Skip to main content
SpringerLink
Log in

Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry

  • Chapter
  • First Online:
Mathematical Modeling in Biomedical Imaging I

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 1983))

Abstract

Understanding the functional brain is one of the top challenges of contemporary science. The challenge is connected with the fact that we are trying to understand how an organized structure works and the only means available for this task is the structure itself. Therefore an extremely complicated scientific problem is combined with a hard philosophical problem.

Under the given conditions it comes as no surprise that so many, apparently simple, physical and mathematical problems in neuroscience are not generally solved today. One of this problems is the electromagnetic problem of a current field inside an arbitrary conductor. We do understand the physics of this problem, but it is very hard to solve the corresponding mathematical problem if the geometry of the conducting medium diverts from the spherical one. Mathematically, the human brain is an approximately 1.5 L of conductive material in the shape of an ellipsoid with average semiaxes of 6, 6.5 and 9 cm [47]. On the outermost layer of the brain, known as the cerebral cortex, most of the 1011 neurons contained within the brain are distributed. The neurons are the basic elements of this complicated network and each one of them possesses 104 interconnections with neighboring neurons. At each interconnection, also known as a synapse, neurons communicate via the transfer of particular ions, the neurotransmitters [38]. Neurons are electrochemically excited and they are able to fire instantaneous currents giving rise to very weak magnetic fields which can be measured with the SQUID (Superconducting QUantum Interference Device). The SQUID is the most sensitive apparatus ever built. It can measure magnetic fields as small as 10−14 T, a sensitivity that is necessary to measure the 10−15 to 10−13 T fields resulting from brain activity. For comparison we mention that the magnetic fields due to brain activity are about 10−9 of the Earth’s average magnetic field, 10−5 of the fluctuations of the Earth’s magnetic field, and about 10−3 of the maximum magnetic field generated by the beating heart.

On leave from the University of Partras and ICE-HT/FORTH, Greece

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ammari H., Bao G., Fleming J.L. (2002). An inverse source problem for Maxwell’s equations in magnetoencephalography. SIAM Journal of Applied Mathematics, vol. 62, pp. 1369–1382.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bronzan J.B. (1971). The magnetic scalar potential. American Journal of Physics, vol. 39, pp. 1357–1359.

    Article  Google Scholar 

  3. Cuffin B.N., Cohen D. (1977). Magnetic fields of a dipole in special volume conductor shapes. IEEE Transactions on Biomedical Engineering, vol. 24, pp. 372–381.

    Article  Google Scholar 

  4. Dassios G. (2005). On the hidden electromagnetic activity of the brain. In: Fotiadis D., Massalas C. (eds) Mathematical Methods in Scattering Theory and Biomedical Technology, Nymfaio, pp. 297–303.

    Google Scholar 

  5. Dassios G. (2006). What is recoverable in the inverse magnetoencephalography problem? In: Ammari H., Kang H.(eds). Inverse Problems, Multi-Scale Analysis and Effective Medium Theory, American Mathematical Society, Contemporary Mathematics 408, Providence, pp. 181–200.

    Google Scholar 

  6. Dassios G. (2007) The magnetic potential for the ellipsoidal MEG problem. Journal of Computational Mathematics, vol. 25, pp. 145–156.

    MathSciNet  Google Scholar 

  7. Dassios G. On the image systems for the electric and the magnetic potential of the brain. In: Alikakos N., DougalisV., (eds). Modern Mathematical Methods in Sciences and Technology, Paros, (in press).

    Google Scholar 

  8. Dassios G., Kariotou F. (2001). The direct magnetoencephalography problem for a genuine 3-D brain model. In: Fotiadis D., Massalas C., (eds) Scattering Theory and Biomedical Engineering. Modeling and Applications, Corfu. World Scientific, London, pp. 289–297.

    Google Scholar 

  9. Dassios G., Kariotou F. (2003). On the Geselowitz formula in biomathematics. Quarterly of Applied Mathematics, vol. 61, pp. 387–400.

    MathSciNet  MATH  Google Scholar 

  10. Dassios G., Kariotou F. (2003). Magnetoencephalography in ellipsoidal geometry. Journal of Mathematical Physics. vol. 44, pp. 220–241.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dassios G., Kariotou F. (2003). The effect on an ellipsoidal shell on the direct EEG problem. In: Fotiadis D., Massalas C., (eds) Mathematical Methods in Scattering Theory and Biomedical Technology, Tsepelovo. World Scientific, London, pp.495–503.

