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Compact dynamical mean-field theory of oscillator networks

Kanishka Reddy*

  • *Contact author: kani@uw.edu
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Phys. Rev. E 113, 034222 – Published 30 March, 2026

DOI: https://doi.org/10.1103/lt71-mfpw

Abstract

We present a compact dynamical mean-field theory (DMFT) for large networks of coupled phase oscillators whose phases live on the circle S1 and interact with both coherent mean-field coupling and quenched randomness. Starting from wrapped Langevin dynamics, we build a path-integral representation that keeps the 2π periodicity of the phases explicit. After averaging over the disorder in the thermodynamic limit, this construction reduces to a single-oscillator stochastic equation driven by a deterministic mean field and a self-consistent colored Gaussian noise, whose covariance is fixed by a circular two-time correlator. In the limit of vanishing disorder, the formalism reproduces the Ott-Antonsen reduction and recovers standard Kuramoto and theta-neuron neural-mass equations. The same framework accommodates arbitrary 2π-periodic coupling functions, including those obtained from infinitesimal phase response curves (IPRCs) of biophysical neuron models. As an example, we show that for adaptive exponential integrate-and-fire neurons, inserting an IPRC-fitted coupling into the compact DMFT yields quantitative predictions for synchronization thresholds, providing a direct route from single-neuron phase response data to network-level mean-field predictions for arbitrary phase-reducible oscillators.

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References (47)

