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 (Demidov, Unique solvability of the inverse MEEG-problem, 2017, to appear). The solution of this problem is the distribution of y↦q(y) current dipoles of brain neurons that occupies the region \(Y \subset \mathbb {R}^3 \). 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 = q 0 + p 0 δ|∂Y, where q 0 is the usual function defined in Y, and p 0 δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p 0. 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.
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Notes
- 1.
The vector E, having the Cartesian coordinates (E 1, E 2, E 3) corresponds to the differential form
$$\displaystyle \begin{aligned} \omega^1_{\mathbf{E}}=E_1dx_1+E_2dx_2+E_3dx_3\,, \end{aligned}$$and to the vector \(\mathrm {rot}\mathbf {E}= \left (\frac {{\partial } E_3}{{\partial }{x_2}}-\frac {{\partial } E_2}{{\partial }{x_3}}, \frac {{\partial } E_1}{{\partial }{x_3}}-\frac {{\partial } E_3}{{\partial }{x_1}}, \frac {{\partial } E_2}{{\partial }{x_1}}-\frac {{\partial } E_1}{{\partial }{x_2}}\right )\)—is the differential form
$$\displaystyle \begin{aligned} \omega^2_{\mathrm{rot}\mathbf{E}}= \left(\frac{{\partial} E_3}{{\partial}{x_2}}-\frac{{\partial} E_2}{{\partial}{x_3}}\right)dx_2\wedge dx_3+ \left(\frac{{\partial} E_1}{{\partial}{x_3}}-\frac{{\partial} E_3}{{\partial}{x_1}}\right)dx_3\wedge dx_1+ \left(\frac{{\partial} E_2}{{\partial}{x_1}}-\frac{{\partial} E_1}{{\partial}{x_2}}\right)dx_1\wedge dx_2\,. \end{aligned}$$We have: \(d\omega ^1_{\mathbf {E}}=\omega ^2_{\mathrm {rot}\mathbf {E}}.\) Therefore, the condition rotE = 0 implies \(d\omega ^1_{\mathbf {E}}=0.\) Consequently, \(\int _{{\partial } \Omega }\omega ^1_{\mathbf {E}}=\int _{\Omega }d\omega ^1_{\mathbf {E}}=0\,,\) where Ω is a surface in \(\mathbb {R}^3,\) limited by the boundary ∂ Ω. If ∂ Ω is a curve (in other words, Ω is a simply connected surface), then the equality \(\int _{{\partial } \Omega }\omega ^1_{\mathbf {E}}=0\) means that the integral from some point P 0 ∈ ∂ Ω to some other point P ∈ ∂ Ω does not depend on which part of the curve ∂ Ω it will be taken. In other words, \(\omega ^1_{\mathbf {E}}=E_1dx_1+E_2dx_2+E_3dx_3\) is the total differential: \(\omega ^1_{\mathbf {E}}=-d\Phi ,\) i.e. E = −∇ Φ. According to physical representations, at infinity the potential Φ of the field E = −∇ Φ is a constant that can be considered equal to zero.
- 2.
If \(\omega ^2_{\mathbf {B}}= B_1dx_2\wedge dx_3+B_2dx_3\wedge dx_1+B_3dx_1\wedge dx_2,\) then \(d\omega ^2_{\mathbf {B}}=\omega ^3_{\mathrm {div}\mathbf {B}}.\) Therefore, the condition divB = 0 implies the equality \(d\omega ^2_{\mathbf {B}}=0,\) i.e. the closed form \(\omega ^2_{\mathbf {B}}.\) By the Poincare lemma, it is exact in a simply-connected domain, that is \(\omega ^2_{\mathbf {B}}=d\omega ^1_{\mathbf {A}},\) in other words B = rotA. We can assume that A|∞ = 0 as B|∞ = 0.
- 3.
It depends on Φρ and therefore on ρ.
- 4.
The formula (12) is an integral version of the Biot–Sawar law: \(\mathbf {B}(\mathbf {x})=\frac {\mathbf {q}\times (\mathbf {x}-\mathbf { y})}{|\mathbf {x}-\mathbf {y}|{ }^3}\) for the field B, that induced by a current dipole q. This formula, which was experimentally established in 1820 by the French physicists Jean-Baptiste Biot (1774–1862) and Felix Sawar (1791–1841), in the process of observing the effect on the magnetic needle of a conductor with the current flowing along it, is a consequence of the equations Maxwell.
- 5.
It has been proved by A.S. Kochurov.
- 6.
A similar result is initiated by the problem of measuring the magnetic field by scanning magnetic microscope, was obtained in [13] using a generalization of the classic decomposition Hodge Laplace operator on a compact orientable manifold in the form of a sum dδ + δd, where δ is the operator conjugate to the operator of exterior differentiation d.
- 7.
This imposes restrictions on the real and imaginary parts of the functions \(\widetilde {B}_j\).
- 8.
- 9.
\({\mathbf {F}}^{-1}_{\xi \to \mathbf {y}}\widetilde {c}(\xi )=\int _0^\infty |\xi |\Bigl (\int _0^1 \widetilde {C}(|\xi |,\omega ) e^{\overset {\circ }{\imath }\rho |\xi |\cos 2\pi (\omega -\phi )}d\omega \Bigr ) d|\xi |\) and \( e^{\overset {\circ }{\imath }\rho |\xi |\cos 2\pi (\omega -\phi )}= \sum \limits _{n\in \mathbb {Z}}J_n(2\pi \rho |\xi |)i^n e^{\overset {\circ }{\imath } n(\omega -\phi )}\) (cf. (25)).
- 10.
Arguments of \(p_n^k(|\xi |),\) \(q_n^k(|\xi |),\) J n(2π|ξ|ρ) for brevity are omitted.
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Acknowledgements
This work is partially supported by grants of Russian Foundation for Basic Research (15-01-03576, 16-01-00781 and 17-01-00809).
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Demidov, A.S., Galchenkova, M.A. (2018). The Inverse Magnetoencephalography Problem and Its Flat Approximation. In: Mondaini, R. (eds) Trends in Biomathematics: Modeling, Optimization and Computational Problems. Springer, Cham. https://doi.org/10.1007/978-3-319-91092-5_10
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