Electroencephalography and Magnetoencephalography: An Analytical-Numerical Approach 9783110547535, 9783110545838

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George Dassios, Athanassios S. Fokas Electroencephalography and Magnetoencephalography

De Gruyter Series in Mathematics and Life Sciences

Edited by Anna Marciniak-Czochra, Heidelberg University, Germany Benoît Perthame, Sorbonne-Université, France Jean-Philippe Vert, Mines ParisTech, France

Volume 7

George Dassios, Athanassios S. Fokas

Electroencephalography and Magnetoencephalography

An Analytical–Numerical Approach

Mathematics Subject Classification 2010 Primary: 35R30, 35Q60, 65M32, 78A46; Secondary: 45Q05, 65N21, 78A70, 78M25, 92C55 Authors Prof. Dr. George Dassios University of Patras Dept. of Chemical Engineering Caratheodory St. 1 265 04 Patras Greece [email protected] Prof. Dr. Athanassios S. Fokas University of Cambridge Department of Applied Mathematics and Theoretical Physics Cambridge CB3 0WA UK [email protected]

ISBN 978-3-11-054583-8 e-ISBN (PDF) 978-3-11-054753-5 e-ISBN (EPUB) 978-3-11-054578-4 ISSN 2195-5530 Library of Congress Control Number: 2020934398 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Introduction

1

Chapter 1 The electromagnetics of brain activity 13 1.1 Maxwell’s equations 13 1.2 Quasi-static theory 16 1.3 The brain as a conductor 17 1.4 Electroencephalography (EEG) and Magnetoencephalography (MEG) for a single conductor 19 1.5 The Geselowitz integral representation 22 1.6 Dyadic unification of EEG and MEG 26 1.7 The inhomogeneous shell model 27 Chapter 2 The definitive nonuniqueness results for deterministic EEG and MEG data 2.1 The electric potential on the scalp 31 2.2 The exterior magnetic field 36 2.3 The radial component of the exterior magnetic field 40 2.4 Calculation of the exterior magnetic field 46 2.5 Convenient Boundary Conditions for A and Ψ 50 2.6 Surface geometry 52 Chapter 3 Distributed current in spherical and ellipsoidal geometry 3.1 Statement of the problem 57 3.2 Spherical geometry 58 3.3 Ellipsoidal geometry 61 Chapter 4 The spherical model 75 4.1 The interior electric potential 75 4.2 The exterior electric potential 78 4.3 Closed expressions of the electric potential 79 4.4 Connection with the method of images 83 4.5 The inhomogeneous model 87 4.6 The exterior magnetic field 92

57

31

VI

Contents

Chapter 5 The ellipsoidal model 95 5.1 EEG for the homogeneous model 95 5.2 MEG for the homogeneous model 98 5.3 The complete magnetic field 102 5.4 EEG for the 3-shell model 106 5.5 Direct solution of the EEG problem 112 5.6 MEG for the 3-shell model 117 5.7 Problems 122 Chapter 6 Determination of localized current via EEG or MEG data 6.1 EEG for isolated dipoles in a sphere 126 6.2 MEG for isolated dipoles in a sphere 128 6.3 EEG for isolated dipoles in an ellipsoid 133 6.4 MEG for isolated dipoles in an ellipsoid 134

125

Chapter 7 A minimisation approach to EEG for the spherical and ellipsoidal geometries 139 7.1 Statement of the problem 139 7.2 Inversion of EEG data for the spherical model 141 7.3 Inversion of EEG data for the ellipsoidal model 146 7.4 Reduction to the sphere 151 Chapter 8 Numerical implementation 153 8.1 Ellipsoidal geometry 153 155 8.1.1 Estimating the geometry-dependent coefficients Clm 8.1.2 Numerical results 156 8.1.3 Numerical verification 157 m 160 8.1.4 The computation of Cl 160 8.1.5 Cross-validation of Clm 8.1.6 EEG inversion matrix 160 8.1.7 EEG reconstructions 164 8.1.8 MEG inversion matrix 165 8.1.9 Conclusion 169 8.2 Arbitrary geometry 171 8.2.1 Regression model of us ðr, τ, ^τ Þ for a realistic head model for EEG 8.2.2 Minimization and a numerical solution of the inverse problem 8.2.3 Expansion of ΨðτÞ using radial basis functions 175

172 174

Contents

8.2.4 8.2.5 8.2.6 8.2.7 8.2.8

Reconstructing ΨðτÞ 177 Numerical results 178 Surrogate model 179 Reconstruction of ΨðτÞ 181 Conclusion 183

Chapter 9 Current confined on the cortical surface References

185

191

For further study Electroencefalography and Magnetoencephalography we refer to… 196 Index

203

VII

Introduction The fundamental cells of the nervous tissue are called neurons.1 A specific mental process is associated with brain activation of a unique form, which in turn expresses itself via the generation of a specific neuronal electric current. Michael Faraday was the first to establish a relation between electricity and magnetism. The precise mathematical form of this relation is expressed by Maxwell’s equations of electromagnetism. According to these equations, an electric current gives rise to both a magnetic field and an electric potential. Once a cortical neuronal current is activated, a magnetic field and an electric potential are generated. Magnetoencephalography (MEG) and Electroencephalography (EEG) are based on measurements of the magnetic field outside the head and of the electric potential on the scalp, respectively. The neuronal current is mainly due to the summation of tiny currents (called ionic currents) flowing across certain pores (called sodium and potassium ionic gates), which are located at the membrane of parts of neurons. The particular type of neurons that are mostly responsible for the current affecting EEG and MEG measurements is called pyramidal cells. In most clinical applications, EEG uses 19 electrodes to measure the electric potential on the scalp (for research purposes there exist high-density, 256-channel EEG, sensory arrays). The measured potential is in the microvolt range. Scalp voltage recordings were first measured in humans in 1924 by Hans Berger (1873–1941), who invented, what he called, the electroencephalogram; Berger extended earlier work by Richard Caton (1842–1926). EEG is an inexpensive, minimally invasive technique, which can provide real time information about the brain’s function. This important modality is widely used in both normal and clinical conditions. For example, it is an indispensable tool in sleep studies as well as for the diagnosis of epilepsy. The latter neurological disorder includes focal seizures and generalized seizures which are further subdivided into tonic-clonic seizures and absence seizures. Tonic-clonic seizures (previously called grand mal) are the seizures most people associate with epilepsy; absence seizures (previously called petit mal) are more common in children than in adults.2 The electroencephalograms obtained during different types of seizure exhibit characteristic patterns, which can aid the diagnosis of epilepsy. For example, in generalized seizures there is a pattern of unison

1 The nervous tissue also contains the so-called glia cells. These cells are also crucial for the function of the central nervous system; however, they do not propagate electric impulses, thus are not directly relevant to EEG and MEG. 2 In absence seizures a child may suddenly stop talking or cease walking; the child stares emptily and does not respond to stimuli. After one to several seconds following these short “absences”, the child regains consciousness and has no memory of the episode. https://doi.org/10.1515/9783110547535-001

2

Introduction

hyperactivity, which can be contrasted with the highly complex pattern of a normal electroencephalogram.3 In general, EEG recordings depend on the location of the electrodes and on the specific mental function performed by the subject during the recordings. In normal subjects there exist several characteristic rhythms that are distinguished by their frequencies: delta rhythm involves high amplitude, slow waves, namely waves of frequency less that 4 Hz (i.e., four cycles per second); this is the characteristic rhythm occurring during deep sleep (these waves are best obtained from recordings on the frontal part of the head). Waves of the theta rhythm have frequency between 4 and 7 Hz; they occur when subjects try to repress a specific response or action (these waves are found in regions not related to the task at hand). The frequency of alpha waves is between 8 and 15 Hz; such waves occur during relaxation. The beta waves have frequency in the range of 16 to 32 Hz; they are associated with active thinking. Finally, the gamma rhythm involves waves of high frequency, namely waves of frequency higher than 32 Hz, typically occur during the processing of at least two different types of sensory information. The magnetic field generated by neuronal currents is extremely weak; it is of the order of femtotesla, that is, approximately 109 times smaller than the magnetic field of the earth field at its surface. Thus, MEG requires the use of an extremely sensitive magnetometer. In the early 1970s, David Cohen built a magnetically shielded room at MIT and measured MEG signals at a single point outside the head using the exquisitely sensitive superconducting quantum interference device (SQUID). This device had just been developed at Ford Motor Company by James Zimmerman [88]. Remarkably, MEG signals are almost as clear as those obtained via EEG. Present-day technology allows the measurement of the magnetic flux at about 300 points in a helmet-shaped apparatus. In a very recent technological breakthrough, a new type of magnetometer has been developed that takes the form of a portable helmet.4 In this new device, the sensors are on the scalp instead of being a few millimeters away from the head as they were in the SQUID, and more importantly, the subject can move. As noted earlier, recordings obtained from EEG can be used to aid the diagnosis of several neurological conditions. Similarly, MEG also has several clinical applications. Later, it was realized that MEG could have an additional, more significant use: it could be employed in order to determine the electric current that gave rise to

3 This suggests that the loss of consciousness in a generalized seizure is the result of a dramatic reduction of complexity of the brain’s neural state; this indicates the inability of the brain to integrate the diverse repertoire of neural substates that are normally available. 4 This device is called spin exchange relaxation-free magnetometer and does not require a bulky cooling system for its operation, thus it is small and portable. Also, the proximity of the sensors to the brain improves the so-called signal to noise ratio.

