Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On a toy network of neurons interacting through their dendrites

Nicolas Fournier, Etienne Tanré, and Romain Veltz

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Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{\mathrm{min}}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_{n}$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{\min },\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N\to \infty $ with $w_{n}\simeq N^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni (Ann. Probab. 23 (1995) 852–878) concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value $v_{\mathrm{max}}$, and find an explicit formula for the (heuristic) mean-field limit.


Considérons un grand nombre $n$ de neurones, chacun étant connecté à environ $N$ autres, choisis au hasard. Quand un neurone décharge, ce qui se produit au hasard à un certain taux en fonction de son potentiel électrique, son potentiel est remis à une valeur minimale $v_{\min }$, ce qui déclenche, après un petit délai, deux fronts sur les dendrites (linéaires) de tous les neurones auxquels il est connecté. Les fronts se déplacent à vitesse constante. Lorsque deux fronts (sur la dendrite du même neurone) entrent en collision, ils s’annihilent. Lorsqu’un front touche le soma d’un neurone, son potentiel est augmenté d’une petite valeur $w_{n}$. Entre les sauts, les potentiels des neurones évoluent dans $[v_{\min },\infty )$, suivant une EDO. Nous prouvons l’existence et l’unicité d’une limite champ moyen du système lorsque $n,N\to \infty $ avec $w_{n}\simeq N^{-1/2}$ obtenue de manière heuristique. Nous utilisons certaines versions récentes des résultats de Deuschel et Zeitouni (Ann. Probab. 23 (1995) 852–878) concernant la taille de la sous-suite croissante la plus longue d’une suite i.i.d. de points du plan. Nous étudions également, dans un cas très particulier, un modèle légèrement différent où les neurones déchargent quand leur potentiel atteint une valeur maximale $v_{\mathrm{max}}$. Nous trouvons heuristiquement une expression explicite pour la limite de champ moyen.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1041-1071.

Received: 17 May 2018
Revised: 22 March 2019
Accepted: 4 April 2019
First available in Project Euclid: 16 March 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J75: Jump processes 92C20: Neural biology

Mean-field limit Propagation of chaos Nonlinear stochastic differential equations Ulam’s problem Longest increasing subsequence Biological neural networks


Fournier, Nicolas; Tanré, Etienne; Veltz, Romain. On a toy network of neurons interacting through their dendrites. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1041--1071. doi:10.1214/19-AIHP993.

