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A Fokker-Planck equation with superlinear drift at infinity for Integrate-and-Fire model

Benoît Perthame, Clément Rieutord, Delphine Salort

Abstract

The Integrate-and-Fire model is a Fokker-Planck equation arising in neuroscience. It describes the evolution of the probability density of the neuronal membrane potential and fitting has shown that the inclusion of a em superlinear drift provides the most realistic description. To make sense of this, we propose to set the equation on the full line, the neural activity being described by the flux at infinity. This framework serves as a model extension of the classical Noisy Integrate-and-Fire model, with a fixed firing potential. We first establish the well-posedness of the solution, establish the boundary condition at infinity which is the major difficulty. Then, state rigorously the entropy dissipation property. Finally, using Doeblin's method, we prove the exponential convergence of the solution toward the unique stationary state in full generality.

A Fokker-Planck equation with superlinear drift at infinity for Integrate-and-Fire model

Abstract

The Integrate-and-Fire model is a Fokker-Planck equation arising in neuroscience. It describes the evolution of the probability density of the neuronal membrane potential and fitting has shown that the inclusion of a em superlinear drift provides the most realistic description. To make sense of this, we propose to set the equation on the full line, the neural activity being described by the flux at infinity. This framework serves as a model extension of the classical Noisy Integrate-and-Fire model, with a fixed firing potential. We first establish the well-posedness of the solution, establish the boundary condition at infinity which is the major difficulty. Then, state rigorously the entropy dissipation property. Finally, using Doeblin's method, we prove the exponential convergence of the solution toward the unique stationary state in full generality.
Paper Structure (21 sections, 15 theorems, 243 equations)

This paper contains 21 sections, 15 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Assumptions
  4. Main results
  5. Existence for the superlinear FP-IF problem
  6. Weak solutions of the superlinear FP-IF problem
  7. The stationary state
  8. A truncated problem
  9. Proof of existence of a solution
  10. Uniqueness of the weak solutions
  11. Regularity of the solution and entropy inequality
  12. Maximum principle
  13. Regularity
  14. The strong form of the condition at infinity
  15. Relative entropy relation
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $h$ satisfies eq:comportement h -infty and eq:comportement h +infty. Then, for any initial data $u_0\in L^1(\mathbb{R})$, there exists a unique weak solution $(u,N_u)\in L^\infty(\mathbb{R}^+;L^1(\mathbb{R}))\times L^1_{loc}(\mathbb{R}^+)$ of Eq. eq:FPSL lin, i.e., that satisfies eq:form

Theorems & Definitions (33)

  • Theorem 1.1: Well-posedness in $L^1$
  • Proposition 1.2: Regularity of the solution
  • Theorem 1.3: Long-term convergence
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Stationary state for the truncated problem
  • Remark 2.5
  • ...and 23 more