Introduction

3

the MEG recordings. Assuming that the brain can be modelled as an appropriate conductor, the above involves the solution of the following mathematical inverse problem: given the magnetic field outside a conductor, determine the electric current generating this field. Similarly, the analogous problem for EEG involves the determination of the electric current from the knowledge of the electric potential measured on the outer surface of a conductor. The main advantage of both MEG and EEG in comparison to other functional imaging techniques is their very high temporal resolution, which is better than 1 millisecond; that is, EEG and MEG essentially provide real-time information about mental processes. This is to be contrasted with the time resolution of other functional imaging techniques, like fMRI, PET and SPECT, which due to slow hemodynamic response, is of the order of 2 s. However, the Achilles heel of MEG and EEG is their spatial resolution. Actually, as it will be discussed in detail in this volume, it is impossible to obtain the continuous distribution of the cortical current from MEG or EEG data without additional physiological or mathematical assumptions. However, there exist particular physiological situations where the current can be computed uniquely from either EEG or MEG data. Such a situation, for example, arises when the current is so well localized that it can be adequately approximated by the assumption that the current is supported only at a single point, or a sector of a curve, or a patch of a surface. In this idealized case, the current can be represented in terms of Dirac’s delta functions supported on zero, one or two-dimensional subsets [1, 29, 30–34, 37, 38]. The associated idealized point-current is called a dipole. Given a set of MEG data, there exists a commercial software that can determine a suitable dipole from the data [73]. The dipole approximation is very useful for the study of focal epilepsy using MEG: in patients with drug-resistant epilepsy,5 a viable treatment option is brain surgery, namely, the removal of the part of the brain that generates seizures. In order to identify this epileptogenic focus, the usual preoperative evaluation is the combination of long-term video EEG and MRI. For patients with no concordance of the EEG results and a lesion on MRI, MEG (under the assumption of a single dipole) is used to identify the focus.6 Due to a recent technological development, it is possible to combine MEG with a newer MRI (called ultra-low field MRI), and this improves significantly the localization capability of MEG [85]. In addition to its important role in the investigation and treatment of epilepsy, MEG also has numerous applications in the study of physiology, psychiatry and cognition: from the investigation of the somatosensory system and the relation of brain-rhythms with cognitive processes, to the study of Alzheimer’s disease and

5 Drug-resistant epilepsy is defined as the failure of adequate control of epilepsy after trials of two appropriately chosen drug schedules with acceptable side effects [70]. 6 In a recent study, 77% of 57 patients who underwent surgery (and continued to receive pharmacological treatment) became seizure-free after one year. In comparison, 93% of those receiving only pharmacological treatment continued to have seizures [44].

4

Introduction

the identification of neural abnormalities in various psychiatric and neurological diseases [51]. Taking into consideration that the conductivities of the cortex, cerebrospinal fluid and scalp are similar, it turns out that the formulation of MEG gives rise to an approximate formalism, which is much simpler than that of EEG. Namely, as it will be explained in this volume, this formalism does not involve the determination of certain auxiliary complicated functions needed for the formulation of EEG; these functions depend on the geometry of the four compartments modelling the brain-head system (cerebrum, cerebrospinal fluid, skull and scalp) as well as the electrical conductivities of these compartments. However, if the current is approximated by a single dipole, the EEG inverse problem can be solved uniquely [29, 30]. Thus, in addition to its wellestablished use for the diagnosis of epilepsy, EEG can also be used (via the one dipole approximation) for the determination of the precise location of an epileptic focus.7 The main purpose of this volume is to elucidate mathematically the nature of the inverse problems arising in EEG and MEG. In addition, efficient numerical algorithms are presented for computing the cortical neuronal current in terms of EEG or MEG data. The emphasis is on the realistic situation of a continuous distribution of current; only one chapter of the volume is dedicated to the analysis of dipoles (Chapter 6). This volume could be of interest to both practitioners of EEG-MEG and affectionatos of exact results. With respect to the former, the most important Chapters are 2, 7, 8 and 9. Chapters 2, 7, and 9, which are based mainly on the work of the second author, present: (i) The definitive answer to the non uniqueness question; namely, in chapter 2, the part of the neuronal current that affects the EEG and MEG measurements is identified. (ii) The solution of the minimisation problem for EEG for the particular cases of the spherical and ellipsoidal geometries; namely, it is shown in chapter 7 that in the case of EEG the assumption of minimising the L2 norm of the current, yields a unique solution. (iii) The definitive solution of the inverse problems for both MEG and EEG in the case that the current is confined to the cerebral surface and is normal to it. Chapter 8, which are based on the joint work of the second author and Parham Hashemzadeh, presents: (i) Numerical algorithms for constructing the components of the current affecting the EEG and MEG measurements in the case of ellipsoidal geometry. (ii) For EEG, an effective numerical algorithm for computing the current in arbitrary geometry by imposing the L2 minimisation. This result is the most complete result presented in this book; the analogous result for MEG will be presented in the near future. Regarding exact results, based mainly on works of the first author and collaborators, the relevant chapters are 3–6; in these chapters’ exact results are presented for the particular cases of spherical and ellipsoidal geometries. More details about the content of the book are presented below.

7 Regarding the determination of epileptic foci via the dipole approximation of the inverse problems of MEG and EEG, comparisons are presented in [75].

Results for an arbitrary geometry

5

Results for an arbitrary geometry This book is based on the standard four compartment brain-head system model: the cerebrum is modelled by a domain denoted by Ωc which is bounded by the cortical surface denoted by Sc ; it is surrounded by cerebrospinal fluid modelled by the domain Ωf which is bounded by the external surface Sf ; the skull (bone) is modeled by the domain Ωb which is bounded by the external surface Sb ; finally, all the above domains are enclosed by the scalp, modeled by the domain Ωs , which is bounded by the external surface Ss . The above domains are distinguished by different electrical conductivities denoted, respectively, by σc, σf, σb, σs. The constant μ0 denotes the permeability of the aforementioned four compartments, as well as of the domain exterior to the head, which is denoted by Ωe . In Chapter 1, the physical and mathematical foundations of EEG and MEG are discussed. In particular, the validity of the quasistatic approximation for electromagnetoencephalography is demonstrated, and the classical integral formulas of Biot –Savart and Geselowitz are derived. In Chapter 2, it is shown that the basic equation relating the electric potential measured on Ss with the primary cortical neuronal current denoted by Jp is the equation ð ð 1 1 p ^ ðτ Þ · Jp ðτÞÞ dSðτÞ. (i) υs ðr, τÞð∇τ · J ðτÞÞ dV ðτÞ + υs ðr, τ Þðn us ðrÞ = − 4π 4π Ωc

Sc

^ denotes the outward unit normal to Ss and υs is an auxiliary funcIn this equation, n tion (a monopolic potential) that depends on the geometry and conductivities, but it is independent of the primary current. This function, as well as the analogous functions υc , υf , υb can be obtained via the solution of a transmission-boundary value problem defined as follows: for the potential υc in Ωc , Δr υc ðr, τÞ = 0, r 2 Ωc  1 1 + υc ðr, τ Þ = υf ðr, τÞ, r 2 Sc σ c jr − τ j      ∂ 1 ∂ υf ðr, τÞ, r 2 Sc ; + υc ðr, τÞ = σf ∂nðrÞ jr − τj ∂nðrÞ 

for the potential υf in Ωf , Δr υf ðr, τÞ = 0, υf ðr, τÞ = υb ðr, τÞ,

r 2 Ωf r 2 Sf

6

Introduction

 σf

   ∂ ∂ υf ðr, τÞ = σb υb ðr, τÞ, ∂nðrÞ ∂nðrÞ

r 2 Sf ;

for the potential υb in Ωb , Δr υb ðr, τÞ = 0,

r 2 Ωb

υb ðr, τÞ = υs ðr, τÞ, r 2 Sb    ∂ ∂ υb ðr, τÞ = σs υs ðr, τÞ, ∂nðrÞ ∂nðrÞ

 σb

r 2 Sb ;

and finally, for the potential υs in Ωs Δr υs ðr, τÞ = 0,

r 2 Ωs

υs ðr, τ Þ = υe ðr, τÞ, r 2 Ss  ∂ υs ðr, τÞ = 0, r 2 Ss : ∂nðrÞ



Further insight from eq. (i) can be gained by representing the primary current in terms of the Helmholtz decomposition: Jp ðτ Þ = ∇τ ΨðτÞ + ∇τ × AðτÞ,

∇τ · AðτÞ = 0:

Then, eq. (i) becomes 1 us ðrÞ = − 4π

ð Ωc

1 υs ðr, τÞΔτ Ψðτ Þ dV ðτ Þ + 4π

ð

^ ðτ Þ · Jp ðτÞÞ dSðτÞ, υs ðr, τ Þðn

r 2 Ss :

Sc

(ii) In order to decouple the EEG and MEG problems we assume that A satisfies the boundary condition: ^ ðτ Þ · ∇τ × AðτÞ = 0, n

τ 2 Sc :

Then, eq. (ii) becomes ð ð 1 1 ^ ðτÞ · ∇τ ΨðτÞÞ dSðτ Þ, υs ðr, τÞΔτ ΨðτÞ dV ðτÞ υs ðr, τÞðn us ðrÞ = − 4π 4π Ωc

(iii)

r 2 Ss :

