Journal of Mathematical Biology

, Volume 78, Issue 6, pp 1771–1820 | Cite as

Synchronization of stochastic mean field networks of Hodgkin–Huxley neurons with noisy channels

  • Mireille Bossy
  • Joaquín Fontbona
  • Héctor OliveroEmail author


In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin–Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean–Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin–Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.


Hodgkin–Huxley neurons Synchronization of neuron networks Mean-field limits Propagation of chaos Stochastic differential equations 

Mathematics Subject Classification

60H99 60K35 82C22 82C32 92B20 92B25 



  1. Ambrosio L, Gigli N, Savaré G (2008) Gradient flows: in metric spaces and in the space of probability measures. Springer, BerlinzbMATHGoogle Scholar
  2. Austin TD (2008) The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann Appl Probab 18(4):1279–1325MathSciNetzbMATHGoogle Scholar
  3. Axmacher N, Mormann F, Fernández G, Elger CE, Fell J (2006) Memory formation by neuronal synchronization. Brain Res Rev 52(1):170–182Google Scholar
  4. Baladron J, Fasoli D, Faugeras O, Touboul J (2012) Mean field description of and propagation of chaos in networks of Hodgkin–Huxley and Fitzhugh–Nagumo neurons. J Math Neurosci 2(1):10MathSciNetzbMATHGoogle Scholar
  5. Berglund N, Gentz B (2004) On the noise-induced passage through an unstable periodic orbit i: two-level model. J Stat Phys 114(5–6):1577–1618MathSciNetzbMATHGoogle Scholar
  6. Berglund N, Gentz B (2014) On the noise-induced passage through an unstable periodic orbit ii: general case. SIAM J Math Anal 46(1):310–352MathSciNetzbMATHGoogle Scholar
  7. Bertini L, Giacomin G, Pakdaman K (2010) Dynamical aspects of mean field plane rotators and the Kuramoto model. J Stat Phys 138(1):270–290MathSciNetzbMATHGoogle Scholar
  8. Bertini L, Giacomin G, Poquet C (2014) Synchronization and random long time dynamics for mean-field plane rotators. Probab Theory Relat Fields 160(3–4):593–653MathSciNetzbMATHGoogle Scholar
  9. Bossy M, Faugeras O, Talay D (2015) Clarification and complement to “mean-field description and propagation of chaos in networks of Hodgkin–Huxley and Fitzhugh–Nagumo neurons”. JMN 5(1):1–23MathSciNetzbMATHGoogle Scholar
  10. Bossy M, Espina J, Morice J, Paris C, Rosseau A (2016) Modeling the wind circulation around mills with a lagrangian stochastic approach. SMAI J Comput Math 2:177–214MathSciNetzbMATHGoogle Scholar
  11. Bressloff PC, Lai YM (2011) Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise. J Math Neurosci 1(1):2MathSciNetzbMATHGoogle Scholar
  12. Burkitt AN (2006a) A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol Cybern 95(1):1–19MathSciNetzbMATHGoogle Scholar
  13. Burkitt AN (2006b) A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol Cybern 95(2):97–112MathSciNetzbMATHGoogle Scholar
  14. Chan T, Golub G, LeVeque R (1983) Algorithms for computing the sample variance: analysis and recommendations. Am Stat 37(3):242–247MathSciNetzbMATHGoogle Scholar
  15. Dangerfield CE, Kay D, Burrage K (2012) Modeling ion channel dynamics through reflected stochastic differential equations. Phys Rev E 85:051907Google Scholar
  16. Delarue F, Inglis J, Rubenthaler S, Tanré E (2015) Global solvability of a networked integrate-and-fire model of Mckean–Vlasov type. Ann Appl Probab 25(4):2096–2133MathSciNetzbMATHGoogle Scholar
  17. Ermentrout GB, Terman DH (2010) Mathematical foundations of neuroscience. Springer, New YorkzbMATHGoogle Scholar
  18. Faugeras O, Touboul J, Cessac B (2009) A constructive mean-field analysis of multi population neural networks with random synaptic weights and stochastic inputs. Front Comput Neurosci 3:1Google Scholar
  19. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466Google Scholar
  20. Fournier N, Guillin A (2015) On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Relat Fields 162(3–4):707–738MathSciNetzbMATHGoogle Scholar
  21. Fournier N, Löcherbach E (2016) On a toy model of interacting neurons. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 52(4):1844–1876MathSciNetzbMATHGoogle Scholar
  22. Friedman A (2006) Stochastic differential equations and applications. Dover Publications Inc., Mineola (Two volumes bound as one, Reprint of the 1975 and 1976 original published in two volumes)zbMATHGoogle Scholar
  23. Gärtner J (1988) On the McKean–Vlasov limit for interacting diffusions. Math Nachr 137:197–248MathSciNetzbMATHGoogle Scholar
  24. Giacomin G, Luçon E, Poquet C (2014) Coherence stability and effect of random natural frequencies in populations of coupled oscillators. J Dyn Differ Equ 26(2):333–367MathSciNetzbMATHGoogle Scholar
