Determining the distribution of calcium channels in the olfactory system

Cilia are long thin cylindrical structures that extend from an olfactory receptor into the nasal mucus. The transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclic-nucleotide-gated (CNG) channels, activated by cyclic adenosine monophosphate (cAMP), allow a depolarizing influx of sodium ions, which initiate the electrical signal. A model for experiments involving the diffusion of cAMP into cilia and the resulting current activity has been developed in [8]. In this paper we study the inverse problem which consists in finding the spatial distribution of CNG ion channels from the experimental current data. This problem comes down to solving a Fredholm integral equation of the first kind. The Mellin transform allows us to write an explicit formula for its solution. Proving observability and continuity inequalities is then a question of estimating the Mellin transform of the kernel of this integral equation on vertical lines. We prove new estimates using arguments in the spirit of the stationary phase method. We complete the continuity and observability results obtained in [4] for an approximated kernel. We also introduce a better approximated kernel for which we prove continuity and observability results, and for which we perform numerical computations using experimental data. Lastly, we prove that no observability inequality, in certain weighted L 2 spaces, can be found for the original model. Throughout this paper we present numerical simulations illustrating the theoretical results