Sc

(iv) This equation implies that only the nonharmonic part of the Ψ component of the current affects the EEG data. Actually, EEG data provides information only about the value of Ψ and of the divergence of Ψ on the boundary of Ωc . Indeed, it turns out that the eq. (iv) can be rewritten in the form

7

Results for an arbitrary geometry

us ðrÞ =

1 4π

ð

^ ðτÞ · ½Ψðτ Þ∇τ υs ðr, τÞdSðτÞ, n

r 2 Ss :

(v)

Sc

This equation captures the definitive nonuniqueness result for EEG. It is also shown in Chapter 2 that the analogue of eq. (ii) for the radial component of the magnetic flux in Ωe , is the equation 4π 1 r · BðrÞ = μ0 4π

ð

ð ðΔτ ΨðτÞÞðr · Hðr, τÞÞdV ðτÞ − Ωc

Ωc

 ð 1 p ^ ðτÞ · Jp ðτÞÞðr · Hðr, τÞÞdSðτ Þ ^ ∇τ · jr − τjðnðτÞ × J ðτÞÞ dSðτÞ − ðn 4π

ð +



1 Δτ ðτ · AðτÞÞdV ðτÞ jr − τ j

Sc

ð −

Sc

Sc

1 ^ ðτÞ × Jp ðτÞÞ dSðτÞ, τ · ðn jr − τ j

r 2 Ωe ,

(vi)

where H is defined in terms of the monopolic potentials υl , l = c, f , b, s by the equation ð Hðr, τÞ = ðσc − σf Þ Sc



+ σf − σb



 ^ υf ð ρ, τÞ nð ρÞ × ∇ρ ð Sf

ð

+ ðσ b − σ s Þ ð + σs

 1 dSð ρÞ jr − ρj

 ^ υb ð ρ, τÞ nð ρÞ × ∇ρ

 1 dSð ρÞ jr − ρj

 ^ ð ρÞ × ∇ρ υs ð ρ, τÞ n

 1 dSð ρÞ jr − ρj

Sb

 ^ ð ρÞ × ∇ρ υs ð ρ, τÞn

Ss

 1 dSð ρÞ, j r − ρj

r 2 Ωs :

(vii)

It turns out that for practical purposes the term involving H can be neglected. Indeed, H involves explicitly the differences of the various electric conductivities, which are small. Thus, the following equation provides a good approximation of the MEG formulation: 4π r · BðrÞ ≈ − μ0

ð Ωc

ð −

Sc

1 Δτ ðτ · AðτÞÞdV ðτÞ + jr − τ j

ð

 ^ ðτÞ × Jp ðτ ÞÞ dSðτÞ ∇τ · jr − τ jðn

Sc

1 ^ ðτÞ × Jp ðτÞÞ dSðτÞ, τ · ðn r − j τj

r 2 Ωe :

(viii)

8

Introduction

Thus, because of (iii) we find  ð 4π 1 1 ^ ðτÞ dSðτÞ ·n r · BðrÞ ≈ − ∇τ ðτ · AðτÞÞ − ðτ · AðτÞÞ∇τ μ0 jr − τ j jr − τ j Sc

ð

+

  ^ ðτ Þ × ∇τ ΨðτÞÞ dSðτ ∇τ · jr − τjðn

Sc

ð

− Sc

1 ^ ðτÞ × ∇τ Ψðτ ÞÞ dSðτ Þ, τ · ðn jr − τ j

For simultaneous EEG and MEG measurements Ψ: ð 4π 1 r · BðrÞ ≈ − Δτ ðτ · AðτÞÞdV ðτ Þ, μ0 jr − τ j

r 2 Ωe :

r 2 Ωe

(ix)

(x)

Ωc

and 4π r · BðrÞ ≈ − μ0

ð Sc

 1 1 ^ ðτÞ dSðτÞ, ·n ∇τ ðτ · AðτÞÞ − ðτ · AðτÞÞ∇τ jr − τj jr − τ j

r 2 Ωe : (xi)

Thus, the radial component of the magnetic field is affected only by the radial component of A. The aforementioned equations provide the definitive answer to the nonuniqueness question of MEG. Since the above equations do not involve any of the auxiliary functions, it is clear that the approximate mathematical formulation of MEG is much simpler than that of EEG. The previously mentioned analysis shows that for a continuously distributed cortical current, EEG and MEG provides some information, respectively, about Ψ and the radial part of A. However, as discussed earlier, there are particular circumstances, where a unique current can be determined. In addition to the dipole case discussed before, another physiologically important situation arises as follows: often, EEG and MEG data are dominated by the pyramidal neurons situated perpendicular to the cortical surface. It turns out that under the assumption that the current is localized on the cortical surface and that it is also normal to this surface, this current can be computed uniquely [51]. In this case the direction of the current vector is known, thus the inverse problem involves determining a single unknown scalar function (the magnitude of the current) supported on a surface.8 It is shown in Chapter 9 that in this case the following equations for EEG and MEG are valid:

8 It is shown in Chapter 1 that the magnetic field outside the head satisfies Laplace’s equation; since the solution of this equation can be obtained from the knowledge of the unknown on the boundary of the domain, it follows that MEG data can always be considered as given on a surface.

Results for an arbitrary geometry

ð

1 us ðrÞ = 4π

J3 ðτÞ Sc

∂υs ðr, τ Þ dSðτÞ, ∂λ

r 2 Ss

9

(xii)

and 4π r · BðrÞ = μ0



ð τ· Sc

−

1 4π

 ∂J3 ðτÞ ∂J3 ðτÞ hλ ð τ Þ 1 τ1 + τ2 dSðτÞ ∂ν ∂μ jτ 1 jjτ2 j jr − τj

ð Sc

  ∂Hðr, τÞ dSðτÞ, J3 ðτÞ r · ∂λ

r 2 Ωe ,

(xiii)

where τ1 , τ2 and hλ are defined as follows: τ1 ðμ, νÞ =

∂τða, μ, νÞ , ∂μ

τ2 ðμ, νÞ =

∂τða, μ, νÞ , ∂ν



∂τða, μ, νÞ

: hλ ðμ, νÞ =



∂λ

Given EEG data, it is shown in Chapter 9 that the aforementioned equation (xii) gives rise to a simple numerical algorithm for determining the normal component of the current. Returning to eqs. (iv) and (x) it follows that it is impossible to determine a continuously distributed cortical current. In this connection, it is shown in Chapter 7 that for the particular cases of spherical and ellipsoidal geometries, the additional assumption of minimizing the L2 norm yields unique answers. Since the brain-system can be approximated by confocal ellipsoidal surfaces, the aforementioned results justify imposing the minimization condition for an arbitrary geometry. For the associated implementation, an effective numerical algorithm is presented in Chapter 8, based on the result of [66] and [67], where: (i) before considering the case of arbitrary geometry (Section 8.2), the case of ellipsoidal geometry is first analyzed (Section 8.1); in this case, it is shown that the direct use of ellipsoidal harmonics is not practical. For both the arbitrary and the ellipsoidal geometries the components of current affecting the data is expanded in terms of multiquadric radial basis functions. (ii) Following [65] a novel approach for the fast and accurate numerical determination of the auxiliary function υs is employed; actually υs can be computed either via the numerical evaluation of a line integral or via the new mathematical technique of deep learning (using an appropriate surrogate model).

Hence, both the unknown current and the MEG data are represented as functions supported on a surface, therefore it is not surprising that in this case there exists a unique relation between them.

10

Introduction

Results for spherical and ellipsoidal geometries Spherical geometry corresponds to the case that the surfaces Sc , Sf , Sb , Ss are spheres of radii c, f , b, s. In this case, Ψ and the radial component of A can be expanded in the form Ψðr0 Þ =

∞ X n X n=1 m= −n

m ^ ym n ðr0 Þ Yn ðr0 Þ

and ∞ X n X

r0 Ar0 ðr0 Þ =

m ^ am n ðr0 Þ Yn ðr0 Þ

n=0 m= −n

denote spherical harmonics. In the aforementioned both for 0 ≤ r0 < c, where particular case, it is shown in Chapter 3 that the basic equations of EEG and MEG become: Ynm

us ðrÞ = −

∞ X n X

 m sn m ^ c n + 1 cy_ m n ðcÞ − nyn ðcÞ Yn ðrÞ, 2n + 1 n=1 m= −n

r=s

(xiv)

and ∞ X n X m 1 cn+2 1  m ^ r · BðrÞ = − ca_ n ðcÞ − ðn − 1Þam n ðcÞ Yn ðrÞ, n + 1 2n + 1 r μ0 n=0 m= −n

r > s:

(xv)

The determination of the constant sn requires the explicit computation of υs ; this is achieved in Chapter 4 where it is shown that: sn = −

~ det N σs n21 ð2n + 1Þ3

,

n = 1, 2, ... .

