| Issue |
ESAIM: M2AN
Volume 52, Number 5, September–October 2018
|
|
|---|---|---|
| Page(s) | 2083 - 2107 | |
| DOI | https://doi.org/10.1051/m2an/2017062 | |
| Published online | 17 January 2019 | |
Determining the distribution of ion channels from experimental data★
1
École Normale Supérieure de Lyon, UMR 5569, Unité de Mathématiques Pures et Appliquées,
69007
Lyon, France.
2
INRIA Numed,
46 allée d’Italie,
69007
Lyon, France.
3
Department of Engineering Mathematics, Center for Mathematical Modelling (CMM), UMI 2807 CNRS-Chile & Center for Biotechnology and Bioengineering (CeBiB), University of Chile,
Santiago, Chile.
4
Departamento de Matemática, Universidad Técnica Federico Santa María,
Avenida España 1680,
Valparaíso, Chile.
* Corresponding author: thibault.bourgeron@ens-lyon.org
Received:
25
October
2016
Accepted:
30
November
2017
The authors study an integral inverse problem arising in the biology of the olfactory system. The transduction of an odor into an electrical signal is accomplished by a depolarising influx of ions through cyclic-nucleotide-gated (CNG for short) channels on the cilium membrane. The inverse problem studied in this paper consists in finding the spatial distribution of the CNG channels from the measured transduce electrical signals. 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. New estimates using arguments in the spirit of the stationary phase method are proven and a numerical scheme is proposed to reconstruct the density of CNG channels from modeled current representing experimental data, for an approximated model. For the original model an identifiability and a non observability (in some weighted L2 spaces) results are proven.
Key words: CNG channel / integral equation / ill-posed problem / Mellin transform
This work has begun during a visit by T.B. to the CMM in Santiago, Chile. This was made possible thanks to Ecos Grant C11E07 and thanks to the CMM. The authors received partial support from Regional Program STIC-AmSud Project Moscow. C.C.’s research is also partially supported by PFBasal-01, PFBasal-03 projects and by Fondecyt Grant 1140773. R.L.’s research was partially supported by PFBasal-03. T.B.’s research is supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 639638).
© EDP Sciences, SMAI 2019
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