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

Annals of Applied Probability

Volumn 25, Issue 4, 2015, Pages 2096-2133

  Delarue, François ^a   Inglis, James ^b   Rubenthaler, Sylvain ^a   Tanré, Etienne ^b  

^a UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

^b INRIA   (France)

Author keywords

First hitting time density estimates; Integrate and fire network; McKean nonlinear diffusion process; Neuroscience; Nonhomogeneous diffusion process; Renewal process

Indexed keywords

EID: 84936791750     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AAP1044     Document Type: Article

References (20)

1. BRUNEL, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 8 183-208.
2. 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.
3. 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. MR2853216
4. CARRILLO, J. A.,GONZÁLEZ, M. D. M.,GUALDANI, M. P.andSCHONBEK, M. E. (2013). Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience. Comm. Partial Differential Equations 38 385-409. MR3019444
5. 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. MR2659772
6. 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 http://hal.archives-ouvertes.fr/hal-00870991.
7. FRIEDMAN, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall International, Englewood Cliffs, NJ. MR0181836
8. GARRONI, M. G. andMENALDI, J. L. (1992). Green Functions for Second Order Parabolic Integro- Differential Problems. Longman, Harlow, UK. MR1202037
9. 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.
10. 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.
11. KRYLOV, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York. Translated from the Russian by A. B. Aries. MR0601776
12. 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. MR0525227
13. LEWIS, T. J. and RINZEL, J. (2003). Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J. Comput. Neurosci. 14 283-309.
14. LIEBERMAN, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ. MR1465184
15. 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. MR2505036
16. PESKIR, G. andSHIRYAEV, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkha¨user, Basel. MR2256030
17. 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. MR2029670
18. 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. MR3051031
19. STROOCK, D. W. and VARADHAN, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin. MR0532498
20. 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. MR1108185