- 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
  • Letter
  • Open Access

Two-phase quadratic integrate-and-fire neurons: Exact low-dimensional description for ensembles of finite-voltage neurons

Rok Cestnik*

  • Centre for Mathematical Sciences, Lund University, Märkesbacken 4, 22362 Lund, Sweden

  • *Contact author: rok.cestnik@math.lth.se
PDF

Phys. Rev. Research 8, L012049 – Published 2 March, 2026

DOI: https://doi.org/10.1103/tq4x-8ny1

Abstract

We introduce a two-phase quadratic integrate-and-fire (QIF) neuron whose membrane potential evolves according to two alternating Riccati equations within finite bounds. This simple extension removes the unphysical voltage divergence of the standard QIF model while producing realistic spike waveforms. Despite this modification, the system retains an exact low-dimensional description in the thermodynamic limit, governed by a single complex Riccati equation. Expressions for collective quantities such as the firing rate and mean voltage remain compact and analytically tractable. Because the formalism preserves the mathematical structure of the standard QIF ensemble, it inherits its many generalizations and can serve as a drop-in replacement in existing mean-field frameworks, providing a more biologically plausible yet still exactly solvable neuronal model.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (62)

  1. G. B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46, 233 (1986).
  2. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comput. 8, 979 (1996).
  3. P. E. Latham, B. J. Richmond, P. G. Nelson, and S. Nirenberg, Intrinsic dynamics in neuronal networks. I. Theory, J. Neurophysiol. 83, 808 (2000).
  4. E. Montbrió, D. Pazó, and A. Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X 5, 021028 (2015).
  5. E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008).
  6. B. Pietras and E. Montbrió, Heterogeneous populations of quadratic integrate-and-fire neurons: On the generality of Lorentzian distributions, arXiv:2409.18278.
  7. H. Bi, M. di Volo, and A. Torcini, Asynchronous and coherent dynamics in balanced excitatory-inhibitory spiking networks, Front. Syst. Neurosci. 15, 752261 (2021).
  8. P. Clusella and E. Montbrió, Exact low-dimensional description for fast neural oscillations with low firing rates, Phys. Rev. E 109, 014229 (2024).
  9. M. di Volo and A. Torcini, Transition from asynchronous to oscillatory dynamics in balanced spiking networks with instantaneous synapses, Phys. Rev. Lett. 121, 128301 (2018).
  10. B. Pietras, F. Devalle, A. Roxin, A. Daffertshofer, and E. Montbrió, Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks, Phys. Rev. E 100, 042412 (2019).
  11. E. Montbrió and D. Pazó, Exact mean-field theory explains the dual role of electrical synapses in collective synchronization, Phys. Rev. Lett. 125, 248101 (2020).
  12. I. Ratas and K. Pyragas, Macroscopic self-oscillations and aging transition in a network of synaptically coupled quadratic integrate-and-fire neurons, Phys. Rev. E 94, 032215 (2016).
  13. S. Coombes, Next generation neural population models, Front. Appl. Math. Stat. 9, 1128224 (2023).
  14. D. S. Goldobin, M. di Volo, and A. Torcini, Reduction methodology for fluctuation driven population dynamics, Phys. Rev. Lett. 127, 038301 (2021).
  15. D. S. Goldobin, Mean-field models of populations of quadratic integrate-and-fire neurons with noise on the basis of the circular cumulant approach, Chaos 31, 083112 (2021).
  16. D. S. Goldobin, E. V. Permyakova, and L. S. Klimenko, Macroscopic behavior of populations of quadratic integrate-and-fire neurons subject to non-Gaussian white noise, Chaos 34, 013121 (2024).
