Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: blow-up and global existence
type: Item
instance of: scholarly article
title: Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: blow-up and global existence
zbMATH Open document ID: 1505.35338
author: Pierre Roux
author: Delphine Salort
published in: Kinetic and Related Models
publication date: 2022-01-03T00:00:00Z
review text: The aim of the article is to review, explain and, to some extent, refine some up-to-date results on Nonlinear Noisy Leaky Integrate \& Fire model, NNLIF. One refers to the model introduced in the papers of \textit{N. Brunel} [J. Comput. Neurosci. 8, No. 3, 183--208 (2000; Zbl 1036.92008)], and \textit{N. Brunel} and \textit{V. Hakim} [``Fast global oscillations in networks of integrate-and-fire neurons with long firing rates'', Neural Comput. 11, No. 7, 1621--1671 (1999; \url{doi:10.1162/089976699300016179})]. This model is described by an initial boundary value problem for a PDE and it writes: \[ \frac {\partial p} {\partial t} (v,t) + \frac{\partial}{\partial v}[(-v + bN(t))p(v,t)]- a \frac{\partial^2 {p}}{\partial {v^2}}(v,t) = N(t)\delta(v-V_R),\text{ for }v\leq V_F, \] \[ N(t) = -a\frac {\partial p} {\partial v}(V_F,t)\geq0,\ a>0 \] \[ p(v,0) =p^0(v)\geq 0,\quad p(V_F,t) =0,\quad p(-\infty,t)=0, \] \[ \int_{-\infty}^{V_F}p^0(v)dv =1 \] where \(p(., t)\) is the probability density of the electric potential of a randomly chosen neuron at time $t, v$ describes the membrane potential, $V_F$ is a threshold value, $\delta$ is the Dirac mass at point 0, $N$ is the firing rate of network, a is a diffusion coefficient and b is the connectivity parameter. One mainly investigates the problem of finite-time blow-up, the possibility to extend the solutions after a blow-up event and the global-in-time existence of classical solutions in the excitatory case, $b>0$. In the second section basics and assumptions are introduced. Blow-up for all initial conditions when b is large enough is discussed in the third section. The global-in-time existence in an excitatory network is largely presented and covers the fourth section. Conclusions are to be found in the fifth section and some technical results in the fifth section of the article.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A09
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 68T07
Mathematics Subject Classification ID: 82C32
Mathematics Subject Classification ID: 82C31
Mathematics Subject Classification ID: 35R07
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 7451609
zbMATH Keywords: leaky integrate and fire models
zbMATH Keywords: noise
zbMATH Keywords: blow-up
zbMATH Keywords: neural networks
zbMATH Keywords: delay
zbMATH Keywords: global existence
MaRDI profile type: Publication
full work available at URL: krm.2021025
OpenAlex ID: W3194169653
cites work: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.
cites work: Analysis of nonlinear noisy integrate \& fire neuron models: blow-up and steady states
cites work: A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
cites work: Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity
cites work: An Understanding of the Physical Solutions and the Blow-Up Phenomenon for Nonlinear Noisy Leaky Integrate and Fire Neuronal Models
cites work: Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay
cites work: Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods
cites work: Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience
cites work: Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience
cites work: Mean-field limit of generalized Hawkes processes
cites work: Microscopic approach of a time elapsed neural model
cites work: Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
cites work: Particle systems with a singular mean-field self-excitation. Application to neuronal networks
cites work: Synchronization of an excitatory integrate-and-fire neural network
cites work: A stochastic McKean-Vlasov equation for absorbing diffusions on the half-line
cites work: A McKean-Vlasov equation with positive feedback and blow-ups
cites work: A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration
cites work: Uniqueness for contagious McKean–Vlasov systems in the weak feedback regime
cites work: On a kinetic FitzHugh-Nagumo model of neuronal network
cites work: Relaxation in time elapsed neuron network models in the weak connectivity regime
cites work: Particle systems with singular interaction through hitting times: application in systemic risk modeling
cites work: Mean field systems on networks, with singular interaction through hitting times
cites work: Dynamics of current-based, Poisson driven, integrate-and-fire neuronal networks
cites work: Dynamics of a structured neuron population
cites work: Relaxation and self-sustained oscillations in the time elapsed neuron network model
cites work: On a voltage-conductance kinetic system for integrate \& fire neural networks
DOI: 10.3934/KRM.2021025
description: scientific article; zbMATH DE number 7451609