~ is a product of specific complicated matrices defined in formulae The matrix N (4.91)–(4.124) of Chapter 4. By expanding the left hand side of eqs. (xiv) and (xv) in terms of spherical harmonics, we find: us ðs^rÞ =

∞ X n X n=1 m= −n

m ^ um n Yn ðrÞ

and r · BðrÞ = μ0

∞ X n X n=0 m= −n

bm n

1 rn + 1

Ynm ð^rÞ

m where um n and bn are known constants. Using the aforementioned equations in (xiv) and (xv), we find the equations

Results for spherical and ellipsoidal geometries

−

11

 sn c n + 1 cy_ nm ðcÞ − nynm ðcÞ = um n 2n + 1

cn+2  m m ca_ n ðcÞ − ðn − 1Þam n ðcÞ = bn : 2n + 1 m It is clear that in order to obtain the unknown functions ym n and an from the aforementioned constants we need some additional information. It is shown in Chapter 7 that in the case of EEG by imposing the minimization of the L2 norm, we find   ð2n + 1Þ um r0 1 n n ym ð r Þ = − r ln − n 0 + 1 km 0 c1 n c2n n 1

where r0 denotes the radial component of the location where the current is evaluated. The numerical implementation of the aforementioned formulae is presented in [7]. Recently, an interesting new treatment of the spherical case is presented in [50], which is not based on Helmholtz decomposition. In ellipsoidal geometry Ψð ρ, μ, νÞ =

∞ 2n +1 X X n=1 m=1

m ~ym n ð ρÞSn ð μ, νÞ

  where ρ0 , μ0 , ν0 are the ellipsoidal coordinates of the point where the current is evaluated and Sm n ðμ, νÞ are the surface ellipsoidal harmonics. Following a similar, but more elaborate program than that of the spherical case, we find 2 cð1 m ~ 1 1 u m m n 4 ~yn ð ρo Þ = En ð ρo Þ Enm ð ρ′ÞFnm ð ρ′Þdρ′ sm 2n + 1 Nnm Csn ρo 3 ρðo cð1 ′ðc1 Þ Fm m m m m m m n + Fn ð ρo Þ En ð ρ′ÞEn ð ρ′Þdρ′ − En ð ρo Þ m En ðρ′ÞEn ð ρ′Þdρ′ 5: En ′ðc1 Þ h2

h2

In Chapter 6, it is assumed that the primary cortical current can be approximated by a single dipole; then, the position and the moment of this dipole are determined in terms of either 6 EEG or 6 MEG measured data. This is implemented for both spherical and ellipsoidal geometries.

Chapter 1 The electromagnetics of brain activity In this chapter we postulate the underlying mathematical model and explain the physical mechanisms that govern the electromagnetic activity of the human brain. We also define the forward and backward problems of electroencephalography (EEG) and magnetoencephalography (MEG). Furthermore, we propose a unified approach for treating these two brain imaging modalities, as well as a generalization of the basic model that includes shells of different conductivities that surround the brain tissue.

1.1 Maxwell’s equations Maxwell’s equations provide a system of partial differential equations that describe electromagnetic phenomena. They connect the vector fields of electric intensity E, magnetic intensity H, electric displacement D and magnetic flux density B, with the electric charge density ρ and the current density J, which constitute the sources of the 4 fields E, H, D, B. The equations of Maxwell are as follows: ∇×E= − ∇×H=

∂ E ∂t

∂ D+J ∂t

(1:1) (1:2)

∇·D=ρ

(1:3)

∇ · B = 0.

(1:4)

Equation (1.1) states Faraday’s law, (1.2) states Ampere-Maxwell’s law, (1.3) Gauss’ law and finally eq. (1.4) ensures the nonexistence of magnetic monopoles. The fields E and H describe the electric and magnetic fields, respectively, in the absence of any kind of matter, that is, in empty space. On the other hand, the fields D and B describe the corresponding electric and magnetic fields in the presence of matter and therefore they depend on the particular characteristic of the hosting medium. In a homogeneous and isotropic medium, the parameters that describe its electromagnetic behavior are the dielectric constant ε, the magnetic permeability μ, and the conductivity σ. The parameters ε and μ appear as proportionality constants in the constitutive equations,

https://doi.org/10.1515/9783110547535-002

D = εE

(1:5)

B = μH

(1:6)

14

Chapter 1 The electromagnetics of brain activity

which characterize the medium of propagation. In particular, for empty space we take ε0 = 1 and μ0 = 1. Therefore, the parameter ε measures the effect of the medium upon the electric field and μ measures the effect of the medium upon the magnetic field. Regarding conductivity it is noted that within a conductive medium with σ > 0, an electric field E generates a secondary current known as induction current Ji , which is expressed by the equation Ji = σE.

(1:7)

Thus the higher the conductivity of the medium, the higher the induction current. An electrical insulator is a medium for which σ = 0, while a medium for which σ ! + ∞ is called a perfect conductor. The current J that appears on the RHS of the Ampere–Maxwell eq. (1.2) can be written in the form J = Jp + Ji = Jp + σE

(1:8)

where Jp represents the primary current, namely the imposed current (as opposed to the current coming from the property of the medium to conduct existing electric charges). Hence, in a homogeneous and isotropic medium characterized by the parameters ε, μ, σ the Maxwell equations assume the form ∇×E= −μ ∇×H=ε

∂ H ∂t

∂ E + Jp + σE ∂t

(1:9) (1:10)

ε ∇·E=ρ

(1:11)

∇ · H = 0.

(1:12)

Taking the curl of eq. (1.9) and using the vector identify ∇ × ð∇ × f Þ = ∇ð∇ · f Þ − Δf

(1:13)

where Δ denotes Laplace’s operator, we obtain   ∂ ∂ ∂ p ε E + J + σE ∇ × ð∇ × EÞ = − μ ð∇ × HÞ = − μ ∂t ∂t ∂t = − εμ

∂2 ∂ ∂ E − μ Jp − μσ E = ∇ð∇ · EÞ − ΔE 2 ∂t ∂t ∂t

1 = ∇ρ − ΔE. ε

(1:14)

15

1.1 Maxwell’s equations

Thus ΔE − εμ

∂2 ∂ 1 ∂ E − μσ E = ∇ρ + μ Jp . ∂t2 ∂t ε ∂t

(1:15)

In an analogous way we obtain the corresponding equation for the magnetic field ΔH − εμ

∂2 ∂ H − μσ H = − ∇ × Jp . ∂t2 ∂t

(1:16)

Comparing eqs. (1.15) and (1.16) we see that the medium propagates both the electric and the magnetic field in the same way, but the two fields have different source terms. In fact, the expression 1 ∂ ∇ρ + μ Jp ε ∂t generates the electric field, whereas the magnetic field is generated by the term − ∇ × Jp . If the conductivity of the medium vanishes, then eqs. (1.15) and (1.16) reduce to the classical wave equation governing electric and magnetic disturbances, which travel with the phase velocity c = ð μεÞ1=2 :

(1:17)

i

The induction current J does not appear as a source for neither field since its appearance in the medium is a consequence and not a prerequisite for the generation of the electric field. Equations (1.2) and (1.3) imply the equation of continuity ∇·J+

∂ρ =0 ∂t

(1:18)

which provides the mathematical formulation of the conservation of the electric charge. In the absence of sources both fields E and H solve the homogeneous wave equation Δf − εμ

∂2 ∂ f − μσ f = 0. ∂t2 ∂t

(1:19)

This equation implies the characteristic polynomial or the dispersion relation k2 = εμω2 + iωμσ.

(1:20)

The vector f takes the spectral form f ðr, tÞ = αeik · r − iωt

(1:21)

16

Chapter 1 The electromagnetics of brain activity

where α is the polarization vector, k is the propagation vector, k = jkj is the wave number, and ω is the angular frequency. In this case, the phase velocity is given by c=

ω Re k

(1:22)

where k is a root of eq. (1.20). We recall the usual relation between the angular frequency ω and the period T ω=

2π T

(1:23)

as well as the relation between the wave number k and the wavelength λ k=

2π . λ

(1:24)

The angular frequency ω defines the temporal density and the wave number k defines the spacial density of the harmonic wave (1.21). Furthermore, c=

ω 2π λ λ = = k T 2π T

(1:25)

as it is expected.

1.2 Quasi-static theory If the propagation space of the wave field is bounded, then we define the characteristic dimension of the medium to be the radius a of the smallest sphere that contains the propagation medium. In the special case where the wavelength is much larger than the characteristic dimension of the medium, that is if a > > > > Ωf : c < r < f > = Ωb : f < r < b > > > Ωs : b < r < s > > > ; Ωe : s < r < + ∞

(3:6)

We recall the Laplace expansion ∞ ∞ X n X X 1 τn 1 r0 n m* ^ ^ P ð Þ = 4π Y ð ^r0ÞYnm ð ^rÞ = r · r n 0 2n + 1 rn + 1 n jr − r0 j n = 0 rn + 1 n=0 m= −n

(3:7)

which holds for r > r0 ,Pn are the Legendre polynomials and Ynm are the complex form of the normalized surface spherical harmonics. For an isolated dipolar current at r0 with moment, Jðr0 Þ we can solve the transmission problem for the harmonic functions υi , i = c, f , b, s, e and obtain the corresponding electric potentials ui , i = c, f , b, s, e via the expressions (2.26) and (2.27). This program, which will be applied in detail in the next chapter, provides the values of υs on the boundary r = s in the form υs ðr, r0 Þ =

∞ X

sn r0 n Pn ð ^r · ^r0Þ

(3:8)

n=1

with r0 < c, where as we have explained earlier, the n = 0 term in (3.8) will be annihi1 Jðr0 Þ · ∇r0 . The coefficients sn , which depend lated by the action of the operator 4π on the radii c, f , b, s and the conductivities σc , σf , σb , σs , will be calculated explicitly in the next chapter. For the spherical model, the dot product r · Hðr, r0 Þ appearing in eq. (3.5) vanishes, since on the surfaces Si , i = c, f , b, s, we have ^ ðr′Þ × ∇r′ r·n

