A theoretical connection between the Noisy Leaky integrate-and-fire and the escape rate models: The non-autonomous case

Grégory Dumont¹ ; Jacques Henry² ; Carmen Oana Tarniceriu^3, 4

¹ École Normale Supérieure, Group for Neural Theory, 45 Rue d’Ulm, 75005 Paris, France.
² INRIA team Carmen, INRIA Bordeaux Sud-Ouest, 200 Avenue de la Vieille Tour, 33405 Talence cedex, France.
³ Department of Mathematics and Informatics, “Gheorghe Asachi” University of Iaşi, Bulevardul Carol I 11, 700506 Iaşi, Romania.
⁴ Interdisciplinary Research Department, Field - Sciences, “Alexandru Ioan Cuza” University of Iaşi, Lascăr Catargi 54, Iaşi, Romania.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 59.

Résumé

Finding a mathematical model that incorporates various stochastic aspects of neural dynamics has proven to be a continuous challenge. Among the different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most popular. These two models are generally thought to express different noise action over the neural cell. In this paper we investigate the link between the two formalisms in the case of a neuron subject to a time dependent input. To this aim, we introduce a new general stochastic framework. As we shall prove, our general framework entails the two already existing ones. Our results have theoretical implications since they offer a general view upon the two stochastic processes mostly used in neuroscience, upon the way they can be linked, and explain their observed statistical similarity.

Détail

DOI : 10.1051/mmnp/2020017

Bibliographie

[1] L.F. Abbott Lapique’s introduction of the integrate-and-fire modelneuron (1907) Brain Res. Bull 1999 303 304

[2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Dordrecht (2000).

[3] N. Brunel, M.C. Van Rossum Lapicque’s 1907 paper: from frogs to integrate-and-fire Biol. Cybern 2007 341 349

[4] N. Brunel Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons J. Comput. Neurosci 2000 183 208

[5] N. Brunel, V. Hakim Fast global oscillations in networks of integrate-and-fire neurons with low firing rates Neural Comput 1999 1621 1671

[6] N. Brunel and V. Hakim, Fokker-planck equation. Encyclopedia of Computational Neuroscience. Springer, New York (2015), 1222–1226, 2015.

[7] N. Brunel, M.C.W. Van Rossum Quantitative investigations of electrical nerve excitation treated as polarization Biol. Cybern. 2007 341 349

[8] A.N. Burkitt. A review of the integrate-and-fire neuron model: I. homogeneous synaptic input Biological Cybernetics 2006 1 19

[9] A.N. Burkitt A review of the integrate-and-fire neuron model: II. inhomogeneous synaptic input and network properties Biol. Cybern 2006 97 112

[10] M. Cáceres, J.A. Carrillo, B. Perthame Analysis of nonlinear noisy integrate fire neuron models: blow-up and steady states J. Math. Neurosci 2011 7

[11] M.J. Cáceres, R. Schneider Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods ESAIM: M2AN 2018 1733 1761

[12] J.A. Cañizo, H. Yoldaş Asymptotic behaviour of neuron population models structured by elapsed-time Nonlinearity 2019 464 495

[13] J.A. Carillo, B. Perthame, D. Salort, D. Smets Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience Nonlinearity 2015 9

[14] J.A. Carrillo, M.D.M. Gonzalez, M.P. Gualdani, M.E. Schonbek Classical solutions for a nonlinear fokker-planck equation arising incomputational neuroscience Commun. Partial Differ. Equ 2013 385 409

[15] J. Chevalier, M.J. Caceres, M. Doumic, P. Reynaud-Bouret Microscopic approach of a time elapsed neural model Math. Models Methods Appl. Sci 2015 2669 2719

[16] F. Delarue, J. Inglis, S. Rubenthaler, E. Tanré Global solvability of a networked integrate-and-fire model of McKean-Vlasov type Ann. Appl. Probab. 2015 2096 2133

[17] N. Dokuchaev, On recovering parabolic diffusions from their time averages. Preprint arXiv: 1609.01890 (2017).

[18] G Dumont, J Henry Synchronization of an excitatory integrate-and-fire neural network Bull. Math. Biol 2013 629 48

[19] G. Dumont, J. Henry, C.O. Tarniceriu Noisy threshold in neuronal models: connections with the noisy leaky integrate - and - firemodel J. Math. Biol 2016 1413 1436

[20] G. Dumont, J. Henry, C.O. Tarniceriu Theoretical connections between neuronal models corresponding to different expressions of noise J. Theor. Biol 2016 31 41

[21] G. Dumont, A. Payeur, A. Longtin A stochastic-field description of finite-size spiking neural networks PLOS Comput. Biol 2017 1 34

[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).

[23] A.A. Faisal, L.P. Selen, D. Wolpert Noise in the nervous system Nat. Rev. Neurosci 2008 292 303

[24] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer, Berlin (1996).

[25] W. Gerstner, J.L. Van Hemmen Associative memory in a network of ’spiking’ neurons Network 1992 139 164

[26] W. Gerstner and W. Kistler, Spiking neuron models. Cambridge University Press, Cambridge (2002).

[27] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics. CNR Applied Mathematics Monographs, Vol. 7. Giardini editori e stampatori, Pisa (1995).

[28] P. Langevin Sur la théorie du mouvement brownien C. R. Acad. Sci. (Paris) 1908 530 533

[29] A. Longtin Neuronal noise Scholarpedia 2013 1618

[30] C. Meyer, C.A. Van Vreeswijk Temporal correlations in stochastic networks of spiking neurons Neural Comput 2002 1321 1372

[31] S. Mischler, C. Quiñinao Weak and strong connectivity regimes for a general time elapsed neuron network model J. Stat. Phys 2018 77 98

[32] K. Pakdaman, B. Perthame, D. Salort Dynamics of a structured neuron population Nonlinearity 2009 23 55

[33] K. Pakdaman, B. Perthame, D. Salort Relaxation and self-sustained oscillations in the time elapsed neuron network model SIAM J. Appl. Math 2013 1260 1279

[34] H.E. Plesser, W. Gerstner Noise in integrate-and-fire neurons: from stochastic input to escape rates Neural Comput 2000 367 384

[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/CRC, Boca Raton (2004).

[36] L.S. Tsimring Noise in biology Rep. Prog. Phys 2014 026601

[37] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985).