Open Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Article Number 59
Number of page(s) 20
DOI https://doi.org/10.1051/mmnp/2020017
Published online 03 December 2020
  1. L.F. Abbott, Lapique’s introduction of the integrate-and-fire modelneuron (1907). Brain Res. Bull. 50 (1999) 303–304. [CrossRef] [PubMed] [Google Scholar]
  2. S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Dordrecht (2000). [Google Scholar]
  3. N. Brunel and M.C. van Rossum, Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol. Cybern. 97 (2007) 341–349. [CrossRef] [PubMed] [Google Scholar]
  4. N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 8 (2000) 183–208. [CrossRef] [PubMed] [Google Scholar]
  5. N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11 (1999) 1621–1671. [CrossRef] [PubMed] [Google Scholar]
  6. N. Brunel and V. Hakim, Fokker-planck equation. Encyclopedia of Computational Neuroscience. Springer, New York (2015), 1222–1226, 2015. [Google Scholar]
  7. N. Brunel and M.C.W. van Rossum, Quantitative investigations of electrical nerve excitation treated as polarization. Biol. Cybern. 97 (2007) 341–349. [CrossRef] [PubMed] [Google Scholar]
  8. A.N. Burkitt. A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biological Cybernetics, 95 (2006) 1–19. [CrossRef] [PubMed] [Google Scholar]
  9. A.N. Burkitt, A review of the integrate-and-fire neuron model: II. inhomogeneous synaptic input and network properties. Biol. Cybern. 95 (2006) 97–112. [CrossRef] [PubMed] [Google Scholar]
  10. M. 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) 7. [CrossRef] [PubMed] [Google Scholar]
  11. M.J. Cáceres and R. Schneider, Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods. ESAIM: M2AN 52 (2018) 1733–1761. [CrossRef] [EDP Sciences] [Google Scholar]
  12. J.A. Cañizo and H. Yoldaş, Asymptotic behaviour of neuron population models structured by elapsed-time. Nonlinearity 32 (2019) 464–495. [CrossRef] [Google Scholar]
  13. J.A. Carillo, B. Perthame, D. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience. Nonlinearity 8 (2015) 9. [Google Scholar]
  14. J.A. Carrillo, M.D.M. Gonzalez, M.P. Gualdani and M.E. Schonbek, Classical solutions for a nonlinear fokker-planck equation arising incomputational neuroscience. Commun. Partial Differ. Equ. 38 (2013) 385– 409. [CrossRef] [Google Scholar]
  15. J. Chevalier, M.J. Caceres, M. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model. Math. Models Methods Appl. Sci. 25 (2015) 2669–2719. [CrossRef] [Google Scholar]
  16. 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. [CrossRef] [Google Scholar]
  17. N. Dokuchaev, On recovering parabolic diffusions from their time averages. Preprint arXiv: 1609.01890 (2017). [Google Scholar]
  18. G Dumont and J Henry, Synchronization of an excitatory integrate-and-fire neural network. Bull. Math. Biol. 75 (2013) 629–48. [CrossRef] [Google Scholar]
  19. G. Dumont, J. Henry and C.O. Tarniceriu, Noisy threshold in neuronal models: connections with the noisy leaky integrate - and - firemodel. J. Math. Biol. 73 (2016) 1413 –1436. [CrossRef] [PubMed] [Google Scholar]
  20. G. Dumont, J. Henry and C.O. Tarniceriu, Theoretical connections between neuronal models corresponding to different expressions of noise. J. Theor. Biol. 406 (2016) 31–41. [CrossRef] [PubMed] [Google Scholar]
  21. G. Dumont, A. Payeur and A. Longtin, A stochastic-field description of finite-size spiking neural networks. PLOS Comput. Biol. 13 (2017) 1–34. [CrossRef] [Google Scholar]
  22. G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states. Preprint arXiv: 1710.05596 (2020). [Google Scholar]
  23. A.A. Faisal, L.P. Selen and D. Wolpert, Noise in the nervous system. Nat. Rev. Neurosci. 9 (2008) 292–303. [CrossRef] [PubMed] [Google Scholar]
  24. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer, Berlin (1996). [Google Scholar]
  25. W. Gerstner and J.L. Van Hemmen Associative memory in a network of ’spiking’ neurons. Network 3 (1992) 139–164. [CrossRef] [Google Scholar]
  26. W. Gerstner and W. Kistler, Spiking neuron models. Cambridge University Press, Cambridge (2002). [CrossRef] [Google Scholar]
  27. M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics. CNR Applied Mathematics Monographs, Vol. 7. Giardini editori e stampatori, Pisa (1995). [Google Scholar]
  28. P. Langevin, Sur la théorie du mouvement brownien. C. R. Acad. Sci. (Paris) 146 (1908) 530–533. [Google Scholar]
  29. A. Longtin, Neuronal noise. Scholarpedia 8 (2013) 1618. [CrossRef] [Google Scholar]
  30. C. Meyer and C.A. van Vreeswijk, Temporal correlations in stochastic networks of spiking neurons. Neural Comput. 10 (2002) 1321–1372. [Google Scholar]
  31. S. Mischler and C. Quiñinao, Weak and strong connectivity regimes for a general time elapsed neuron network model. J. Stat. Phys. 173 (2018) 77–98. [CrossRef] [Google Scholar]
  32. K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population. Nonlinearity 23 (2009) 23–55. [Google Scholar]
  33. K. Pakdaman, B. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM J. Appl. Math. 73 (2013) 1260–1279. [CrossRef] [Google Scholar]
  34. H.E. Plesser and W. Gerstner, Noise in integrate-and-fire neurons: from stochastic input to escape rates. Neural Comput. 12 (2000) 367–384. [CrossRef] [Google Scholar]
  35. A. Renart, N. Brunel and X.-J Wang, Mean-Field Theory of Irregularly Spking Neuronal Populations and Working Memory in Recurrent Cortical Networks, Chapter 15 in Computational Neuroscience: A comprehensive Approach, Mathematical Biology and Medicine Series. Chapmann&Hall/CRC, Boca Raton (2004). [Google Scholar]
  36. L.S. Tsimring, Noise in biology. Rep. Prog. Phys. 77 (2014) 026601. [CrossRef] [PubMed] [Google Scholar]
  37. G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985). [Google Scholar]

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