Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On a toy model of interacting neurons

Nicolas Fournier and Eva Löcherbach

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We continue the study of a stochastic system of interacting neurons introduced in De Masi, Galves, Löcherbach and Presutti (J. Stat. Phys. 158 (2015) 866–902). The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to $0$ and all other neurons receive an additional amount $1/N$ of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its average potential. We prove propagation of chaos of the system, as $N\to\infty $, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of (J. Stat. Phys. 158 (2015) 866–902), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/\sqrt{N}$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique nontrivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.

Résumé

Cet article continue l’étude du système stochastique de neurones en interaction introduit par De Masi, Galves, Löcherbach et Presutti (J. Stat. Phys. 158 (2015) 866–902). Le système est composé de $N$ neurones. Chaque neurone décharge un potentiel d’action à des instants aléatoires, à un taux qui dépend de son potentiel de membrane. Ce potentiel est alors remis à $0$, et tous les autres neurones reçoivent une charge supplémentaire de $1/N$. De plus, des synapses électriques induisent une dérive déterministe qui attire le système vers sa valeur moyenne. Nous établissons la propriété de propagation du chaos lorsque $N\to\infty$, vers la solution d’une équation différentielle stochastique non-linéaire à sauts. Nous améliorons les résultats obtenus dans (J. Stat. Phys. 158 (2015) 866–902) puisque (i) nous levons la condition de support compact imposée aux données initiales, (ii) nous obtenons une vitesse de convergence en $1/\sqrt{N}$. Enfin, nous proposons une étude de l’équation limite : nous décrivons la forme de ses lois marginales (en temps), nous démontrons l’existence d’une unique loi invariante non-triviale et montrons que la mesure invariante triviale n’est pas attractive. Enfin, nous obtenons la convergence vers l’équilibre dans un cas particulier.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1844-1876.

Dates
Received: 13 October 2014
Revised: 3 May 2015
Accepted: 22 July 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373251

Digital Object Identifier
doi:10.1214/15-AIHP701

Mathematical Reviews number (MathSciNet)
MR3573298

Zentralblatt MATH identifier
1355.92014

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes 60F17: Functional limit theorems; invariance principles

Keywords
Piecewise deterministic Markov processes Mean-field interaction Biological neural nets Interacting particle systems Nonlinear stochastic differential equations

Citation

Fournier, Nicolas; Löcherbach, Eva. On a toy model of interacting neurons. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1844--1876. doi:10.1214/15-AIHP701. https://projecteuclid.org/euclid.aihp/1479373251


Export citation

References

  • [1] F. Bolley, J. A. Cañizo and J. A. Carrillo. Stochastic mean-field limit: Non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21 (11) (2011) 2179–2210.
  • [2] F. Bolley, A. Guillin and F. Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. ESAIM Math. Model. Numer. Anal. 44 (2010) 867–884.
  • [3] M. J. Cáceres, J. A. Carrillo and B. Perthame. Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states. J. Math. Neurosci. 1 (2011) Article 7.
  • [4] 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.
  • [5] 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.
  • [6] A. De Masi, A. Galves, E. Löcherbach and E. Presutti. Hydrodynamic limit for interacting neurons. J. Stat. Phys. 158 (2015) 866–902.
  • [7] R. J. DiPerna and P. L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–548.
  • [8] O. Faugeras, J. Touboul and B. Cessac. A constructive mean-field analysis of multi-population neural networks with random synaptic weights and stochastic inputs. Front. Comput. Neurosci. 3 (2009) 1–28.
  • [9] N. Fournier. On some stochastic coalescents. Probab. Theory Related Fields 136 (2006) 509–523.
  • [10] N. Fournier and A. Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 (2015) 707–738.
  • [11] N. Fournier and E. Löcherbach. Stochastic coalescence with homogeneous-like interaction rates. Stochastic Process. Appl. 119 (2009) 45–73.
  • [12] N. Fournier and S. Mischler. Rate of convergence of the Nanbu particle system for hard potentials. Ann. Probab. To appear, 2016.
  • [13] A. Galves and E. Löcherbach. Infinite systems of interacting chains with memory of variable length – a stochastic model for biological neural nets. J. Stat. Phys. 151 (5) (2013) 896–921.
  • [14] N. Hansen, P. Reynaud-Bouret and V. Rivoirard. Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21 (2015) 83–143.
  • [15] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1989.
  • [16] J. Inglis and D. Talay. Mean-field limit of a stochastic particle systems smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component. SIAM J. Math. Anal. 47 (5) (2015) 3884–3916.
  • [17] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003.
  • [18] P. Jahn, R. Berg, J. Hounsgaard and S. Ditlevsen. Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process. J. Comput. Neurosci. 31 (3) (2011) 563–579.
  • [19] 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, Berkeley.
  • [20] E. Luçon and W. Stannat. Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab. 24 (5) (2014) 1946–1993.
  • [21] H. P. McKean. Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21 (1966) 347–367.
  • [22] H. P. McKean. Propagation of chaos for a class of nonlinear parabolic equations. In Stochastic Differential Equations 41–57. Washington, Lecture Series in Differential Equations (Session 7, Catholic University). Air Force Office Sci. Res., Arlington, VA, 1967.
  • [23] F. Malrieu. Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stochastic Process. Appl. 95 (2001) 109–132.
  • [24] K. D. Schmidt. On inequalities for moments and the covariance of monotone functions. Insurance Math. Econom. 55 (2014) 91–95.
  • [25] A.-S. Sznitman. Equations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66 (1984) 559–592.
  • [26] 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.
  • [27] H. Tanaka. On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 409–425. Wiley, New York, 1978.
  • [28] H. Tanaka. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 (1978/79) 67–105.