  1. Y. Kuramoto, in International Symposium on Mathematical Problems in Theoretical Physics, edited by H. Araki, Lecture Notes in Physics Vol. 39 (Springer, Berlin, 1975), pp. 420–422.
  2. S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143, 1 (2000).
  3. J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  4. F. A. Rodrigues, T. K. D. M. Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Phys. Rep. 610, 1 (2016).
  5. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57, 1563 (1998).
  6. G. Buzsáki and X.-J. Wang, Mechanisms of gamma oscillations, Annu. Rev. Neurosci. 35, 203 (2012).
  7. E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008).
  8. S. Watanabe and S. H. Strogatz, Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).
  9. E. Ott and T. M. Antonsen, Long time evolution of phase oscillator systems, Chaos 19, 023117 (2009).
  10. E. Montbrió, D. Pazó, and A. Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X 5, 021028 (2015).
  11. D. Pazó and E. Montbrió, Low-dimensional dynamics of populations of pulse-coupled oscillators, Phys. Rev. X 4, 011009 (2014).
  12. C. R. Laing, Derivation of a neural field model from a network of theta neurons, Phys. Rev. E 90, 010901(R) (2014).
  13. 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).
  14. B. Ottino-Löffler and S. H. Strogatz, Volcano transition in a solvable model of frustrated oscillators, Phys. Rev. Lett. 120, 264102 (2018).
  15. D. Pazó and R. Gallego, Volcano transition in populations of phase oscillators with random nonreciprocal interactions, Phys. Rev. E 108, 014202 (2023).
  16. Y. Kati, J. Ranft, and B. Lindner, Self-consistent autocorrelation of a disordered Kuramoto model in the asynchronous state, Phys. Rev. E 110, 054301 (2024).
  17. H. Hong and E. A. Martens, First-order like phase transition induced by quenched coupling disorder, Chaos 32, 063125 (2022).
  18. I. León and D. Pazó, Enlarged Kuramoto model: Secondary instability and transition to collective chaos, Phys. Rev. E 105, L042201 (2022).
  19. I. León and D. Pazó, Dynamics and chaotic properties of the fully disordered Kuramoto model, Chaos 35, 073140 (2025).
  20. C. Bick, M. J. Panaggio, and E. A. Martens, Chaos in Kuramoto oscillator networks, Chaos 28, 071102 (2018).
  21. S. Olmi, A. Politi, and A. Torcini, Collective chaos in pulse-coupled neural networks, Europhys. Lett. 92, 60007 (2010).
  22. A. Prüser, S. Rosmej, and A. Engel, Nature of the volcano transition in the fully disordered Kuramoto model, Phys. Rev. Lett. 132, 187201 (2024).
  23. A. Prüser and A. Engel, Role of coupling asymmetry in the fully disordered Kuramoto model, Phys. Rev. E 110, 064214 (2024).
  24. J. Villain, Theory of one- and two-dimensional magnets with an easy magnetization plane. II. The planar, classical, two-dimensional magnet, J. Phys. (Paris) 36, 581 (1975).
  25. L. M. Sieberer, G. Wachtel, E. Altman, and S. Diehl, Lattice duality for the compact Kardar-Parisi-Zhang equation, Phys. Rev. B 94, 104521 (2016).
  26. T. B. Luke, E. Barreto, and P. So, Complete classification of the macroscopic behavior of a heterogeneous network of theta neurons, Neural Comput. 25, 3207 (2013).
  27. E. Brown, J. Moehlis, and P. Holmes, On the phase reduction and response dynamics of neural oscillator populations, Neural Comput. 16, 673 (2004).
  28. M. Augustin, J. Ladenbauer, F. Baumann, and K. Obermayer, Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation, PLoS Comput. Biol. 13, e1005545 (2017).
  29. Y. Zerlaut, S. Chemla, F. Chavane, and A. Destexhe, Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons, J. Comput. Neurosci. 44, 45 (2018).
  30. M. di Volo, A. Romagnoni, C. Capone, and A. Destexhe, Biologically realistic mean-field models of conductance-based networks of spiking neurons with adaptation, Neural Comput. 31, 653 (2019).
  31. M. Carlu, O. Chehab, L. Dalla Porta, D. Depannemaecker, C. Héricé, M. Jedynak, E. K. Ersöz, P. Muratore, S. Souihel, C. Capone, Y. Zerlaut, A. Destexhe, and M. di Volo, A mean-field approach to the dynamics of networks of complex neurons, from nonlinear integrate-and-fire to Hodgkin–Huxley models, J. Neurophysiol. 123, 1042 (2020).
  32. H. Daido, Multibranch entrainment and scaling in large populations of coupled oscillators, Phys. Rev. Lett. 77, 1406 (1996).
  33. P. S. Skardal, E. Ott, and J. G. Restrepo, Cluster synchrony in systems of coupled phase oscillators with higher-order coupling, Phys. Rev. E 84, 036208 (2011).
  34. P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dynamics of classical systems, Phys. Rev. A 8, 423 (1973).
  35. J. A. Hertz, Y. Roudi, and P. Sollich, Path integral methods for the dynamics of stochastic and disordered systems, J. Phys. A: Math. Theor. 50, 033001 (2017).
  36. H. Sompolinsky, A. Crisanti, and H.-J. Sommers, Chaos in random neural networks, Phys. Rev. Lett. 61, 259 (1988).
  37. A. Crisanti and H. Sompolinsky, Path-integral approach to random neural networks, Phys. Rev. E 98, 062120 (2018).
  38. A. Crisanti, H. Horner, and H.-J. Sommers, The spherical p-spin interaction spin-glass model, Z. Phys. B 92, 257 (1993).
  39. G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Interdisciplinary Applied Mathematics Vol. 35 (Springer, New York, 2010).
  40. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15 (1967).
  41. T. I. Netoff, M. A. Schwemmer, and T. J. Lewis, in Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis, edited by N. W. Schultheiss, A. A. Prinz, and R. J. Butera, Springer Series in Computational Neuroscience Vol. 6 (Springer, New York, 2012), pp. 95–129.
  42. N. W. Schultheiss, J. R. Edgerton, and D. Jaeger, Phase response curve analysis of a full morphological globus pallidus neuron model reveals distinct perisomatic and dendritic modes of synaptic integration, J. Neurosci. 30, 2767 (2010).
  43. G. B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46, 233 (1986).
  44. R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol. 94, 3637 (2005).
  45. J. P. L. Hatchett and T. Uezu, Mean field and cavity analysis for coupled oscillator networks, Phys. Rev. E 78, 036106 (2008).
  46. F. L. Metz, Dynamical mean-field theory of complex systems on sparse directed networks, Phys. Rev. Lett. 134, 037401 (2025).
  47. K. Reddy, Simulation code for “Compact dynamical mean-field theory of oscillator networks,” Zenodo (2026), https://doi.org/10.5281/zenodo.18796357.

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Phys. Rev. E 113, 034222– Published 30 March, 2026
  • Received 22 November 2025
  • Accepted 24 February 2026
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