INVERSE PROBLEMS IN MAGNETO-ELECTROSCANNING (IN ENCEPHALOGRAPHY, FOR MAGNETIC MICROSCOPES, ETC.)

Alexre Sergeevich Demidov

doi:10.11948/2018.915

2018 Volume 8 Issue 3

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Alexre Sergeevich Demidov. INVERSE PROBLEMS IN MAGNETO-ELECTROSCANNING (IN ENCEPHALOGRAPHY, FOR MAGNETIC MICROSCOPES, ETC.)[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 915-927. doi: 10.11948/2018.915

Citation:

Alexre Sergeevich Demidov. INVERSE PROBLEMS IN MAGNETO-ELECTROSCANNING (IN ENCEPHALOGRAPHY, FOR MAGNETIC MICROSCOPES, ETC.)[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 915-927. doi: 10.11948/2018.915

INVERSE PROBLEMS IN MAGNETO-ELECTROSCANNING (IN ENCEPHALOGRAPHY, FOR MAGNETIC MICROSCOPES, ETC.)

Alexre Sergeevich Demidov

Department of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskije Gory, 119991 Moscow, Russia Moscow Institute of Physics and Technology(State University), Dolgoprudny, Moscow Region, 141700, Russia

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Abstract

Abstract

Contrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove its unique solvability in the framework of the system of Maxwell's equations[3]. The solution of this problem is the distribution of y ↦ q(y) current dipoles of brain neurons that occupies the region Y ⊂ R³. It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient's head. The solution can be represented in the form q=q₀ + p₀^δ|_∂Y', where q₀ is the usual function defined in Y,and p₀^δ|_∂Y is a δ-function on the boundary of the domain Y with a certain density p₀. It is essential that p₀ and p₀ are interrelated. This ensures the correctness of the inverse MEEGproblem. However, the components of the required 3-dimensional distribution q must turn out to be linearly dependent if only the magnetic field B is taken into account. This question is considered in detail in a flat model of the situation.
- Inverse problems /
- integral equations /
- pseudo-differential operators /
- magneto-electroscanning
MSC: 31A25;31B10;78A30

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References

[1]	M. H. Acuna, G. Kletetschka and J. E. P. Connerney, Mars crustal magnetization:a window into the past?, The Martian Surface:Composition, Mineralogy and Physical Properties, ed. J.F. Bell, Cambridge University Press, 2008, 242-262. Google Scholar
[2]	L. Baratchart et al., Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions, Inverse Problems, 2013, 29, 1-29. Google Scholar
[3]	A. S. Demidov, The inverse problem of magneto-electroencephalography is wellposed:it has a unique solution that is stable with respect to perturbations, Fundamental and Applied Mathematics, 2016, 21(4), 17-22. Google Scholar
[4]	A. S. Demidov, Elliptic pseudodifferential boundary value problems with a small parameter in the coefficient of the leading operator, Math. USSR-Sb., 1973, 20(3), 439-463. Google Scholar
[5]	A. S. Demidov, Asymptotics of the solution of the boundary value problem for elliptic pseudodifferential equations with a small parameter with the highest operator, Proceedings of the Moscow Mathematical Society, 32, Publishing house of Moscow University, 1975, 119-146(In Russian). See also:A. S. Demidov, Les problèmes elliptiques pseudo-différentiels, à petit paramètre dans l'opérateur principal, Lecture Notes in Math., Springer-Verlag, 1977, 594, 108-122. Google Scholar
[6]	A. S. Demidov and M. A. Galchenkova, The inverse magnitoencephalograhy problem and its flat approximation, Proceedings BIOMATH Moscow 2017, to appear. Google Scholar
[7]	A. S. Demidov and M. A. Galchenkova, Inverse magneto-electroscanning problems (in encephalography, for magnetic microscopes, etc.), to appear. Google Scholar
[8]	M:Hämäläinen et al, Magnetoencephalography\|theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, 1993, 65(2), 413-497. Google Scholar
[9]	H. Helmholtz, Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern, mit Anwendung auf die thierisch-elektrischen Versuche, Ann. Phys. Chem., 1853, 89, 211-233, 353-377. Google Scholar
[10]	M. A. Lavrentiev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable, Nauka, Moscow, 1973. Google Scholar
[11]	Y. Martin et al, Magnetic imaging by "force microscopy" with 1000 A resolution, Appl. Phys. Lett., 1987, 50, 1455-1547. Google Scholar
[12]	T. A. Stroganova et al, EEG alpha activity in the human brain during perception of an illusory kanizsa square, Neuroscience and Behavioral Physiology, 2011, 41(2), 130-139. Google Scholar
[13]	D. Sheltraw and E. Coutsias, Invertibility of current density from near-field electromagnetic data, Journal of Applied Physics, 2003, 94(8), 5307-5315. Google Scholar
[14]	A. N. Shestakova, A.V. Butorina, A. E. Ossadtchi and Yu.Yu. Shtyrov, Magnetoencephalography-a new method of functional mapping of the human brain, Experimental Psychology, 2012, 5(2), 119-134(in Russian). Google Scholar
[15]	V. S. Vladimirov Equations of mathematical physics, Mir, Moscow, 1984. Google Scholar
[16]	B. P. Weiss, E. A. Lima, L. E. Fong and F. J. Baudenbacher, Paleomagnetic analysis using SQUID microscopy, J. Geophys Res., 2007, 112, B09105. Google Scholar