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

TL;DR

This survey reviews mathematical methods for nonlinear, non-local PDE models in neuroscience, covering derivations, analysis, and applications to neural dynamics and decision-making.

Contribution

It provides a comprehensive overview of mathematical techniques for analyzing complex PDE models in neuroscience, including parabolic, hyperbolic, and kinetic equations.

Findings

Analysis of nonlinear Fokker-Planck equations in neural models

Mathematical treatment of hyperbolic transport equations in neuroscience

Discussion of kinetic mesoscopic models like Voltage-Conductance and FitzHugh-Nagumo systems

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.