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On the Inverse MEG Problem with a 1-D Current Distribution

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DOI: 10.4236/am.2015.61010    2,111 Downloads   2,441 Views  

ABSTRACT

The inverse problem of magnetoencephalography (MEG) seeks the neuronal current within the conductive brain that generates a measured magnetic flux in the exterior of the brain-head system. This problem does not have a unique solution, and in particular, it is not even possible to identify the support of the current if it extends over a three-dimensional set. However, a localized current supported on a zero-, one- or two-dimensional set can in principle be identified. In the present work, we demonstrate an analytic algorithm that is able to recover a one-dimensional distribution of current from the knowledge of the exterior magnetic flux field. In particular, we consider a neuronal current that is supported on a small line segment of arbitrary location and orientation in space, and we reduce the identification of its characteristics to a nonlinear algebraic system. A series of numerical tests show that this system has a unique real solution. A special case is easily solved via the use of trivial algebraic operations.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Dassios, G. and Satrazemi, K. (2015) On the Inverse MEG Problem with a 1-D Current Distribution. Applied Mathematics, 6, 95-105. doi: 10.4236/am.2015.61010.

References

[1] Hamalainen, M., Hari, R., Ilmoniemi, R.J., Knuutila, J. and Lounasmaa, O. (1993) Magnetoencephalography—Theory, Instrumentation, and Applications to Noninvasive Studies of the Working Human Brain. Reviews of Modern Physics, 65, 413.
http://dx.doi.org/10.1103/RevModPhys.65.413
[2] Malmivuo, J. and Plonsey, R. (1995) Bioelectromagnetism. Oxford University Press, New York.
[3] Helmholtz, H. (1853) Ueber einige Gesetze der Vertheilung elektrischer Str ome in k orperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche. Annalen der Physik und Chemie, 89, 211-233, 353-377.
[4] Dassios, G. and Fokas, A.S. (2013) The Definitive Non Uniqueness Results for Deterministic EEG and MEG Data. Inverse Problems, 29, 1-10.
http://dx.doi.org/10.1088/0266-5611/29/6/065012
[5] Albanese, R. and Monk, P.B. (2006) The Inverse Source Problem for Maxwell’s Equations. Inverse Problems, 22, 1023-1035.
http://dx.doi.org/10.1088/0266-5611/22/3/018
[6] Landau, L.D. and Lifshitz, E.M. (1960) Electrodynamics of Continuous Media. Pergamon Press, London.
[7] Plonsey, R. and Heppner, D.B. (1967) Considerations of Quasi-Stationarity in Electrophysiological Systems. Bulletin of Mathematical Biophysics, 29, 657-664.
http://dx.doi.org/10.1007/BF02476917
[8] Sarvas, J. (1987) Basic Mathematical and Electromagnetic Concepts of the Biomagnetic Inverse Problems. Physics in Medicine and Biology, 32, 11-22.
http://dx.doi.org/10.1088/0031-9155/32/1/004
[9] Geselowitz, D.B. (1970) On the Magnetic Field Generated outside an Inhomogeneous Volume Conductor by Internal Current Sources. IEEE Transactions in Biomagnetism, 6, 346-347.
http://dx.doi.org/10.1109/TMAG.1970.1066765
[10] Dassios, G. (2009) Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry. Mathematical Modeling in Biomedical Imaging I Lecture Notes in Mathematics, 133-202.
[11] Dassios, G. (2012) Ellipsoidal Harmonics. Theory and Applications. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781139017749
[12] Dassios, G. and Fokas, A.S. (2009) Electro-Magneto-Encephalography and Fundamental Solutions. Quarterly of Applied Mathematics, 67, 771-780.
[13] Dassios, G. and Fokas, A.S. (2009) Electro-Magneto-Encephalography for the Three-Shell Model: Dipoles and Beyond for the Spherical Geometry. Inverse Problems, 25, Article ID: 035001.
http://dx.doi.org/10.1088/0266-5611/25/3/035001
[14] Morse, P.M. and Feshbach, H. (1953) Methods of Theoretical Physics, Volume I. McGraw-Hill, New York.
[15] Brand, L. (1947) Vector and Tensor Analysis. John Wiley and Sons, New York.

  
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