhu, J.: Modelling and synchronization of pulse-coupled non-identical oscillators for wireless sensor networks. In 2018 IEEE 16th International Conference on Industrial Informatics. (INDIN) (pp. 101-107). IEEE

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Acknowledgements

The work of D. Ko was supported by The Catholic University of Korea, Research Fund, 2022, and by National Research Foundation of Korea (NRF-2021R1G1A1008559). The work of S.-Y. Ha was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881). The work of J. Yoon is supported by DFG-NRF International Research Training Group IRTG 2235 supporting the Bielefeld-Seoul graduate exchange programme.

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Authors and Affiliations

  1. Department of Mathematics, The Catholic University of Korea, Jibongro 43, Bucheon, Gyeonggido, 14662, Republic of Korea

    Dongnam Ko

  2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  3. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Jaeyoung Yoon

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  1. Dongnam Ko
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  2. Seung-Yeal Ha
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Correspondence to Jaeyoung Yoon.

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Ko, D., Ha, SY. & Yoon, J. Emerging Asymptotic Patterns in a Winfree Ensemble with Higher-Order Couplings. Acta Appl Math 185, 2 (2023). https://doi.org/10.1007/s10440-023-00571-9

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