An Inverse Problem in the Mathematical Modelling of our Sense of Smell

Abstract

The first step in our sensing of smell is the conversion of chemical odorants into electrical signals. This happens when odorants stimulate ion channels along cilia, which are long thin cylindrical structures in our olfactory system. Determining how the ion channels are distributed along the length of a cilium is beyond current experimental methods. Here we describe how this can be approached as a mathematical inverse problem. Precisely, two integral equations based mathematical models are studied for the inverse problem of deter- mining the distribution of ion channels in cilia of olfactory neurons from experimental data. The Mellin transform allows us to write an explicit formula for their solutions. Proving observability and continuity inequalities for the second integral equation is then a question of estimating the Mellin transform of the kernel on vertical lines. For the first integral model, an identifiability and a non observability (in some weighted spaces) results are proven.