Numerical approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Cilia are long thin structures that extend from the olfactory receptor neurons into the nasal mucus. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclic-nucleotide-gated (CNG) channels which reside in the ciliary membrane and are activated by adenosine 3′, 5′-cyclic monophosphate (cAMP) allow a depolarizing influx of Ca2+ and Na+ and are thought to initiate the electrical signal.In this paper, a mathematical model consisting of two nonlinear differential equations and a constrained Fredholm integral equation of the first kind is developed to model experiments involving the diffusion of cAMP into cilia and the resulting electrical activity. The unknowns in the problem are the concentration of cAMP, the membrane potential and, the quantity of most interest in this work, the distribution of CNG channels along the length of a cilium. A simple numerical method is derived that can be used to obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. Certain computations indicate that this mathematical problem is ill-conditioned.