1 r′ r − r′ =r· × =0 ri jr − r′j3 j r − r′ j

(3:9)

for every ri = c, f , b, s. The main results of this section are stated and proved in the next proposition. Proposition 3.2 Consider the 3-shell spherical model described in (3.6) and the distributed current Jðr0 Þ, 0 ≤ r0 < c. Then, for r = s, we have ð 1 us ðrÞ = Jðr0 Þ · ∇r0 υs ðr, r0 Þdυðr0 Þ (3:10) 4π Ωc

3.2 Spherical geometry

where υs ðr, r0 Þ is given by (3.8). Hence ð X ∞ 1 sn ðΔΨðr0 ÞÞr0 n Pn ð ^r, ^r0Þdυðr0 Þ: us ðrÞ = − 4π n = 1

59

(3:11)

Ωc

Also, we have 4π r · BðrÞ = − μ0

ð X ∞ Ωc

Δðr0 Ar0 ðr0 ÞÞ

n=0

r0 n Pn ð ^r, ^r0Þdυðr0 Þ, rn + 1

r>s

(3:12)

where Jðr0 Þ = ∇Ψðr0 Þ + ∇ × Aðr0 Þ

(3:13)

with ∇ · Aðr0 Þ = 0, Ar0 denotes the radial part of the vector potential A and sn are known constants. Furthermore, the expansions (3.11) and (3.12) can be simplified into the following expressions us ðrÞ = −

∞ X n X

 m sn m cn + 1 cy_ m rÞ, n ðcÞ − nyn ðcÞ Yn ð ^ 2n + 1 n=1 m= −n

r=s

(3:14)

and ∞ X n X m 1 cn + 2 1  m rÞ, r · BðrÞ = − ca_ n ðcÞ − ðn − 1Þam n ðcÞ Yn ð ^ n+1 2n + 1 μ0 r n=0 m= −n

r > s;

(3:15)

m m _m where y_ m n, a n denote derivatives with respect to r0 and yn ðr0 Þ, an ðr0 Þ are the r0 − r0 dependent parts of the expansions of Ψðr0 Þ and A ðr0 Þ in spherical harmonics, namely, 0 ≤ r < c

Ψðr0 Þ =

∞ X n X n=1 m= −n

m ym r0Þ n ðr0 ÞYn ð ^

(3:16)

m am r0Þ: n ðr0 ÞYn ð ^

(3:17)

and Ar0 ðr0 Þ =

∞ X n X n=0 m= −n

Proof. First we observed that the expansion for Ψ starts with n = 1 since the expansion of υs starts from n = 1, while the expansion for Ar0 starts from n = 0. Utilizing (3.8), we obtain expression (3.14). Then using (3.7) and (3.9), we obtain (3.12). Inserting (3.16) in (3.11), we find

60

Chapter 3 Distributed current in spherical and ellipsoidal geometry

 ð ∞ X ∞ X n 1 X ′ m m ^ us ðrÞ = − s′ Δyn ðr0 ÞYn ð r0Þ r0 n Pn′ ð ^r, ^r0Þdυðr0 Þ: 4π ′ n = 1 m = − n n n =1

(3:18)

Ωc

If IB denotes the Betrami operator (the angular part of the Laplacian), then   1 d 1 m ^ m ^ 2 d m ð r ÞY ð Þ = r y ð r Þ Ynm ð ^r0Þ + ym Δym r 0 0 0 n 0 n n ðr0 Þ 2 IBYn ð r0Þ r0 2 dr0 dr0 n r0     1 d d m m ^ ð Þ r0 2 yn ðr0 Þ − nðn + 1Þym r = 2 n 0 Yn ð r0Þ: r0 dr0 dr0

(3:19)

Furthermore, we have the integral mapping þ Ynm ð ^r0ÞPn′ ð ^r, ^r0Þdωð ^r0Þ jr0 j = 1 n′ X

=

m′ = − n n′ X

=

m′ = − n

4π m′ Y ð ^rÞ ′ + 1 n′ 2n ′

þ

′

Ynm ð ^r0ÞYnm′ ð ^r0Þdωð ^r0Þ

(3:20)

j ^r0 j = 1

4π m′ 4π m Y ð ^rÞδnn′ δmm′ = Y ð ^rÞδnn′: ′ + 1 n′ 2n + 1 n′ 2n ′

Substituting (3.19) and (3.20) in (3.18), we obtain us ðrÞ = −

∞ X n X

sn m Y ð ^rÞ 2n +1 n n=1 m= −n

ðc 

   d d m r0 2 yn ðr0 Þ − nðn + 1Þym ð r Þ r0 n dr0 : n 0 dr0 dr0

0

(3:21) Next, we integrate by parts twice to obtain ðc 

  ðc  d n+1 _m m 2 d m n n r0 y ðr0 Þ r0 dr0 = c cyn ðcÞ − nyn ðcÞ + nðn + 1Þ ym n ðr0 Þ r0 dr0 dr0 dr0 n

0

0

(3:22) which implies that the integral in (3.21) is equal to  m cn + 1 cy_ m n ðcÞ − nyn ðcÞ : Finally, substituting this value of the integral in (3.21), we obtain (3.14). The proof of expansion (3.15) follows exactly the same steps used above where m ym n is replaced by r0 an . Hence, the proof of Proposition 3.2 is completed. The aforementioned proposition shows that the electric potential, as given by (3.14), depends on the boundary expressions

3.3 Ellipsoidal geometry

m cy_ m n ðcÞ − nyn ðcÞ

61

(3:23)

whereas the radial component of the magnetic field r · B, as given by (3.15), depends on the boundary expressions m ca_ m n ðcÞ − ðn − 1Þ an ðcÞ

(3:24)

where the functions ym n ðr0 Þ provide the r0 − dependence of the expansions for Ψ and Ar0 , respectively, in spherical harmonics. Since the vector potential is considered to be divergence free it only has two independent functional components. One of these is the radial component of A and the other is a tangential component of A that does not affect the MEG recordings. Corollary 3.1 Given that 1 ∂ r · BðrÞ = r · ∇U ðrÞ = r U ðrÞ μ0 ∂r

(3:25)

where U is the magnetic potential, which vanishes at infinity, we find the following expansions of U in the spherical geometry of Proposition 3.2: ð ∞ 1 X 1 Δðr0 Ar0 ðr0 ÞÞr0 n Pn ð ^r · ^r0Þdυðr0 Þ, r > s (3:26) U ðrÞ = − 4π n = 0 ðn + 1Þrn + 1 Ωc

and U ðrÞ = −

∞ X n X

 m 1 cn + 2 m ca_ n ðcÞ − ðn − 1Þam rÞ, n ðcÞ n + 1 Yn ð ^ ð n + 1 Þ ð 2n + 1 Þ r n=0 m= −n

r > s:

(3:27)

Proof. Integrate expressions (3.12) and (3.15) with respect to r along the ray that goes from r to ∞.

3.3 Ellipsoidal geometry The ellipsoidal model provides a much better approximation to the realistic brain geometry. However, the mathematical analysis is much harder than the corresponding analysis of the spherical model. In the present chapter, we present details of the mathematical analysis of the ellipsoidal model using a confocal ellipsoidal system. This system is the only ellipsoidal system that admits spectral analysis for the Laplace’s operator. The results obtained for the general triaxial ellipsoid can be reduced to the corresponding results for the prorate spheroid, the oblate spheroid, their asymptotic forms of needle and disc, as well as for the sphere. However, the implementation of

62

Chapter 3 Distributed current in spherical and ellipsoidal geometry

this reduction is quite complicated. For the geometry as well as for the analysis of ellipsoidal harmonic functions we refer to [26]. We start with a short description of the ellipsoidal system and the related ellipsoidal harmonics. The spherical coordinate system represents quantities exhibiting isotropic structure, whereas the ellipsoidal system can be adjusted to represent quantities with anisotropic structure. In this sense, the ellipsoid can be considered as the sphere of the anisotropic space. In an isotropic space, distances from a center are measured by the radius of a sphere. On the other hand, in an anisotropic space, distances from a center are measured in terms of the three different semiaxes of the ellipsoid that incorporates the given special anisotropy. The coordinate system that allows spectral decomposition of harmonic functions is the one that preserves the position of the six foci of the system. Such a system is termed confocal ellipsoidal system. In analogy with the spherical system, where we start with a reference sphere involving its center and its unit radius in the ellipsoidal system, we start with the reference ellipsoid specified by its center, three semiaxes and three Euler angles, defining the ellipsoidal analogue of the unit sphere. Given the reference ellipsoid x12 x22 x23 + + = 1; a21 a22 a23

(3:28)

in its canonical form, with 0 < a3 < a2 < a1 < + ∞, three semifocal distances h1 , h2 , h3 are defined by h21 = a22 − a23 , h22 = a21 − a23 , h23 = a21 − a22 :

(3:29)

The semifocal distances satisfy the relation h21 − h22 + h23 = 0:

(3:30)

The six foci are located at the pvoints ð ± h2 , 0, 0Þ, ð ± h3 , 0, 0Þ, ð0, ± h1 , 0Þ:

(3:31)

By extending eq. (3.28) to one parameter family of equations x2 x21 x2 + 2 2 + 2 3 =1 − λ a2 − λ a3 − λ

a21

(3:32)

  we obtain a family of confocal ellipsoids when λ 2 − ∞, a23 . The focal ellipse x12 x22 + = 1, h22 h21

x3 = 0

(3:33)

