Determination of the calcium channel distribution in the olfactory system

Carlos Conca (1, 2) , Rodrigo Lecaros (1, 2) , Jaime Ortega (1, 2) , Lionel Rosier (3, 4)

1 DIM - Departamento de Ingeniería Matemática [Santiago]
2 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
3 CAS - Centre Automatique et Systèmes
4 IECL - Institut Élie Cartan de Lorraine

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 functions, we obtain identifiability, stability and reconstruction results. Furthermore, 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 invertibility 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.

Mots clés: Partial Differential Equations; Kernel Determination; Olfactory System; Inverse Problems; Numerical Reconstruction

Domaines: Automatique / Robotique

Carlos Conca, Rodrigo Lecaros, Jaime Ortega, Lionel Rosier. Determination of the calcium channel distribution in the olfactory system. Journal of Inverse and Ill-posed Problems, 2014, 22 (5), pp.671--711. ⟨10.1515/jip-2013-0033⟩. ⟨hal-01112038⟩