Riv. Mat. Univ. Parma, Vol. 10, No. 2, 2019

María J. Cáceres [a]

A review about nonlinear noisy leaky integrate and fire models for neural networks

Pages: 269-298
Received: 20 December 2018
Accepted in revised form: 4 June 2019
Mathematics Subject Classification (2010): 35K60, 35Q92, 82C31, 82C32, 92B20.
Keywords: Neural network, Leaky integrate and fire models, Fokker-Planck equations, Blow-up, entropy methods.
Author address:
[a]: University of Granada, Campus de Fuentenueva, Granada, 18071, Spain

Full Text (PDF)

Abstract: In these notes we analyse a family of self-contained mean field systems, called nonlinear noisy leaky integrate and fire (NNLIF) models, which describe the activity of neural networks by means of the membrane potential. These models are based on nonlinear systems of PDEs of Fokker-Planck type. We study the wide range of phenomena that appear in this kind of models: blow-up/global existence, asynchronous/synchronous solutions, instability/stability of the steady states ... . This review is intended to collect the main results of papers [8, 10, 11, 12, 13] in collaboration with José A. Carrillo, Benoit Perthame, Pierre Roux, Ricarda Schneider and Delphine Salort; as well as other results presented in [6,7,14,15,18,19].

J. A. Acebrón, A. R. Bulsara and W.-J. Rappel, Noisy FitzHugh-Nagumo model: from single elements to globally coupled networks, Phys. Rev. E (3) 69 (2004), 026202, 9 pp. MR2098593
L. Albantakis and G. Deco, The encoding of alternatives in multiple-choice decision making, Proc. Natl. Acad. Sci. USA 106 (2009), 10308-10313. DOI
A. Arnold, J. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh. Math. 142 (2004), 35-43. MR2065020
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26 (2001), 43-100. MR1842428
R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol. 94 (2005), 3637-3642. DOI
N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks, J. Comput. Neurosci. 8 (2000), 183-208. DOI
N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with long firing rates, Neural Comput. 11 (1999), 1621-1671. DOI
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), Art. 7, 33 pp. MR2853216
M. J. Cáceres, J. A. Carrillo and L. Tao, A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics, J. Comput. Phys. 230 (2011), 1084-1099. MR2753350
M. J. Cáceres and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity, J. Theoret. Biol. 350 (2014), 81-89. MR3190511
M. J. Cáceres, P. Roux, D. Salort and R. Schneider, Avoiding the blow-up: global-in-time classical solutions for the excitatory NNLIF model with delay, submitted, 2017.
M. J. Cáceres and R. Schneider, Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models, Kinet. Relat. Models 10 (2017), 587-612. MR3591125
M. J. Cáceres and R. Schneider, Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods, ESAIM Math. Model. Numer. Anal. 52 (2018), 1733-1761. MR3878608
J. A. Carrillo, B. Perthame, D. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience, Nonlinearity 28 (2015), 3365-3388. MR3403402
J. A. Carrillo, M. D. M. González, M. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience, Comm. Partial Differential Equations 38 (2013), 385-409. MR3019444
J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl. 127 (2017), 3870-3912. MR3718099
J. Chevallier, M. J. Cáceres, M. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci. 25 (2015), 2669-2719. MR3411353
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. MR3322871
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. MR3349003
G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states, arXiv:1710.05596, preprint, 2017.
G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: well-posedness, J. Math. Biol. 67 (2013), 453-481. MR3084360
G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol. 75 (2013), 629-648. MR3039916
R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961), 445-466. DOI
C. M. Gray and W. Singer, Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc. Natl. Acad. Sci. USA 86 (1989), 1698-1702. DOI
P. Grazieschi, M. Leocata, C. Mascart, J. Chevallier, F. Delarue and E. Tanré, Network of interacting neurons with random synaptic weights, ESAIM: ProcS 65 (2019), 445-475. DOI
J. A. Henrie and R. Shapley, LFP power spectra in V1 cortex: the graded effect of stimulus contrast, J. Neurophysiol. 94 (2005), 479-490. DOI
S. Mischler, C. Quininao and J. Touboul, On a kinetic Fitzhugh-Nagumo model of neuronal network, Comm. Math. Phys. 342 (2016), 1001-1042. MR3465438
K. A. Newhall, G. Kovačič, P. R. Kramer and D. Cai, Cascade-induced synchrony in stochastically driven neuronal networks, Phys. Rev. E (3) 82 (2010), 041903, 17 pp. MR2788032
K. A. Newhall, G. Kovačič, P. R. Kramer, D. Zhou, A. V. Rangan and D. Cai, Dynamics of current-based, Poisson driven, integrate-and-fire neuronal networks, Commun. Math. Sci. 8 (2010), 541-600. MR2664463
A. Omurtag, B. W. Knight and L. Sirovich, On the simulation of large populations of neurons, J. Comput. Neurosci. 8 (2000), 51-63. DOI
K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity 23 (2010), 55-75. MR2576373
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. MR3071416
K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci. 4 (2014), Art. 14, 26 pp. MR3246939
B. Perthame and D. Salort, On a voltage-conductance kinetic system for integrate & fire neural networks, Kinet. Relat. Models 6 (2013), 841-864. MR3177631
B. Perthame and D. Salort, Derivation of an integrate&fire equation for neural networks from a voltage-conductance kinetic model, hal-01881950, preprint, 2018. Article
A. V. Rangan, G. Kovačič and D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Phys. Rev. E (3) 77 (2008), 041915, 13 pp. MR2495462
A. Renart, N. Brunel and X.-J. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks, in ''Computational Neuroscience: A comprehensive approach'', J. Feng, ed., Chapman & Hall/CRC Math. Biol. Med. Ser., Boca Raton, FL, 2004, 431-490. MR2029670
P. Robert and J. Touboul, On the dynamics of random neuronal networks, J. Stat. Phys. 165 (2016), 545-584. MR3562424
C. Villani, Entropy production and convergence to equilibrium, in ''Entropy methods for the Boltzmann equation'', Lecture Notes in Math., 1916, Springer, Berlin, 2008, 1-70. MR2409050

Home Riv.Mat.Univ.Parma