3.3 Ellipsoidal geometry

63

is the particular surface where λ = a23 . If a23 < λ < a22 we obtain a family of confocal hyperboloids of one sheet. The focal hyperbola x12 x23 − = 1, h23 h21

x2 = 0

(3:34)

is the particular surface where λ = a22 . If a22 < λ < a21 , we obtain a family of confocal hyperboloids of two sheets. Finally, when λ ≥ a21 eq. (3.32) does not represents a real surface. We define the ellipsoidal coordinates by ðρ, μ, νÞ where ρ 2 ðh2 , + ∞Þ represents the family of ellipsoids. Similarly, μ 2 ðh3 , h2 Þ represents the family of hyperboloids of one sheet and ν 2 ð − h3 , h3 Þ represents the family of hyperboloids of two sheets. The ellipsoidal system (3.32) is written in the axes system defined by its principal directions. Therefore, the eight octants of the Cartesian system ðx1 , x2 , x3 Þ represent similar ellipsoidal characteristics. In particular, in the first octant, where x1 > 0, x2 > 0, x3 > 0, the ellipsoidal coordinates are related with the Cartesian ones the following equations: ρμν h2 h3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 − h23 μ2 − h23 h23 − ν2 x1 =

x2 =

h 1 h3 pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 − h22 h22 − μ2 h22 − ν2 x3 = h1 h2

(3:35)

(3:36)

(3:37)

where h2 < ρ < + ∞, h3 < μ < h2 , 0 < ν < h3 . The confocal ellipsoidal system allows separation of variables for the Laplace equation. All three separated functions satisfy the same Lamé equation  2     x − h23 x2 − h22 E′′ðxÞ + x 2x2 − h23 − h22 E′ðxÞ  2  (3:38) h2 + h23 p − nðn + 1Þx2 EðxÞ = 0 where n = 0, 1, 2, ... is the first separation constant and p is the second separation constant which is specified through the solvability of a linear algebraic system, different for each n. In particular, there are 2n + 1 different values of p for each n denoted by pm n , m = 1, 2, 3, ..., 2n + 1. For any pair ðn, mÞ, where n defines the degree and m the order of the eq. (3.38), we obtain two linearly independent solutions Enm ðxÞ and Fnm ðxÞ of the first (interior) kind and the second (exterior) kind. When x = ρ 2 ðh2 , + ∞Þ, we obtain the Lamé functions Enm ðρÞ and Fnm ðρÞ which define the ρ − dependence of the generated harmonic functions. Similarly, when x = μ 2 ðh3 , h2 Þ, Enm ðμÞ and Fnm ðμÞ define the μ − dependence, and when x = ν 2 ð − h3 , h3 Þ, Enm ðνÞ and Fnm ðνÞ define the ν − dependence of the corresponding harmonic function. The function F ðρÞ is defined by

64

Chapter 3 Distributed current in spherical and ellipsoidal geometry

Fnm ðρÞ = ð2n + 1ÞEnm ðρÞInm ðρÞ

(3:39)

where Inm ðxÞ is the elliptic integral +ð∞

Inm ðρÞ =

 ρ

dx 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi . Enm ðxÞ x2 − h23 x2 − h22

(3:40)

In terms of these Lamé functions, we define the interior ellipsoidal harmonics IEnm ðρ, μ, νÞ = Enm ðρÞEnm ðμÞEnm ðνÞ,

(3:41)

and the exterior ellipsoidal harmonics IFnm ðρ, μ, νÞ = Fnm ðρÞEnm ðμÞEnm ðνÞ.

(3:42)

The reason why we consider only the F − functions of the variable ρ is due to the fact that interior and exterior harmonics are defined with respect to domains inside and outside the ellipsoids. On the ellipsoid ρ = constant, the angular part of the harmonic functions E and F is given by the surface ellipsoidal harmonics m m Sm n ðμ, νÞ = En ðμÞEn ðνÞ.

(3:43)

The surface harmonics Sm n are orthogonal on the surface of any ellipsoid with respect to the weighting function 1 lρ ðμ, νÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ρ − μ ρ2 − ν2

(3:44)

where ρ denotes the ellipsoidal surface. In fact, the solid angle element in the ellipsoidal system is given by dΩðμ, νÞ = lρ ðμ, νÞdSρ ðμ, νÞ

(3:45)

where dSρ is the surface element on the ellipsoid ρ. The solid angle element satisfies the identity þ dΩðμ, νÞ = 4π (3:46) Sρ

for any ρ 2 ðh2 , + ∞Þ. The orthgonality relations satisfied by Sm n ðμ, νÞ are given by þ m′ m Sm (3:47) n ðμ, νÞSn′ ðμ, νÞdΩðμ, νÞ = γn δnn′ δmm′ S0

3.3 Ellipsoidal geometry

65

where S0 represents the full solid angle and γm n are the normalization constants. It is impossible to obtain all of the ellipsoidal harmonics in closed form. Analytic solutions can be obtained only for degree n ≤ 7. This is due to the lack of recurrence formulae for the Lamé functions, which in turn is due to the fact that as we move from one n to the next, all values of the pm n m = 1, 2, ..., 2n + 1, change in an irregular way. The metric coefficients of the ellipsoidal system are given as follows: ffipffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − μ2

∂r

ρ ρ2 − ν2 (3:48) hρ =



= qffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ρ ρ2 − h23 ρ2 − h22 ffipffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffi

∂r

ρ2 − μ2 μ2 − ν2 hμ =



= qffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂μ μ2 − h23 h22 − μ2

(3:49)

pffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffi

∂r

ρ2 − ν2 μ2 − ν2



hν = = qffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi : ∂ν h23 − ν2 h22 − ν2

(3:50)

The corresponding unit vectors of the local system are given by ^= ρ

3 ρX xi ^i x hρ i = 1 ρ2 − a21 + a2i

(3:51)

^= μ

3 μX xi ^i x hμ i = 1 μ2 − a21 + a2i

(3:52)

^ν =

3 νX xi ^i : x hν i = 1 ν2 − a21 + a2i

(3:53)

Also, the position vector is represented as ρ μ ν ^+ ^+ ^ν μ ρ hρ hμ hν

(3:54)

^ ∂ ^ ∂ ^ν ∂ μ ρ + + : hρ ∂ρ hμ ∂μ hν ∂ν

(3:55)

r= and the gradient operator as ∇=

Consequently, the outward normal derivative on the ellipsoid ρ is given by ∂ 1 ∂ ^·∇= = ρ ∂n hρ ∂ρ which, on the surface of the reference ellipsoid becomes

(3:56)

66

Chapter 3 Distributed current in spherical and ellipsoidal geometry

∂

a2 a3 ∂ ∂ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi = a2 a3 la1 ðμ, νÞ :

2 2 2 2 ∂n ρ = a1 ∂ρ ∂ρ a1 − μ a1 − ν

(3:57)

We now turn to the 3-shell ellipsoidal model of the head-brain system. The surfaces Sj , j = c, f , b, s are confocal ellipsoidal surfaces with the following characteristics: Sc : 0 < c3 < c2 < c1 < + ∞ Sf : 0 < f3 < f2 < f1 < + ∞

(3:58)

Sb : 0 < b3 < b2 < b1 < + ∞ Ss : 0 < s3 < s2 < s1 < + ∞ These surfaces define the regions Ωc = fðρ, μ, νÞjh2 ≤ ρ < c1 g Ωf = fðρ, μ, νÞjc1 < ρ < f1 g Ωb = fðρ, μ, νÞjf1 < ρ < b1 g

(3:59)

Ωs = fðρ, μ, νÞjb1 < ρ < s1 g Ωe = fðρ, μ, νÞjs1 < ρ < + ∞g This means that the surface Sc is defined by the equation x12 x22 x23 + + =1 c21 c22 c23

(3:60)

and similarly for the other surfaces. The aforementioned surfaces have the same semifocal distances h1 , h2 , h3 , where h21 = c22 − c23 = f22 − f32 = b22 − b23 = s22 − s23 h22 = c21 − c23 = f12 − f32 = b21 − b23 = s21 − s23 .

(3:61)

h23 = c21 − c22 = f12 − f22 = b21 − b22 = s21 − s22 The Cartesian coordinates are related to their ellipsoidal coordinates by eqs. (3.35)– (3.37).   If we set r0 = ρ0 , μ0 , ν0 and r = ðρ, μ, νÞ, then the fundamental solution has the expansion ∞ 2X n+1 X 1 4π 1 m IEn ðr0 ÞIFnm ðrÞ, = jr − r0 j n = 0 m = 1 2n + 1 γm n

ρ0 < c 1 .

(3:62)

3.3 Ellipsoidal geometry

67

IEnm and IFnm are the interior and exterior ellipsoidal harmonics as given in (3.41) and (3.42), respectively. The numbers γm n are the normalization constants defined in (3.47). By solving the transmission problem, where υi , i = c, f , b, s, e are harmonic functions, we obtain five expansions in ellipsoidal harmonics for the aforementioned five monopolic potential fields. The exact form of these fields are derived in Chapter 5. In contrast to the case of spherical geometry, the dot product of r and H does not vanish in the ellipsoidal case. Thus, it is now necessary to evaluate the surface integrals appearing in the definition of H, given in (3.62). In this connection, we need the form of the monopolic potentials υi , i = c, f , b, s on the surfaces Sk which are given by υi ðr, r0 Þ =

∞ 2n +1 X X 4π

γm n=1 m=1 n

km m Cin IEn ðr0 ÞSm n ðμ, νÞ

(3:63)

km depend only on where, i and k take the values c, f , b, s, τ 2 Ωc . The constants Cin the geometry, via c1 , f1 , b1 , s1 , and the physics, via σc , σf , σb , σs . The explicit values of km will be calculated in Chapter 5. Cin

Lemma 3.1 The function Hðr, r0 Þ, defined in (2.62) is written as Hðr, r0 Þ = ð4πÞ2

∞ 2n +1 +1 X X X IFnm ðrÞ 2n l Hml n IEn ðr0 Þ ð2n + 1Þγm n l=1 n=1 m=1

(3:64)

l≠m

where r 2 Ωe , r0 2 Ωc and   cl ml ml ^ i · Hml x n = Hni = c1 c2 c3 σc − σf Ccn cni   fl ml + f1 f2 f3 σf − σb Cfn fni bl ml + b1 b2 b3 ðσb − σs ÞCbn bni sl ml + s1 s2 s3 σs Csn sni

(3:65)

ml ml ml with i = 1, 2, 3. The constants cml ni , fni , bni and sni are specified by the following formulae:

^ ðrÞ × ∇IEnm ðrÞjr2Sc n 2n +1 X c1 c2 c3 ^ 1 + cml ^ 2 + cml ^ 3 ÞSln ðμ, νÞ = qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi ðcml n1 x n2 x n3 x 2 2 2 2 c1 − μ c1 − ν l = 1 l≠m

(3:66)

68

Chapter 3 Distributed current in spherical and ellipsoidal geometry

^ ðrÞ × ∇IEnm ðrÞjr2S n

f

2n +1 X f1 f2 f3 ml ^ ml ^ ml ^ = qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi ðfn1 x1 + fn2 x2 + fn3 x3 ÞSln ðμ, νÞ f12 − μ2 f12 − ν2 l = 1

(3:67)

l≠m

2n +1 X b1 b2 b3 ^ ðrÞ × ∇IEnm ðrÞjr2S = qffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 1 + bml ^ 2 + bml ^ 3 ÞSln ðμ, νÞ ffiqffiffiffiffiffiffiffiffiffiffiffiffiffi ðbml n n1 x n2 x n3 x b b21 − μ2 b21 − ν2 l = 1

(3:68)

l≠m

s1 s2 s3 ^ ðrÞ × ∇IEnm ðrÞjr2Ss = qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi n s21 − μ2 s21 − ν2

2n +1 X

^ 1 + sml ^ 2 + sml ^3 ÞSln ðμ, νÞ . ðsml n1 x n2 x n3 x

(3:69)

2n +1 X a1 a2 a3 ml ^ ml ^ l ^ ðrÞ × ∇IEnm ðrÞjρ = a = pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ðaml n n1 x1 + an2 x2 + an3 x3 ÞSn ðμ, νÞ 1 a21 − μ2 a21 − ν2 l = 1

(3:70)

l=1 l≠m

Proof. The proof is based on the identity

l≠m

where ρ = a1 is the ellipsoidal surface x12 x22 x32 + + =1 a21 a22 a23

(3:71)

ðρ, μ, νÞ are the ellipsoidal coordinates of the position pvoint r and Sm n ðμ, νÞ are the surface ellipsoidal harmonics. Formulae (3.70) expresses the tangential surface gradient operator on the surface (3.71). The proof of (3.70) is rather technical and can be found in [30]. Using formulae (3.43), (3.44), (3.45), (3.21), (3.62) and (3.63), we can write the first integral on the right hand side of (2.62a) as follows  ð X ð ∞ 2n +1 X 1 4π cm m m ′ ′ ^ ðr′Þ × ∇r′ υf ðr′, r0 Þ n C IE ð r ÞS ð μ , ν Þ dsðr′Þ = 0 n γm fn n jr − r′j n=1 m=1 n Sc

Sc

2

3

" # ′+1 3 ∞ 2n X X X c1 c2 c3 x ′i 4π 6 7 ′ ′ m m ^ i × ∇r′ 5 · 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi IE ðr′ÞIFn′ ðrÞ x ′ n′ 2 2 c2 ð2n′ + 1Þγm n′ = 0 m′ = 1 c21 − μ′ c21 − ν′ i = 1 i n′ ·

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 c21 − μ′ c21 − ν′ dΩðμ′, ν′Þ

= ð4πÞ2 c1 c2 c3

′ + 1 cm m ∞ 2n +1 X ∞ 2n X X X Ccn IEn ðr0 ÞIF m′ ′ ðrÞ n

n = 1 m = 1 n′ = 0 m′ = 1

m′ ð2n′ + 1Þγm n γn′

 ð X 3 x ′i m′ ′ ′ ′ ′ ′ ^ ð Þ · × ∇ IE r x Sm r′ n′ n ðμ , ν ÞdΩðμ , ν Þ 2 i c i=1 i Sc

(3:72)

69

3.3 Ellipsoidal geometry

where we have used the fact that because of continuity cm cm = Ccn : Cfn

(3:73)

Inserting (3.66) in (3.72) and using the orthogonality property (3.47) we write (3.72) as ð 1 ^ ðr′Þ × ∇r′ dsðr′Þ υf ðr′, r0 Þ n jr − r′j Sc

= ð4πÞ2 c1 c2 c3

′ + 1 2n ′ + 1 cm m ∞ 2n +1 X ∞ 2n X X X X Ccn IEn ðr0 ÞIF m′ ′ ðrÞ n

n = 1 m = 1 n′ = 0 m′ = 1 l = 1 l≠m′

ð

m′ ð2n′ + 1Þγm nγ ′

′l m′l ^ m′l ^ ^ ðcm n′1 x1 + cn′2 x2 + cn′3 x3 Þ

n

′ ′ ′ ′ · Slm′ ðμ′, ν′Þ Sm n ðμ , ν ÞdΩðμ , ν Þ Sc

= ð4πÞ2 c1 c2 c3

′ + 1 2n ′ + 1 cm m ∞ 2n +1 X ∞ 2n 3 X X X X Ccn IEn ðr0 ÞIF m′ ′ ðrÞ X n

n = 1 m = 1 n′ = 0 m′ = 1 l = 1 l≠m′

= ð4πÞ2 c1 c2 c3

m′ ð2n′ + 1Þγm n γn′

∞ 2X n + 1 2X n + 1 cl m 3 X Ccn IEn ðr0 ÞIF m′ ′ ðrÞ X n

′ ð2n′ + 1Þγm n

n = 1 m′ = 1 l = 1 l≠m′

i=1

i=1

′

l^ l cm xγ δ δ n′i i n nn′ ml

′

l^ cm n′i xi .

(3:74)

Replacing m′ by m in (3.74) and analyzing the second, third and fourth integrals on the right hand side of (2.62a) in exactly the same way as the first integral we finally obtain Hðr, r0 Þ = ð4πÞ2

∞ 2X n + 1 2X n+1 l X IE ðr0 ÞIF m ðrÞ n

n=1 m=1 l=1 l≠m

n

ð2n + 1Þγm n

3 h X

   fl ml cl ml cni + σf − σb f1 f2 f3 Cfn fni σc − σf c1 c2 c3 Ccn i=1 i bl ml sl ml ^ + ðσb − σs Þb1 b2 b3 Cbn bni + σs s1 s2 s3 Csn sni xi (3:75) +

which coincides with (3.64) and (3.65). Hence, the proof of Lemma 3.1. In view of (3.64), we obtain that 2 3 ∞ 2X n+1 2X n+1 X 1 IFnm ðrÞ l 5 Bðr, r0 Þ = · 4Jðr0 Þ × ∇r0 IEnm ðr0 Þ − Jðr0 Þ · ∇r0 Hnl n IEn ðr0 Þ , μ0 ð2n + 1Þγm n n=1 m=1 l=1 l≠m

r 2 Ωe ,

r0 2 Ωc .

(3:76)

70

Chapter 3 Distributed current in spherical and ellipsoidal geometry

Proposition 3.3 Consider the 3-shell ellipsoidal model specified by the inequalities (3.59) and let the neuronal current Jðr0 Þ, r0 2 Ωc , be expressed in the form (3.1) and (3.2). Then ð ∞ 2n +1 X 1 X 1 sm m C S ð μ, ν Þ ðΔΨðr0 ÞÞIEnm ðr0 Þdυðr0 Þ, r 2 Ss (3:77) us ðrÞ = − sn n 4π n = 1 m = 1 γm n Ωc

and ∞ 2n +1 X X 1 1 r · BðrÞ = − IFnm ðrÞ μ0 ð 2n + 1Þγm n n=1 m=1 2 3 ð 2n +1 X 6 7 l r · Hml · 4ðΔðr0 · Aðr0 ÞÞÞIEnm ðr0 Þ − ðΔΨðr0 ÞÞ n IEn ðr0 Þ5 dυðr0 Þ,

r 2 Ωe . (3:78)

l=1 l≠m

Ωc

In eq. (3.77), the ellipsoidal coordinates of the pvoint r 2 Ss are ðs1 , μ, νÞ, while the sm as well as the constant vectors Hml constants Csn n can be explicitly computed in terms of the conductivities σc , σf , σb , σs and the ellipsoidal parameters ci , fi , bi , si , i = 1, 2, 3. The expressions in eqs. (3.77) and (3.78) can be simplified into the following expressions: us ðrÞ = −