    Google Scholar 

  12. Dassios G., Kariotou F. (2004). On the exterior magnetic field and silent sources in magnetoencephalography. Abstract and Applied Analysis, vol. 2004, pp. 307–314.

    Article  MathSciNet  MATH  Google Scholar 

  13. Dassios G., Kariotou F. (2005). The direct MEG problem in the presence of an ellipsoidal shell inhomogeneity. Quarterly of Applied Mathematics, vol. 63, pp. 601–618.

    MathSciNet  MATH  Google Scholar 

  14. Dassios G., Kleinman R. (2000). Low frequency scattering. Oxford University Press. Oxford.

    MATH  Google Scholar 

  15. Dassios G., Kariotou F., Kontzialis K., Kostopoulos V. (2002). Magnetoencephalography for a realistic geometrical model of the brain. Third European Symposium in Biomedical Engineering and Medical Physics, Patras, p.22.

    Google Scholar 

  16. Dassios G., Giapalaki S., Kariotou F., Kandili A. (2005). Analysis of EEG Images. In: Fotiadis D., Massalas C., (eds). Mathematical Methods in Scattering Theory and Biomedical Technology, Nymfaio, pp. 304–311.

    Google Scholar 

  17. Dassios G., Fokas A.S., Kariotou F. (2005). On the non-uniqueness of the inverse MEG problem. Inverse Problems, vol. 21, pp. L1–L5.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dassios G., Giapalaki S., Kandili A., Kariotou F. (2007). The exterior magnetic field for the multilayer ellipsoidal model of the brain. Quarterly Journal of Mechanics and Applied Mathematics vol. 60, pp. 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  19. de Munck J.C. (1988). The potential distribution in a layered anisotropic spheroidal volume conductor. Journal of Applied Physics, vol. 2, pp. 464–470.

    Article  Google Scholar 

  20. Fokas A.S., Gelfand I.M., Kurylev Y. (1996). Inversion method for magnetoencephalography. Inverse Problems, vol. 12, pp. L9–L11.

    Article  Google Scholar 

  21. Fokas A.S., Kurylev Y., Marikakis V. (2004). The unique determination of neuronal current in the brain via magnetoencephalography. Inverse Problems, vol. 20, pp. 1067–1082.

    Article  MathSciNet  MATH  Google Scholar 

  22. Frank E. (1952). Electric potential produced by two point current sources in a homogeneous conducting sphere. Journal of Applied Physics, vol. 23, pp. 1225–1228.

    Article  MATH  Google Scholar 

  23. Garmier R., Barriot J.P. (2001). Ellipsoidal harmonic expansions of the gravitational potential: Theory and applications. Celestial Mechanics and Dynamical Astronomy, vol. 79, pp. 235–275.

    Article  MathSciNet  MATH  Google Scholar 

  24. Geselowitz D.B. (1967). On bioelectric potentials in an inhomogeneous volume conductor. Biophysical Journal, vol. 7, pp. 1–11.

    Article  Google Scholar 

  25. 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.

    Article  Google Scholar 

  26. Giapalaki S. (2006). A study of models in Medical Physics through the analysis of mathematical problems in Neurophysiology. Doctoral Dissertation, Department of Chemical Engineering, University of Patras.

    Google Scholar 

  27. Giapalaki S., Kariotou F. (2006). The complete ellipsoidal shell model in EEG Imaging. Abstract and Applied Analysis, vol. 2006, pp. 1–18.

    Article  MathSciNet  Google Scholar 

  28. Gutierrez D., Nehorai A., Preissl H. (2005). Ellipsoidal head model for fetal magnetoencephalography: forward and inverse solutions. Physics in Medicine and Biology, vol. 50, pp. 2141–2157.

    Article  Google Scholar 

  29. Hämäläinen M., Hari R., Ilmoniemi RJ., 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.

    Google Scholar 

  30. Hobson E.W. (1931). The theory of spherical and ellipsoidal harmonics. Cambridge University Press. Cambridge.

    Google Scholar 

  31. Jerbi K., Mosher J.C., Baillet S., Leahy R.M. (2002). On MEG forward modeling using multipolar expansions. Physics in Medicine and Biology, vol. 47, pp. 523–555.

    Article  Google Scholar 

  32. Kamvyssas G., Kariotou F. (2005). Confocal ellipsoidal boundaries in EEG modeling. Bulletin of the Greek Mathematical Society, vol. 50, pp. 119–133.

    MathSciNet  Google Scholar 

  33. Kariotou F. (2001). On the electroencephalography (EEG) problem for the ellipsoidal brain model. Proceedings of the 6th International conference of the HSTAM, Thessaloniki, pp. 222–226.

    Google Scholar 

  34. Kariotou F. (2002). Mathematical problems of the electromagnetic activity in the neurophysiology of the brain. Doctoral Dissertation, Department of Chemical Engineering, University of Patras.

    Google Scholar 

  35. Kariotou F. (2003). On the mathematics of EEG and MEG in spheroidal geometry. Bulletin of the Greek Mathematical Society, vol 47, pp. 117–135.

    MathSciNet  Google Scholar 

  36. Kariotou F. (2004). Electroencephalography in ellipsoidal geometry. Journal of Mathematical Analysis and Applications, vol. 290, pp. 324–342.

    Article  MathSciNet  MATH  Google Scholar 

  37. MacMillan W.D. (1930).The theory of the potential. MacGraw-Hill, London.

    Google Scholar 

  38. Malmivuo J., Plonsey R. (1995). Bioelectromagnetism. Oxford University Press. New York.

    Google Scholar 

  39. Nolte G. (2002). Brain current multipole expansion in realistic volume conductors: calculation, interpretation and results. Recent Research Developments in Biomedical Engineering vol.1, pp. 171–211.

    Google Scholar 

  40. Nolte G. (2003). The magnetic lead field theorem in the quasi-static approximation and its use for MEG forward calculation in realistic volume conductors. Physics in Medice and Biology, vol. 48, pp. 3637–3652.

    Article  Google Scholar 

  41. Nolte G., Curio G. (1999). Perturbative analytical solutions of the forward problem for realistic volume conductors. Journal of Applied Physics, vol. 86, pp. 2800–2811.

    Article  Google Scholar 

  42. Nolte G., Dassios G. (2004). Semi-analytic forward calculation for the EEG in multi-shell realistic volume conductors based on the lead field theorem. Poster presentation in BIOMAG 2004, Boston.

    Google Scholar 

  43. Nolte G., Dassios G. (2005). Analytic expansion of the EEG lead field for realistic volume conductors. Physics in Medicine and Biology, vol. 50, pp. 3807–3823.

    Article  Google Scholar 

  44. Nolte G., Fieseler T., Curio G. (2001). Perturbative analytical solutions of the magnetic forward problem for realistic volume conductors. Journal of Applied Physics, vol. 89, pp. 2360–2369.

    Article  Google Scholar 

  45. Plonsey R., Heppner D.B. (1967). Considerations of quasi-stationarity in electrophysiological systems. Bulletin of Mathematical Biophysics, vol. 29, pp. 657–664.

    Article  Google Scholar 

  46. Sarvas J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Physics in Medicine and Biology, vol. 32, pp. 11–22.

    Article  Google Scholar 

  47. Snyder W.S., Ford M.R., Warner G.G., Fisher H.L. (1969, 1978). Estimates of absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of a heterogeneous phantom. Journal of Nuclear Medicine, Supplement Number 3 (August 1969),vol. 10, Pamphlet No.5, Revised 1978.

    Google Scholar 

  48. Whittaker E.T., Watson G.N. (1920). A course of modern analysis. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  49. Wilson F.N., Bayley R.H. (1950). The electric field of an eccentric dipole in a homogeneous spherical conducting medium. Circulation, vol. 1, pp. 84–92.

    Article  Google Scholar 

  50. Yeh G.C., Martinek J. (1957). The potential of a general dipole in a homogeneous conducting prolate spheroid. Annals of the New York Academy of Sciences, vol. 65, pp. 1003–1006.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

    George Dassios

Authors
  1. George Dassios
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Dassios .

Editor information

Editors and Affiliations

  1. CNRS UMR 7641, Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau CX, 91128, France

    Habib Ammari

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dassios, G. (2009). Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry. In: Ammari, H. (eds) Mathematical Modeling in Biomedical Imaging I. Lecture Notes in Mathematics(), vol 1983. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03444-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation