Determining the distribution of ion channels from experimental data
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2083-2107.

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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017062
Mots-clés : CNG channel, integral equation, ill-posed problem, Mellin transform
Bourgeron, Thibault 1 ; Conca, Carlos 1 ; Lecaros, Rodrigo 1

1
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     title = {Determining the distribution of ion channels from experimental data},
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Bourgeron, Thibault; Conca, Carlos; Lecaros, Rodrigo. Determining the distribution of ion channels from experimental data. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2083-2107. doi : 10.1051/m2an/2017062. http://archive.numdam.org/articles/10.1051/m2an/2017062/

[1] L.F.A. Arbogast, Du calcul des dérivations. Levrault frères (1800).

[2] D. Badamdorj, D.A. Edwards, D.A. French and S.J. Kleene, Identification of cl(ca) channel distributions in olfactory cilia. Math. Meth. Appl. Sci. 31 (2008) 1860–1873. | DOI | MR | Zbl

[3] T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel. Inverse Probl. 30 (2014) 025007–025028. | DOI | MR | Zbl

[4] C. Chen, T. Nakamura and Y. Koutalos, Cyclic AMP diffusion coefficient in frog olfactory cilia. Biophys. J. 76 (1999) 2861–2867. | DOI

[5] C. Conca, R. Lecaros, J.H. Ortega and L. Rosier, Determination of the calcium channel distribution in the olfactory system. J. Inverse Ill-Posed Probl. 22 (2014) 671–711. | DOI | MR | Zbl

[6] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Vol. 375 of Mathematics and its Applications. Springer (1996). | MR | Zbl

[7] G.L. Fain, M.J. Fain and T.J. O’Dell, Molecular and Cellular Physiology of Neurons. Harvard University Press, 2nd edition (2014).

[8] R.J. Flannery, D.A. French and S.J. Kleene, Clustering of cyclic-nucleotide-gated channels in olfactory cilia. Biophys. J. 91 (2006) 179–188. | DOI

[9] D.A. French and D.A. Edwards, Perturbation approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation. J. Math. Biol. 55 (2007) 745–765. | DOI | MR | Zbl

[10] D.A. French and C.W. Groetsch, Integral equation models for the inverse problem of biological ion channel distributions. J. Phys.: Conf. Ser. 73 (2007) 1742–6596.

[11] D.A. French, R.J. Flannery, C.W. Groetsch, W.B. Krantz and S.J. Kleene, Numerical approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation. Math. Comput. Model. 43 (2006) 945–956. | DOI | MR | Zbl

[12] S.J. Kleene, Origin of the chloride current in olfactory transduction. Neuron 11 (1993) 123–132. | DOI

[13] S.J. Kleene, Both external and internal calcium reduce the sensitivity of the olfactory cyclic-nucleotide-gated channel to camp. J. Neurophysiol. 81 (1999) 2675–2682. | DOI

[14] S.J. Kleene and R.C. Gesteland, Calcium-activated chloride conductance in frog olfactory cilia. J. Neurosci. 11 (1991) 3624–3629. | DOI

[15] S.J. Kleene and R.C. Gesteland, Transmembrane currents in frog olfactory cilia. J. Membr. Biol. 120 (1991) 75–81. | DOI

[16] S.J. Kleene, R.C. Gesteland and S.H. Bryant, An electrophysiological survey of frog olfactory cilia. J. Exp. Biol. 195 (1994) 307–328. | DOI

[17] Y. Koutalos, K. Nakatani and K.-W. Yau, Cyclic GMP diffusion coefficient in rod photoreceptor outer cyclic gmp diffusion coefficient in rod photoreceptor outer cyclic gmp diffusion coefficient in rod photoreceptor outer segments. Biophys. J. (1995) 373–382.

[18] E. Lindelöf, Robert Hjalmar Mellin. Adv. Math. 61 (1933) i–vi. | JFM | MR

[19] S.M. Riad, The deconvolution problem: an overview. Proc. IEEE 74 (1986) 82–85. | DOI

[20] W. Rudin. Fourier Analysis on Groups. Interscience Publishers, Inc., New York (1962). | MR | Zbl

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