  17. H. Taher, A. Torcini, and S. Olmi, Exact neural mass model for synaptic-based working memory, PLoS Comput. Biol. 16, e1008533 (2020).
  18. S. Coombes and Á. Byrne, Next generation neural mass models, in Nonlinear Dynamics in Computational Neuroscience, edited by F. Corinto and A. Torcini (Springer International Publishing, Cham, 2019), pp. 1–16.
  19. A. Byrne, R. D. O'Dea, M. Forrester, J. Ross, and S. Coombes, Next-generation neural mass and field modeling, J. Neurophysiol. 123, 726 (2020).
  20. 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).
  21. P. So, T. B. Luke, and E. Barreto, Networks of theta neurons with time-varying excitability: Macroscopic chaos, multistability, and final-state uncertainty, Physica D 267, 16 (2014).
  22. C. R. Laing, Derivation of a neural field model from a network of theta neurons, Phys. Rev. E 90, 010901(R) (2014).
  23. C. R. Laing, Exact neural fields incorporating gap junctions, SIAM J. Appl. Dyn. Syst. 14, 1899 (2015).
  24. D. Pazó and E. Montbrió, Low-dimensional dynamics of populations of pulse-coupled oscillators, Phys. Rev. X 4, 011009 (2014).
  25. I. V. Tyulkina, D. S. Goldobin, L. S. Klimenko, and A. Pikovsky, Dynamics of noisy oscillator populations beyond the Ott-Antonsen ansatz, Phys. Rev. Lett. 120, 264101 (2018).
  26. R. Cestnik and A. Pikovsky, Hierarchy of exact low-dimensional reductions for populations of coupled oscillators, Phys. Rev. Lett. 128, 054101 (2022).
  27. R. Cestnik and A. Pikovsky, Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories, Chaos 32, 113126 (2022).
  28. A. Pikovsky and M. Rosenblum, Dynamics of globally coupled oscillators: Progress and perspectives, Chaos 25, 097616 (2015).
  29. P. S. Skardal, Low-dimensional dynamics of the Kuramoto model with rational frequency distributions, Phys. Rev. E 98, 022207 (2018).
  30. V. Pyragas and K. Pyragas, Mean-field equations for neural populations with q-Gaussian heterogeneities, Phys. Rev. E 105, 044402 (2022).
  31. A. Pikovsky and M. Rosenblum, Dynamics of heterogeneous oscillator ensembles in terms of collective variables, Physica D 240, 872 (2011).
  32. S. A. Marvel, R. E. Mirollo, and S. H. Strogatz, Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action, Chaos 19, 043104 (2009).
  33. E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So, and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distribution, Phys. Rev. E 79, 026204 (2009).
  34. O. E. Omel'chenko, Periodic orbits in the Ott–Antonsen manifold, Nonlinearity 36, 845 (2023).
  35. C. R. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D 238, 1569 (2009).
  36. O. E. Omel'chenko, M. Wolfrum, S. Yanchuk, Y. L. Maistrenko, and O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators, Phys. Rev. E 85, 036210 (2012).
  37. P. Ashwin, S. Coombes, and R. Nicks, Mathematical frameworks for oscillatory network dynamics in neuroscience, J. Math. Neurosci. 6, 2 (2016).
  38. 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).
  39. R. Gast, H. Schmidt, and T. R. Knösche, A mean-field description of bursting dynamics in spiking neural networks with short-term adaptation, Neural Comput. 32, 1615 (2020).
  40. R. Gast, T. R. Knösche, and H. Schmidt, Mean-field approximations of networks of spiking neurons with short-term synaptic plasticity, Phys. Rev. E 104, 044310 (2021).
  41. A. Ferrara, D. Angulo-Garcia, A. Torcini, and S. Olmi, Population spiking and bursting in next-generation neural masses with spike-frequency adaptation, Phys. Rev. E 107, 024311 (2023).
