Determination of the calcium channel distribution in the olfactory system

TL;DR

This paper investigates a linear inverse problem modeled by a Fredholm integral equation to determine calcium channel distribution in the olfactory system, providing theoretical results and a numerical reconstruction method.

Contribution

It offers new identifiability, stability, and reconstruction results for step function kernels and introduces a numerical algorithm using a non-regular mesh for kernel estimation.

Findings

Established identifiability and stability for step function kernels

Developed a numerical reconstruction algorithm with non-regular mesh

Proved identifiability for polynomial kernel approximations under degree nine

Abstract

In this paper we study a linear inverse problem with a biological interpretation, which is modeled by a Fredholm integral equation of the first kind. When the kernel in the Fredholm equation is represented by step func- tions, we obtain identifiability, stability and reconstruction results. Further- more, we provide a numerical reconstruction algorithm for the kernel, whose main feature is that a non-regular mesh has to be used to ensure the invert- ibility of the matrix representing the numerical discretization of the system. Finally, a second identifiability result for a polynomial approximation of degree less than nine of the kernel is also established.