  25. Goldwyn J, Shea-Brown E (2011) The what and where of adding channel noise to the Hodgkin–Huxley equations. PLoS Comput Biol 7(11):e1002247MathSciNetGoogle Scholar
  26. Goldwyn J, Imennov Nikita S, Famulare M, Shea-Brown E (2011) Stochastic differential equation models for ion channel noise in Hodgkin–Huxley neurons. Phys Rev E 83(4):041908Google Scholar
  27. Hansel D, Mato G (1993) Patterns of synchrony in a heterogeneous Hodgkin–Huxley neural network with weak coupling. Phys A Stat Mech Appl 200(1–4):662–669Google Scholar
  28. Hansel D, Mato G, Meunier C (1993) Phase dynamics for weakly coupled Hodgkin–Huxley neurons. EPL 23(5):367Google Scholar
  29. Hodgkin A, Huxley A (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544Google Scholar
  30. Hormuzdi SG, Filippov MA, Mitropoulou G, Monyer H, Bruzzone R (2004) Electrical synapses: a dynamic signaling system that shapes the activity of neuronal networks. BBA Biomembr 1662(1–2):113–137Google Scholar
  31. Izhikevich EM (2007) Dynamical systems in neuroscience. The MIT Press, CambridgeGoogle Scholar
  32. Jiruska P, de Curtis M, Jefferys JGR, Schevon CA, Schiff SJ, Schindler K (2013) Synchronization and desynchronization in epilepsy: controversies and hypotheses. J Physiol 591(4):787–797Google Scholar
  33. Karatzas I, Shreve S (1991) Brownian motion and stochastic calculus. Graduate texts in mathematics, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  34. Kopell Nancy, Ermentrout Bard (2004) Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proc Natl Acad Sci 101(43):15482–15487Google Scholar
  35. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence. Springer, BerlinzbMATHGoogle Scholar
  36. Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarization. J Physiol Pathol Gen (Paris) 9:620–635Google Scholar
  37. Luçon E, Poquet C (2017) Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model. Ann Inst Henri Poincaré Probab Stat 53(3):1196–1240MathSciNetzbMATHGoogle Scholar
  38. Marella S, Ermentrout GB (2008) Class-ii neurons display a higher degree of stochastic synchronization than class-i neurons. Phys Rev E 77(4):041918MathSciNetGoogle Scholar
  39. Méléard S (1996) Asymptotic behaviour of some interacting particle systems; Mckean–Vlasov and Boltzmann models. In: Probabilistic models for nonlinear partial differential equations. Springer, pp 42–95Google Scholar
  40. Mischler S, Quiñinao C, Touboul J (2016) On a kinetic Fitzhugh–Nagumo model of neuronal network. Commun Math Phys 342(3):1001–1042MathSciNetzbMATHGoogle Scholar
  41. Morris C, Lecar H (1981) Voltage oscillations in the barnacle gian muscle fiber. Biophys J 31(1):193–213Google Scholar
  42. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070Google Scholar
  43. Ostojic S, Brunel N, Hakim V (2008) Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J Comput Neurosci 26(3):369MathSciNetGoogle Scholar
  44. Pakdaman K, Thieullen M, Wainrib G (2010) Fluid limit theorems for stochastic hybrid systems with aplication to neuron models. Adv Appl Probab 42(3):761–794MathSciNetzbMATHGoogle Scholar
  45. Perthame B, Salort D (2013) On a voltage-conductance kinetic system for integrate and fire neural networks. Kinet Relat Models 6(4):841–864MathSciNetzbMATHGoogle Scholar
  46. Pikovskii AS (1984) Synchronization and stochastization of nonlinear oscillations by external noise. In: Nonlinear and turbulent processes in physics, vol 1, p 1601Google Scholar
  47. Pikovsky Arkady, Rosenblum Michael, Kurths Jürgen (2003) Synchronization: a universal concept in nonlinear sciences, vol 12. Cambridge University Press, CambridgezbMATHGoogle Scholar
  48. Sacerdote L, Giraudo M (2013) Stochastic integrate and fire models: a review on mathematical methods and their applications. Springer, Berlin, pp 99–148zbMATHGoogle Scholar
  49. Sznitman A-S (1991) Topics in propagation of chaos. In: Ecole d’été de probabilités de Saint-Flour XIX—1989. Springer, pp 165–251Google Scholar
  50. Villani C (2009) Optimal transport, old and new, vol 338. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]. Springer, BerlinGoogle Scholar
  51. Wainrib G (2010) Randomness in neurons: a multiscale probabilistic analysis. PhD thesis, École PolytechniqueGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.INRIA Sophia Antipolis MéditerranéeSophia AntipolisFrance
  2. 2.Department of Mathematical Engineering and Center for Mathematical ModelingUMI(2807) UCHILE-CNRS, University of ChileSantiagoChile
  3. 3.CIMFAV, Facultad de IngenieríaUniversidad de ValparaísoValparaísoChile

Personalised recommendations