  42. R. Gast, S. A. Solla, and A. Kennedy, Macroscopic dynamics of neural networks with heterogeneous spiking thresholds, Phys. Rev. E 107, 024306 (2023).
  43. L. Chen and S. A. Campbell, Exact mean-field models for spiking neural networks with adaptation, J. Comput. Neurosci. 50, 445 (2022).
  44. B. Duchet, C. Bick, and Á. Byrne, Mean-field approximations with adaptive coupling for networks with spike-timing-dependent plasticity, Neural Comput. 35, 1481 (2023).
  45. N. Fennelly, A. Neff, R. Lambiotte, A. Keane, and Á. Byrne, Mean-field approximation for networks with synchrony-driven adaptive coupling, Chaos 35, 013152 (2025).
  46. This type of distribution (3) can in practice be sampled by drawing standard Cauchy random numbers corresponding to parameter Q: C∼Cauchy(Re[Q],Im[Q]) (for example, by taking the ratio of two Gaussian random numbers) and if the sampled value falls outside the interval [vmin,vmax], it is transformed via Eq. (6): −vminvmax/C+vmin+vmax.
  47. Defining spikes as voltage threshold crossings can, in rare cases, lead to counting the same spike multiple times if the voltage during phase II exceeds vmax, effectively looping back through the firing threshold. This artifact is only relevant for small threshold values of vmax.
  48. Second moment: V2=〈vj2〉=1πIm[F2(Q)+F2(QII)], where F2(Q)=Q(vmax−vmin)+Q2ln(Q−vmaxQ−vmin). Third moment: V3=〈vj3〉=1πIm[F2(Q)+F3(QII)], where F3(Q)=Q(vmax2−vmin2)2+Q2(vmax−vmin)+Q3ln(Q−vmaxQ−vmin).
  49. R. Cestnik, Simulation code for the micro vs macro dynamics of the two-phase QIF ensemble, Zenodo (2025), https://doi.org/10.5281/zenodo.17497834.
  50. B. Pietras, Pulse shape and voltage-dependent synchronization in spiking neuron networks, Neural Comput. 36, 1476 (2024).
  51. B. Pietras, R. Cestnik, and A. Pikovsky, Exact finite-dimensional description for networks of globally coupled spiking neurons, Phys. Rev. E 107, 024315 (2023).
  52. B. Pietras, P. Clusella, and E. Montbrió, Low-dimensional model for adaptive networks of spiking neurons, Phys. Rev. E 111, 014422 (2025).
  53. B. Pietras and A. Daffertshofer, Ott-Antonsen attractiveness for parameter-dependent oscillatory systems, Chaos 26, 103101 (2016).
  54. J. R. Engelbrecht and R. Mirollo, Is the Ott-Antonsen manifold attracting? Phys. Rev. Res. 2, 023057 (2020).
  55. R. Tönjes and A. Pikovsky, Low-dimensional description for ensembles of identical phase oscillators subject to cauchy noise, Phys. Rev. E 102, 052315 (2020).
  56. T. Tanaka, Low-dimensional dynamics of phase oscillators driven by Cauchy noise, Phys. Rev. E 102, 042220 (2020).
  57. V. Pyragas and K. Pyragas, Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities, Phys. Lett. A 480, 128972 (2023).
  58. V. V. Klinshov and S. Y. Kirillov, Shot noise in next-generation neural mass models for finite-size networks, Phys. Rev. E 106, L062302 (2022).
  59. S. Watanabe and S. H. Strogatz, Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).
  60. S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74, 197 (1994).
  61. R. Cestnik and E. A. Martens, Integrability of a globally coupled complex Riccati array: Quadratic integrate-and-fire neurons, phase oscillators, and all in between, Phys. Rev. Lett. 132, 057201 (2024).
  62. D. Pazó and R. Cestnik, Low-dimensional dynamics of globally coupled complex Riccati equations: Exact firing-rate equations for spiking neurons with clustered substructure, Phys. Rev. E 111, L052201 (2025).

Outline

Information

Phys. Rev. Research 8, L012049– Published 2 March, 2026
  • Received 2 November 2025
  • Accepted 10 February 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. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation