Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Jahresbericht der Deutschen Mathematiker-Vereinigung
Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung 1/2019

11-07-2018 | Survey Article

Review on Mathematical Modelling of Electroencephalography (EEG)

Authors: Marion Darbas, Stephanie Lohrengel

Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Issue 1/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The paper reviews mathematical and numerical aspects in EEG modelling and gives researchers new in this field an overview about the state-of-the-art results and techniques on the topic. The classical dipolar source model is presented for modelling the electrical activity of the brain and several discretization methods for solving the forward model are described. Theoretical results from the mathematical analysis of the forward and inverse problem are given and an overview of the most popular numerical methods for solving the inverse problem is presented. A specific case study for EEG modelling in neonates highlights current questions that are actually asked by the clinicians.
previous article Preface
next article Mechthild Koreuber: „Emmy Noether, die Noether-Schule und die moderne Algebra“

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Jahresbericht der Deutschen Mathematiker-Vereinigung

Der „Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)“ versteht sich als ein Schaufenster für Mathematik. In Übersichtsartikeln und Berichten aus der Forschung soll für möglichst viele LeserInnen verständlich und interessant über aktuelle und wichtige Entwicklungen der Mathematik berichtet werden.

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Show more products
Literature
1.
Adde, G.: Méthodes de traitement d’images appliquées au problème inverse en Magnéto-Electro-Encéphalographie. Thèse de doctorat (2005)
2.
Adde, G., Clerc, M., Faugeras, O., Keriven, R., Kybic, J., Papadopoulo, T.: Symmetric BEM formulation for the M/EEG forward problem. In: Information Processing in Medical Imaging, pp. 524–535. Springer, Berlin (2003) CrossRef
3.
Akalin Acar, Z., Makeig, S.: Effects of forward model errors on EEG source localization. Brain Topogr. 26, 378–396 (2013) CrossRef
4.
5.
Azizollahi, H., Aarabi, A., Wallois, F.: Effects of uncertainty in head tissue conductivity and complexity on EEG forward modelling in neonates. Hum. Brain Mapp. 37(10), 3604–3622 (2016) CrossRef
6.
Baillet, S.: Toward Functional Brain Imaging of Cortical Electrophysiology Markovian Models for Magneto and Electroencephalogram Source Estimation and Experimental Assessments. Ph.D. thesis, University of Paris Orsay, France (1998)
7.
Baillet, S., Mosher, J.C., Leahy, R.M.: Electromagnetic brain mapping. IEEE Signal Process. Mag. 18(6), 14–30 (2001) CrossRef
8.
Baillet, S., Tadel, F., Leahy, R.M., Mosher, J.C., Delorme, A., Makeig, S., Oostenveld, R., Hämäläinen, M., Dalal, S.S., Zumer, J., Clerc, M., Wolters, C.H., Kiebel, S., Jensen, O.: Academic Software Toolboxes for the Analysis of MEG Data. In: IFMBE Proceedings BIOMAG2010, Dubrovnik, Croatia, vol. 28 (2010)
9.
Baratchart, L., Leblond, J., Mandrea, F., Saff, E.B.: How can meromorphic approximation help to solve some 2D inverse problems for the Laplacian? Inverse Probl. 15, 79–90 (1999) MathSciNetCrossRefMATH
10.
Baratchart, L., Ben Abda, A., Ben Hassen, F., Leblond, J.: Recovery of pointwise sources or small inclusions in 2D domains and rational approximation. Inverse Probl. 21, 51–74 (2005) MathSciNetCrossRefMATH
11.
Baratchart, L., Leblond, J., Marmorat, J.-P.: Inverse source problem in a 3D ball from best meromorphic approximation on 2D slices. Electron. Trans. Numer. Anal. 25, 41–53 (2006) MathSciNetMATH
12.
13.
Berg, P., Scherg, M.: A fast method for forward computation of multiple-shell spherical head models. Electroencephalogr. Clin. Neurophysiol. 90, 58–64 (1994) CrossRef
14.
Chen, H., Halleza, F., Staelens, S.: Influence of skull conductivity perturbations on EEG dipole source analysis. Med. Phys. 37(8), 4475–4484 (2010) CrossRef
15.
Cho, J.-H., Vorweck, J., Wolters, C.H., Knösche, T.R.: Influence of the head model on EEG and MEG source connectivity analyses. NeuroImage 10, 60–77 (2015) CrossRef
16.
Christiansen, S., Nédélec, J.C.: A preconditioner for the electric field integral equation based on Calderon formulas. SIAM J. Numer. Anal. 40, 1100–1135 (2002) MathSciNetCrossRefMATH
17.
Clerc, M., Kybic, J.: Cortical mapping by Laplace?Cauchy transmission using a boundary element method. Inverse Probl. 23, 2589–2601 (2007) MathSciNetCrossRefMATH
18.
Clerc, M., Leblond, J., Marmorat, J.-P., Papadopoulo, T.: Source localization in EEG using rational approximation on plane sections. Inverse Probl. 28, 055018 (2012) CrossRefMATH
19.
Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics. Wiley, New York (1983) MATH
20.
Dannhauer, M., Lanfer, B., Wolters, C.H., Knösche, T.R.: Modelling of the human skull in EEG source analysis. Hum. Brain Mapp. 32(9), 1383–1399 (2011) CrossRef
21.
Darbas, M., Diallo, M.M., El Badia, A., Lohrengel, S.: An inverse dipole source problem in inhomogeneous media: application to the EEG source localization in neonates. Submitted
22.
Diallo, M.M.: Problème inverse de sources en Electro-Encéphalo-Graphie chez le nouveau-né. Ph.D. thesis, Université de Picardie Jules Verne, Amiens, France (2017)
23.
El Badia, A., El Hajj, A.: Hölder stability estimates for some inverse pointwise source problems. C. R. Math. Acad. Sci. (Paris) 350(23–24), 1031–1035 (2012) MathSciNetCrossRefMATH
24.
El Badia, A., Farah, M.: Identification of dipole sources in an elliptic equation from boundary measurements: application to the inverse EEG problem. J. Inverse Ill-Posed Probl. 14, 331–353 (2006) MathSciNetCrossRefMATH
25.
El Badia, A., Farah, M.: A stable recovering of dipole sources from partial boundary measurements. Inverse Probl. 26, 115006 (2010) MathSciNetCrossRefMATH
26.
27.
Engwer, C., Vorwerk, J., Ludewig, J., Wolters, C.H.: A discontinuous Galerkin method for the EEG forward problem. SIAM J. Sci. Comput. 39(1), B138–B164 (2017) CrossRefMATH
28.
Farah, M.: Problèmes inverses de sources et lien avec l’Electro-encéphalo-graphie. Ph.D. thesis, Université de Technologie de Compiègne, France (2007)
29.
Gargiulo, P., Belfiore, P., Friogeirsson, E.A., Vanhalato, S., Ramon, C.: The effect of fontanel on scalp EEG potentials in the neonate. Clin. Neurophysiol. 126, 1703–1710 (2015) CrossRef
30.
Geddes, L.A., Baker, L.E.: The specific resistance of biological material - a compendium of data for the biomedical engineer and physiologist. Med. Bio. Eng 5(3), 271–293 (1967) CrossRef
31.
Geselowitz, D.B.: On bioelectric potentials in an homogeneous volume conductor. Biophys. J. 7, 1–11 (1967) CrossRef
32.
Gramfort, A., Papadopoulo, T., Olivi, E., Clerc, M.: OpenMEEG: opensource software for quasistatic bioelectromagnetics. Biomed. Eng. Online 9, 45 (2010) CrossRef
33.
Grech, R., Cassar, T., Muscat, J., Camilleri, K.P., Fabri, S.G., Zervakis, M., Xanthopoulos, P., Sakkalis, V., Vanrumste, B.: Review on solving the inverse problem in EEG source analysis. J. Neuro. Rehabil. 5(25), 1–33 (2008). https://​doi.​org/​10.​1186/​1743-0003-5-25
34.
Hallez, H., Vanrumste, B., Van Hese, P., D’Asseler, Y., Lemahieu, I., Van de Walle, R.: A finite difference method with reciprocity used to incorporate anisotropy in electroencephalogram dipole source localization. Phys. Med. Biol. 50, 3787–3806 (2005) CrossRef
35.
Hallez, H., Vanrumste, B., Grech, R., Muscat, J., De Clercq, W., Vergult, A., D’Asseler, Y., Camilleri, K.P., Fabri, S.G., Van Huffel, S.: Review on solving the forward problem in EEG source analysis. J. NeuroEng. Rehabil. 4(1), 46 (2007) CrossRef
36.
Hämäläinen, M.S., Hari, R., Ilmoniemi, R.J., Knuutila, J., Lounasamaa, O.V.: Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65(2), 413–497 (1993) CrossRef
37.
Hämäläinen, M.S., Ilmoniemi, R.J.: Interpreting Measured Magnetic of the Brain: Estimates of Current Distributions. Technical Report TKK-F-A559, Helsinki University of Technology, Espoo (1984)
38.
Hämäläinen, M.S., Sarvas, J.: Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data. IEEE Trans. Biomed. Eng. 36(2), 165–171 (1989) CrossRef
39.
Haueisen, J., Tuch, D., Ramon, C., Schimpf, P., Wedeen, V., George, J., Belliveau, J.: The influence of brain tissue anisotropy on human EEG and MEG. NeuroImage 15, 159–166 (2002) CrossRef
40.
Hecht, F., Pironneau, O., Le Hyaric, A., Ohtsuka, K.: FreeFem++ Manual (2014)
41.
Hédou, V.: A finite difference method to solve the forward problem in electroencephalography (EEG). J. Comput. Appl. Math. 167, 35–58 (2004) MathSciNetCrossRefMATH
42.
Kellogg, O.D.: Foundations of Potential Theory. Dover, New York (1953) MATH
43.
Kohn, R.V., Vogelius, M.: Relaxation of a variational method of impedance computed tomography. Commun. Pure Appl. Math. 40, 745–777 (1987) MathSciNetCrossRefMATH
44.
Kozlov, V.A., Maz’ya, V.G., Fomin, A.V.: An iterative method for solving the Cauchy problem for elliptic equations. Comput. Math. Phys. 31, 45–52 (1991) MathSciNetMATH
45.
Kress, R.: Linear Integral Equations. Applied Mathematical Sciences, vol. 82. Springer, Berlin (1999) MATH
46.
Kybic, J., Clerc, M., Abboud, T., Faugeras, O., Keriven, R., Papadopoulo, T.: A common formalism for the integral formulations of the forward EEG problem. IEEE Trans. Med. Imaging 24(1), 12–28 (2005) CrossRef
47.
Kybic, J., Clerc, M., Faugeras, O., Keriven, R., Papadopoulo, T.: Fast multipole acceleration of the MEG/EEG boundary element method. Phys. Med. Biol. 50, 4695–4710 (2005) CrossRef
48.
Kybic, J., Clerc, M., Faugeras, O., Keriven, R., Papadopoulo, T.: Generalized head models for MEG/EEG: boundary element method beyond nested volumes. Phys. Med. Biol. 51, 1333–1346 (2006) CrossRef
49.
Lanfer, B., Scherg, M., Dannhauer, M., Knösche, T.R., Burger, M., Wolters, C.H.: Influences of skull segmentation inaccuracies on EEG source analysis. NeuroImage 62(1), 418–431 (2014) CrossRef
50.
Lew, S., Wolters, C.H., Dierkes, T., Röer, C., MacLeod, R.S.: Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis. Appl. Numer. Math. 59(8), 1970–1988 (2009) MathSciNetCrossRefMATH
51.
Mordecai, A.: Nonlinear Programming: Analysis and Methods. Dover, New York (2003) MATH
52.
Lew, S., Silva, D.D., Choe, M., Ellen Grant, P., Okada, Y., Wolters, C.H., Hämäläinen, M.S.: Effects of sutures and fontanels on MEG and EEG source analysis in a realistic infant head model. NeuroImage 76, 282–293 (2013) CrossRef
53.
Love, A.E.H.: A Treatise of Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927) MATH
54.
Meijs, J.W.H., Weier, O.W., Peters, M.J., van Oosterom, A.: On the numerical accuracy of the boundary element method. IEEE Trans. Biomed. Eng. 36(10), 1038–1049 (1989) CrossRef
55.
Montes-Restrepo, V., van Mierlo, P., Strobbe, G., Staelens, S., Vandenberghe, S., Hallez, H.: Influence of skull modeling approaches on EEG source localization. Brain Topogr. 27(1), 95–111 (2014) CrossRef
56.
Mosher, J.C., Lewis, P.S., Leahy, R.M.: Multiple dipole modeling and localization from spatio-temporal MEG data. IEEE Trans. Biomed. Eng. 39(6), 541–557 (1992) CrossRef
57.
de Munck, J.C., Peters, M.J.: A fast method to compute the potential in the multisphere model. IEEE Trans. Biomed. Eng. 40, 1166–1174 (1993) CrossRef
58.
Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Applied Mathematical Sciences, vol. 144. Springer, New York (2001) MATH
59.
Odabaee, M., Tokariev, A., Layeghy, S., Mesbah, M., Colditz, P.B., Ramon, C., Vanhatalo, S.: Neonatal EEG at scalp is focal and implies high skull conductivity in realistic neonatal head models. NeuroImage 96, 73–80 (2014) CrossRef
60.
Ortiz G, J.E., Pillain, A., Rahmouni, L., Andriulli, F.P.: A Calderon regularized symmetric formulation for the electroencephalography forward problem. J. Comput. Phys., 1–21 (2018)
61.
Papageorgakis, C.: Modèles de conductivité patient-spécifiques: caractérisation de l’os du crâne. Ph.D. thesis, INRIA Sophia, Université Côte d’Azur (2017)
62.
Pascual-Marqui, R.D.: Review of methods for solving the EEG inverse problem. Int. J. Bioelectromagn. 1, 75–86 (1999)
63.
Pascual-Marqui, R.D.: Standardized low resolution brain electromagnetic tomography (sLORETA): technical details. Methods Find. Exp. Clin. Pharmacol. 24D, 5–12 (2002). (Author’s version)
64.
Pursiainen, S., Lew, S., Wolters, C.H.: Forward and inverse effects of the complete electrode model in neonatal EEG. J. Neurophysiol. 117(3), 876–884 (2017) CrossRef
65.
Roche-Labarbe, N., Aarabi, A., Kongolo, G., Gondry-Jouet, C., Dümpelmann, M., Grebe, R., Wallois, F.: High-resolution electroencephalography and source localization in neonates. Hum. Brain Mapp. 29, 167–176 (2008) CrossRef
66.
Rush, S., Driscoll, D.A.: Current distribution in the brain from surface electrodes. Anesth. Analg. 47, 717–723 (1967)
67.
Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS-Kent, Boston (1996) MATH
68.
de Saint-Venant, A.J.C.B.: Mémoire sur la torsion des prismes. Mem. Divers Savants 14, 233–560 (1855)
69.
Salo, M.: Unique continuation for elliptic equations Notes Fall (2014)
70.
71.
Steinbach, O., Wendland, W.L.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998) MathSciNetCrossRefMATH
72.
Vorwerk, J.: Comparison of Numerical Approaches to the EEG Forward Problem. Master’s thesis, University of Münster, Germany (2011)
73.
Vorwerk, J.: New Finite Element Methods to Solve the EEG/MEG Forward Problem. Ph.D. thesis, University of Münster, Münster, Germany (2016)
74.
Wolters, C.H., Köstler, H., Möller, C., Härdtlein, J., Grasedyck, L., Hackbusch, W.: Numerical mathematics of the subtraction method for the modelling of a current dipole in EEG source reconstruction using finite element head models. SIAM J. Sci. Comput. 30, 24–45 (2007) MathSciNetCrossRefMATH
75.
Wolters, C.H., Anwander, A., Tricoche, X., Weinstein, D., Koch, M.A., MacLeod, R.S.: Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: a simulation and visualization study using high-resolution finite element modelling. NeuroImage 30(3), 818–826 (2006) CrossRef
76.
Xu, X., Xu, B., He, B.: An alternative subspace approach to EEG dipole source localization. Phys. Med. Biol. 49, 327–343 (2004) CrossRef
77.
Yitembe, B.R.: Reduction of Conductivity Uncertainty Propagations in the Inverse Problem of EEG Source Analysis. Ph.D. thesis, University of Gent (2011)
78.
Zhang, S.: A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres. Phys. Med. Biol. 40, 335–349 (1995) CrossRef
Metadata
Title
Review on Mathematical Modelling of Electroencephalography (EEG)
Authors
Marion Darbas
Stephanie Lohrengel
Publication date
11-07-2018
Publisher
Springer Berlin Heidelberg
Published in
Jahresbericht der Deutschen Mathematiker-Vereinigung / Issue 1/2019
Print ISSN: 0012-0456
Electronic ISSN: 1869-7135
DOI
https://doi.org/10.1365/s13291-018-0183-z

Other articles of this Issue 1/2019

Jahresbericht der Deutschen Mathematiker-Vereinigung 1/2019 Go to the issue

Book Review

Mechthild Koreuber: „Emmy Noether, die Noether-Schule und die moderne Algebra“

Book Review

Dan Burghelea: “New Topological Invariants for Real- and Angle-Valued Maps”

Preface

Preface

Book Review

M. Lapidus, G. Radunović, D. Žubrinić: Fractal Zeta Functions and Fractal Drums

Premium Partner

    Image Credits