The Annals of Applied Probability

Global solvability of a networked integrate-and-fire model of McKean–Vlasov type

François Delarue, James Inglis, Sylvain Rubenthaler, and Etienne Tanré

Full-text: Open access


We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 2096-2133.

Received: April 2014
First available in Project Euclid: 21 May 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 92C20: Neural biology 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

McKean nonlinear diffusion process renewal process first hitting time density estimates integrate-and-fire network nonhomogeneous diffusion process neuroscience


Delarue, François; Inglis, James; Rubenthaler, Sylvain; Tanré, Etienne. Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 (2015), no. 4, 2096--2133. doi:10.1214/14-AAP1044.

Export citation


  • Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 8 183–208.
  • Brunel, N. and Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11 1621–1671.
  • Cáceres, M. J., Carrillo, J. A. and Perthame, B. (2011). Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states. J. Math. Neurosci. 1 Art. 7, 33.
  • Carrillo, J. A., González, M. D. M., Gualdani, M. P. and Schonbek, M. E. (2013). Classical solutions for a nonlinear Fokker–Planck equation arising in computational neuroscience. Comm. Partial Differential Equations 38 385–409.
  • Delarue, F. and Menozzi, S. (2010). Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259 1577–1630.
  • Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2013). First hitting times for general non-homogeneous 1d diffusion processes: Density estimates in small time. Technical report. Available at
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall International, Englewood Cliffs, NJ.
  • Garroni, M. G. and Menaldi, J. L. (1992). Green Functions for Second Order Parabolic Integro-Differential Problems. Longman, Harlow, UK.
  • Jolivet, R., Lewis, T. J. and Gerstner, W. (2004). Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. J. Neurophysiol. 92 959–976.
  • Kistler, W., Gerstner, W. and van Hemmen, J. L. (1997). Reduction of the Hodgkin–Huxley equations to a single-variable threshold model. Neural Comput. 5 1015–1045.
  • Krylov, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York. Translated from the Russian by A. B. Aries.
  • Krylov, N. V. and Safonov, M. V. (1979). An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245 18–20.
  • Lewis, T. J. and Rinzel, J. (2003). Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J. Comput. Neurosci. 14 283–309.
  • Lieberman, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ.
  • Ostojic, S., Brunel, N. and Hakim, V. (2009). Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J. Comput. Neurosci. 26 369–392.
  • Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • Renart, A., Brunel, N. and Wang, X.-J. (2004). Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. In Computational Neuroscience 431–490. Chapman & Hall, Boca Raton, FL.
  • Sacerdote, L. and Giraudo, M. T. (2013). Stochastic integrate and fire models: A review on mathematical methods and their applications. In Stochastic Biomathematical Models. Lecture Notes in Math. 2058 99–148. Springer, Heidelberg.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.
  • Sznitman, A.-S. (1991). Topics in propagation of chaos. In École d’Été de Probabilités de Saint–Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.