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Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

Electro-magneto-encephalography and fundamental solutions


Authors: G. Dassios and A. S. Fokas
Journal: Quart. Appl. Math. 67 (2009), 771-780
MSC (2000): Primary 35J25, 35J55, 92B05, 92C55
Published electronically: May 27, 2009
MathSciNet review: 2588236
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Abstract | References | Similar Articles | Additional Information

Abstract: Electroencephalography (EEG) and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brain. For the case of the spherical homogeneous model, we elucidate the mathematical relations of these two methods. In particular, we derive and analyse three different representations for the electric and magnetic potentials, as well as the corresponding electric and magnetic induction fields, namely: integral representations involving the Green and the Neumann kernels, representations in terms of eigenfunction expansions, and closed form expressions. We show that the parts of the EEG and MEG fields in the interior of the brain that are due to the induction current are related via Kelvin's inversion transformation. We also derive closed form expressions for the interior and exterior vector potentials of the corresponding magnetic induction fields.


References [Enhancements On Off] (What's this?)

  • 1. Habib Ammari, Gang Bao, and John L. Fleming, An inverse source problem for Maxwell’s equations in magnetoencephalography, SIAM J. Appl. Math. 62 (2002), no. 4, 1369–1382 (electronic). MR 1898525 (2003f:35282), http://dx.doi.org/10.1137/S0036139900373927
  • 2. Bronzan J.B. (1971). The magnetic scalar potential. American Journal of Physics, vol. 39, pp. 1357-1359.
  • 3. George Dassios, What is recoverable in the inverse magnetoencephalography problem?, Inverse problems, multi-scale analysis and effective medium theory, Contemp. Math., vol. 408, Amer. Math. Soc., Providence, RI, 2006, pp. 181–200. MR 2262357 (2007h:35347), http://dx.doi.org/10.1090/conm/408/07693
  • 4. Dassios G. Electric and magnetic activity of the brain in spherical and ellipsoidal geometry. Lecture Notes from the Mini-course on Mathematics of Emerging Biomedical Imaging (H.Ammari, ed.), Paris 2007, Springer-Verlag (in press).
  • 5. Dassios G., Fokas A.S. Electro-magneto-encephalography for a three-shell model: Dipoles and beyond for the spherical geometry. Inverse Problems 25, 2009. doi: 10.1088/0266-5611/25/3/025001
  • 6. Dassios G., Fokas A.S. Methods for solving elliptic PDEs in spherical coordinates. SIAM Journal of Applied Mathematics 68, 1080-1096, 2008.
  • 7. Dassios G., Fokas A.S., Hadjiloizi D. On the complementarity of electro-encephalography and magnetoencephalography, Inverse Problems 23, 2007. doi: 10.1088/0266-5611/23/6/016
  • 8. G. Dassios, A. S. Fokas, and F. Kariotou, On the non-uniqueness of the inverse MEG problem, Inverse Problems 21 (2005), no. 2, L1–L5. MR 2146267 (2006b:78033), http://dx.doi.org/10.1088/0266-5611/21/2/L01
  • 9. A. S. Fokas, I. M. Gel-fand, and Y. Kurylev, Inversion method for magnetoencephalography, Inverse Problems 12 (1996), no. 3, L9–L11. MR 1391533, http://dx.doi.org/10.1088/0266-5611/12/3/001
  • 10. A. S. Fokas, Y. Kurylev, and V. Marinakis, The unique determination of neuronal currents in the brain via magnetoencephalography, Inverse Problems 20 (2004), no. 4, 1067–1082. MR 2087980 (2005c:92006), http://dx.doi.org/10.1088/0266-5611/20/4/005
  • 11. Geselowitz D.B. (1967). On bioelectric potentials in an inhomogeneous volume conductor. Biophysical Journal, vol. 7, pp. 1-11.
  • 12. Geselowitz D.B. (1970). On the magnetic field generated outside an inhomogeneous volume conductor by internal current sources. IEEE Transactions in Magnetism, vol. 6, pp. 346-347.
  • 13. Hamalainen M., Hari R., Ilmoniemi R.J., Knuutila J., Lounasmaa O. (1993). Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain. Reviews of Modern Physics, vol. 65, pp. 413-497.
  • 14. Heller L., Ranken D., Best E. (2004) The magnetic field inside special conducting geometries due to internal current. IEEE Transactions on Biomedical Engineering, vol. 51, pp. 1310-1318.
  • 15. Kellogg O.D. (1953) Foundations of Potential Theory. Dover Publications, New York.
  • 16. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Course of Theoretical Physics, Vol. 8. Translated from the Russian by J. B. Sykes and J. S. Bell, Pergamon Press, Oxford, 1960. MR 0121049 (22 #11796)
  • 17. Malmivuo J., Plonsey R. (1995). Bioelectromagnetism. Oxford University Press. New York.
  • 18. Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York, 1953. MR 0059774 (15,583h)
  • 19. Plonsey R., Heppner D.B. (1967). Considerations of quasi-stationarity in electrophysiological systems. Bulletin of Mathematical Biophysic, vol. 29, pp. 657-664.
  • 20. Sarvas J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetical inverse problem. Physics in Medicine and Biology, vol. 32, pp. 11-22.

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Additional Information

G. Dassios
Affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

A. S. Fokas
Affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

DOI: http://dx.doi.org/10.1090/S0033-569X-09-01144-7
PII: S 0033-569X(09)01144-7
Received by editor(s): September 4, 2008
Published electronically: May 27, 2009
Additional Notes: The first author is on leave from the University of Patras and ICEHT/FORTH, Greece.
Article copyright: © Copyright 2009 Brown University



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