Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear partial differential equations in neuroscience: from modelling to mathematical theory

José A Carrillo, Pierre Roux

Abstract

Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the mathematical methods developed to handle the unique features of these models, which are often nonlinear and non-local. The first part focuses on parabolic Fokker-Planck equations: the Nonlinear Noisy Leaky Integrate and Fire neuron model, stochastic neural fields in PDE form with applications to grid cells, and rate-based models for decision-making. The second part concerns hyperbolic transport equations, namely the model of the Time Elapsed since the last discharge and the jump-based Leaky Integrate and Fire model. The last part covers some kinetic mesoscopic models, with particular attention to the kinetic Voltage-Conductance model and FitzHugh-Nagumo kinetic Fokker-Planck systems.

Nonlinear partial differential equations in neuroscience: from modelling to mathematical theory

Abstract

Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the mathematical methods developed to handle the unique features of these models, which are often nonlinear and non-local. The first part focuses on parabolic Fokker-Planck equations: the Nonlinear Noisy Leaky Integrate and Fire neuron model, stochastic neural fields in PDE form with applications to grid cells, and rate-based models for decision-making. The second part concerns hyperbolic transport equations, namely the model of the Time Elapsed since the last discharge and the jump-based Leaky Integrate and Fire model. The last part covers some kinetic mesoscopic models, with particular attention to the kinetic Voltage-Conductance model and FitzHugh-Nagumo kinetic Fokker-Planck systems.
Paper Structure (136 sections, 127 theorems, 654 equations, 13 figures)

This paper contains 136 sections, 127 theorems, 654 equations, 13 figures.

Table of Contents

  1. Notation:
  2. Parabolic equations of Fokker-Planck type
  3. The Nonlinear Noisy Leaky Integrate & Fire model for neural networks
  4. Derivation of the model à la physicienne
  5. Mathematical setting for the NNLIF type models
  6. Characterisation of the stationary states
  7. Construction of solutions
  8. Local-in-time existence via a Stefan-like free boundary problem
  9. Global-in-time existence and uniform bounds in the inhibitory case
  10. Finite time blow-up in non-delayed networks (d=0)
  11. Desynchronisation in the weakly connected case
  12. A priori convergence in relative entropy
  13. Linear case $b=0$.
  14. Weakly nonlinear case.
  15. $L^q$ estimates on the firing rate and global existence for weak interaction
  16. ...and 121 more sections

Key Result

Theorem 1.5

Let $(\rho_\infty,N_\infty)$ be a stationary state of NNLIF. It must be of the form where $N_\infty$ satisfies the fixed-point equation

Figures (13)

  • Figure 1: Simulation of the PDE \ref{['NNLIF']} with $V_F=1,V_R=0,a=0.2,d=1$ and $b=-45$; (A.) solution at three different times; (B.) evolution in time of the firing rate $N(t)$. Ikeda, Roux, Salort, Smets IRSS.
  • Figure 2: Simulation of the PDE \ref{['NNLIF']} with $V_F=0,V_R=-2,a=0.2,d=1$ and $b=-35$; (A.) solution at three different times; (B.) evolution in time of the firing rate $N(t)$. Ikeda, Roux, Salort, Smets IRSS.
  • Figure 3: Numerical simulation of the four population model for grid cells \ref{['eq:4PDE']} for different values of $\sigma$. Top Row is the repartition of the density for $s=0.5$; middle row is the same for $s=0$; bottom row represents in grey the path of a real rat and in red the prediction of the model of where in physical space one specific grid cell fires -- the one at $x=(0,0)$. Carrillo, Holden, Solem CHS
  • Figure 4: Numerical simulation of \ref{['eqn:2']} starting from a small perturbation of a constant function, $t=40, 220, 1500, 1810, 2190,$ and $2400$ m$s$. Carrillo, Roux, Solem CRS.
  • Figure 5: Dynamics in the unbiased ($\lambda_i=0$) case of \ref{['ODE']} in the deterministic system $\beta=0$ (straight lines) and the stochastic system $\beta>0$ (wiggled line). Straight lines emphasize the location of the equilibrium points. The black bars are the equilibrium points of the deterministic differential system. The single particle realization of the stochastic system starts its dynamics at point $(5,5)$, moves almost straight forward towards a slow-manifold nearby the spontaneous state, and then slowly oscillates towards one of the decision states. Carrillo, Cordier, Mancini CCM.
  • ...and 8 more figures

Theorems & Definitions (149)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4: Initial datum for classical solutions
  • Theorem 1.5: Cáceres, Carrillo, Perthame CCP
  • Theorem 1.6: Cáceres, Carrillo, Perthame, CCP
  • Theorem 1.7: Cáceres, Carrillo, Perthame CCP
  • Theorem 1.8: Carrillo, Gonzalez, Gualdani, Schonbeck CGGS
  • Definition 1.9: universal super-solutions
  • Lemma 1.10
  • ...and 139 more