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  • [1] D. Aldous and P. Diaconis. Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 (1995) 199–213.
  • [2] A. L. Basdevant, L. Gerin, J. B. Gouéré and A. Singh. From Hammersley’s lines to Hammersley’s trees. Probab. Theory Related Fields 171 (2018) 1–51.
  • [3] B. Bollobás and P. Winkler. The longest chain among random points in Euclidean space. Proc. Amer. Math. Soc. 103 (1988) 347–353.
  • [4] M. Bossy, O. Faugeras and D. Talay. Clarification and complement to “Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons”. J. Math. Neurosci. 5 (2015) 23.
  • [5] M. Cáceres, J. Carrillo and B. Perthame. Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states. J. Math. Neurosci. 1 (2011) 33.
  • [6] J. Calder, S. Esedoḡlu and A. O. Hero. A Hamilton–Jacobi equation for the continuum limit of nondominated sorting. SIAM J. Math. Anal. 46 (2014) 603–638.
  • [7] J. Carrillo, B. Perthame, D. Salort and D. Smets. Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience. Nonlinearity 28 (2015) 3365–3388.
  • [8] E. Cator and P. Groeneboom. Hammersley’s process with sources and sinks. Ann. Probab. 33 (2005) 879–903.
  • [9] J. Chevallier. Mean-field limit of generalized Hawkes processes. Stochastic Process. Appl. 127 (2017) 3870–3912.
  • [10] J. Chevallier, M. Cáceres, M. Doumic and P. Reynaud-Bouret. Microscopic approach of a time elapsed neural model. Math. Models Methods Appl. Sci. 25 (2015) 2669–2719.
  • [11] J. Chevallier, A. Duarte, E. Löcherbach and G. Ost. Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels. Stochastic Process. Appl. 129 (2019) 1–27.
  • [12] A. De Masi, A. Galves, E. Löcherbach and E. Presutti. Hydrodynamic limit for interacting neurons. J. Stat. Phys. 158 (2015) 866–902.
  • [13] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré. Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 (2015) 2096–2133.
  • [14] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré. Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Process. Appl. 125 (2015) 2451–2492.
  • [15] J. D. Deuschel and O. Zeitouni. Limiting curves for i.i.d. records. Ann. Probab. 23 (1995) 852–878.
  • [16] S. Ditlevsen and E. Löcherbach. Multi-class oscillating systems of interacting neurons. Stochastic Process. Appl. 127 (2017) 1840–1869.
  • [17] N. Fournier and E. Löcherbach. On a toy model of interacting neurons. Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1844–1876.
  • [18] T. Górski, R. Veltz, M. Galtier, H. Fragnaud, B. Telenczuk and A. Destexhe. Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity. J. Comput. Neurosci. 45 (2018) 223–234.
  • [19] J. M. Hammersley. A few seedlings of research. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 345–394. Univ. California, Berkeley, Calif., 1970/1971. Vol. I: Theory of Statistics. Univ. California Press, Berkeley, CA, 1972.
  • [20] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500–544.
  • [21] J. Inglis and D. Talay. Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component. SIAM J. Math. Anal. 47 (2015) 3884–3916.
  • [22] M. Kac. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III (Berkeley and Los Angeles, 1956) 171–197. University of California Press.
  • [23] E. R. Kandel. Principles of Neural Science, 5th edition. McGraw-Hill, New York, 2013.
  • [24] C. Koch. Biophysics of Computation: Information Processing in Single Neurons Oxford Univ. Press Paperback. Computational Neuroscience. Oxford Univ. Press, New York, 2004.
  • [25] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Adv. Math. 26 (1977) 206–222.
  • [26] E. Luçon. Quenched limits and fluctuations of the empirical measure for plane rotators in random media. Electron. J. Probab. 16 (2011) 792–829.
  • [27] E. Luçon and W. Stannat. Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab. 24 (2014) 1946–1993.
  • [28] E. Luçon and W. Stannat. Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction. Ann. Appl. Probab. 26 (2016) 3840–3909.
  • [29] H. P. McKean. Propagation of chaos for a class of non-linear parabolic equations. In Lecture Series in Differential equations 7 41–57. Catholic University, Washington, 1967.
  • [30] S. Méléard. Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations 42–95. Lecture Notes in Math. 1627. Fond. CIME, Springer, Berlin, 1996.
  • [31] S. Ostojic, N. Brunel and V. Hakim. Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J. Comput. Neurosci. 26 (2009) 369–392.
  • [32] K. Pakdaman, M. Thieullen and G. Wainrib. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. in Appl. Probab. 42 (2010) 761–794.
  • [33] A. Renart, N. Brunel and X.-J. Wang. Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. In Computational Neuroscience: A Comprehensive Approach 431–490. Chapman & Hall/CRC Mathematical Biology and Medicine Series, 2004.
  • [34] M. Riedler, M. Thieullen and G. Wainrib. Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models. Electron. J. Probab. 17 (2012) 48.
  • [35] G. Stuart, N. Spruston and M. Häusser. Dendrites, 3rd edition. Oxford University Pres, New York, 2015.
  • [36] A.-S. Sznitman. Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989 165–251. Lecture Notes in Math. 1464. Springer, Berlin, 1991.
  • [37] S. M. Ulam. Monte Carlo calculations in problems of mathematical physics. In Modern mathematics for the engineer: Second series 261–281. McGraw-Hill, New York, 1961.
  • [38] A. M. Versik and S. V. Kerov. Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233 (1977) 1024–1027.
  • [39] I. Yakupov and M. Buzdalov. Incremental non-dominated sorting with O(N) insertion for the two-dimensional case. In IEEE Congress on Evolutionary Computation (CEC) 1853–1860, 2015.