∞ 2n +1 X  c2 c3 X m m _m Csm Sm ðμ, νÞ y_ m n ðc1 ÞEn ðc1 Þ − yn ðc1 ÞEn ðc1 Þ , 4π n = 1 m = 1 sn n

r 2 Ss

(3:79)

and  ∞ 2n +1 m X X 1 IFn ðrÞ  m _m r · BðrÞ = − c2 c3 a_ n ðc1 ÞEnm ðc1 Þ − am n ðc1 ÞEn ðc1 Þ μ0 2n + 1 n=0 m=1 2n +1 i

h X ml l l l l _ _ − r · Hn yn ðc1 ÞEn ðc1 Þ − yn ðc1 ÞEn ðc1 Þ , r 2 Ωe (3:80) l=1 l≠m m where ym n ðr0 Þ and an ðr0 Þ are the ρ − parts of the expansions of Ψðr0 Þ and r0 · Aðr0 Þ in surface ellipsoidal harmonics, namely,

Ψðr0 Þ =

∞ 2n +1 X X n=0 m=1

  m  ym n ρ0 Sn μ0 , ν0

(3:81)

and r0 · Aðr0 Þ =

∞ 2n +1 X X n=0 m=1

with r0 2 Ωc .

  m  am n ρ0 Sn μ0 , ν0

(3:82)

71

3.3 Ellipsoidal geometry

Proof. Replacing in equation (3.3), the function υs as it is given by (3.63) with i = k = s we obtain (3.77). Similarly, replacing in eq. (3.5), the function 1=jr − r′j by the right hand side of (3.62) and the function H by the right hand side of (3.64), we obtain eq. (3.78). We recall that ð ρ2 − μ2 Þð ρ2 − ν2 Þ dυ = hρ hμ hν dρ dμdν = hρ dρ ds = qffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi dρ dΩðμ, νÞ ρ2 − h23 ρ2 − h22

(3:83)

where dΩ denotes the elementary solid angle in ellipsoidal coordinates, that is, μ 2 − ν2 dΩðμ, νÞ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi dμdν: μ2 − h23 h22 − μ2 h23 − ν2 h22 − ν2

(3:84)

The Laplace’s operator in ellipsoidal coordinates has the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  1 ∂ ∂ ρ2 − h23 ρ2 − h22 Δ= 2 ρ2 − h23 ρ2 − h22 2 2 2 ðρ − μ Þðρ − ν Þ ∂ρ ∂ρ +

1 IBe ðρ2 − μ2 Þðρ2 − ν2 Þ

(3:85)

where IBe is the following ellipsoidal Beltrami operator or the ellipsoidal surface Laplacian: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ρ2 − ν2 ∂ ∂ μ2 − h23 h22 − μ2 μ2 − h23 h22 − μ2 IBe = 2 2 μ −ν ∂μ ∂μ qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  ρ2 − μ 2 ∂ ∂ h23 − ν2 h22 − ν2 h23 − ν2 h22 − ν2 : (3:86) + 2 2 μ −ν ∂ν ∂ν The Beltrami operator satisfies the spectral equations  2  2 m 2 m IBe Sm n ðμ, νÞ = h3 + h2 pn − nðn + 1Þρ Sn ðμ, νÞ

(3:87)

where pm n is the second separation constant of the Lamé functions appearing in eq. (3.38). Furthermore, it is straightforward to show that   m m  m ðρ2 − μ2 Þðρ2 − ν2 Þ  m qffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi Δ ym n ðρÞSn ðμ, νÞ = Ln yn ðρÞ Sn ðμ, νÞ ρ2 − h23 ρ2 − h22 where

(3:88)

72

Chapter 3 Distributed current in spherical and ellipsoidal geometry

Lm n

d = dρ

 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h2 + h23 pm d n − nðn + 1Þρ 2 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffiffiffiffiffiffiffiffi ρ − h3 ρ − h2 + dρ ρ2 − h23 ρ2 − h22

(3:89)

is the Lamé operator. Replacing in eq. (3.77) the expression (3.81) for Ψ, (3.83) for dυ, then making use of eq. (3.88) and the orthogonality condition (3.47), we obtain ′

us ðrÞ = −

∞ 2n +1 X ∞ 2n +1 X X 1 X 1 sm m C S ðμ, νÞ m sn n 4π γ n ′ ′ n=1 m=1 n =0 m =1

   þ   m′   m   m   ρ0 2 − μ0 2 ρ0 2 − ν0 2   m′ Δ yn′ ρ0 Sn′ μ0 , ν0 En ρ0 Sn μ0 , ν0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dρ0 dΩ μ0 , ν0 · ρ0 2 − h23 ρ0 2 − h22 h S cð1

2 0

′

=−

∞ 2X n+1 X ∞ 2n +1 X 1 X 1 sm m C S ðμ, νÞ m sn n 4π γ n ′ ′ n=1 m=1 n =0 m =1

cð1

·

þ

  m′   m     ′ m′   ′ Lm Sm n′ yn′ ρ0 n′ μ0 , ν0 En′ ρ0 Sn μ0 , ν0 dρ0 dΩ μ0 , ν0

h2 S0 ′

∞ 2n +1 X ∞ 2n +1 X X 1 X 1 sm m =− Csn Sn ðμ, νÞ 4π γm n ′ ′ n=1 m=1 n =0 m =1

=−

∞ 2X n+1 1 X Csm Sm ðμ, νÞ 4π n = 1 m = 1 sn n

cð1



cð1



  ′ m′   ′ Lm y ρ Enm′ ρ0 dρ0 γm 0 n δnn′ δmm′ n′ n′

h2

  m   m Lm n yn ρ0 En ρ0 dρ0 :

(3:90)

h2

Two successive integrations by parts give cð1

h2

    d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi d m   ρ 0 2 − h 3 ρ 0 2 − h2 Enm ρ0 yn ρ0 dρ0 dρ0 dρ0  m _m = c2 c3 Enm ðc1 Þy_ m n ðc1 Þ − En ðc1 Þyn ðc1 Þ  2     h22 + h23 pm n − nðn + 1Þρ0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Enm ρ0 ym n ρ0 dρ0 : ρ0 2 − h23 ρ0 2 − h22

(3:91)

 m   m   m m _m _m Lm n yn ρ0 En ρ0 dρ0 = c2 c3 En ðc1 Þy n ðc1 Þ − En ðc1 Þyn ðc1 Þ :

(3:92)

cð1

− h2



In view of (3.89), we obtain cð1



h2

Inserting eq. (3.92) in (3.90), we obtain (3.79).

73

3.3 Ellipsoidal geometry

Finally, for the proof of formula (3.80), we follow the same steps as with (3.88) and obtain  2       m   ρ0 − μ0 2 ρ0 2 − ν0 2  m   m  m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ an ρ0 Sn μ0 , ν0 = Lm (3:93) n an ρ0 Sn μ0 , ν0 ρ0 2 − h23 ρ0 2 − h22 where Lm n is defined in (3.89) and we have use of the expansion (3.82). Then, ∞ 2n +1 X X 1 IFnm ðrÞ r · BðrÞ = − μ0 ð2n + 1Þγm n n=0 m=1 " ð · ðΔðr0 · Aðr0 ÞÞÞIEnm ðr0 Þdυðr0 Þ Ωc 2X n+1 

−

l=1 l≠m

=−

n=0 m=1

−

ðΔΨðr0 ÞÞIEnl ðr0 Þ

dυðr0 Þ

Ωc

∞ 2X n+1 X

2n +1  X

#

ð



r · Hml n

IFnm ðrÞ

" cð1

2n + 1

cð1

r · Hml n

 

l=1 l≠m



  m   m Lm n an ρ0 En ρ0 dρ0

h2 m Lm n yn

  m   ρ0 En ρ0 dρ0

#

h

2 " ∞ 2n +1 m X X  IFn ðrÞ  m m _m = − c2 c3 En ðc1 Þa_ m n ðc1 Þ − En ðc1 Þan ðc1 Þ 2n + 1 n=0 m=1

−

2X n+1 

r · Hml n



Enl ðc1 Þy_ ln ðc1 Þ − E_ nl ðc1 Þyln ðc1 Þ



# (3:94)

l=1 l≠m

which is exactly (3.80). Hence, the proof of Proposition 3.3 is completed. Remark 3.1 The general identity (3.70) can also be replaced by its ellipsoidal analogue which is   ^ρ ∂ ^ ∂ ^ν ∂ μ ^ ðrÞ × ∇IEm ^ IEnm ðrÞ ð r Þ = ρ × + + n n hρ ∂ρ hμ ∂μ hν ∂ν   ^ν ∂ ^ ∂ μ IEnm ðrÞ = − + hμ ∂μ hν ∂ν   ^ m ^ν μ = IEnm ðρÞ En ðμÞE_ nm ðνÞ − E_ nm ðμÞEnm ðνÞ : (3:95) hν hμ

74

Chapter 3 Distributed current in spherical and ellipsoidal geometry

However, this form does not involve the surface functions Sm n ðμ, νÞ and therefore we can not apply the orthogonality condition as we did in (3.74). We observe here the unusual case where a combination of an ellipsoidal with a Cartesian form is more effective than the case where every expression is written in ellipsoidal form. Remark 3.2 Using the expansions of the potentials us , ub , uf and uc in ellipsoidal harmonics, as they are given analytically in Chapter 5, and the corresponding potentials given in (3.63), after we apply the source dependent operator 1 Jðr0 Þ · ∇r0 4π we obtain the relations sm = Csn

1 m _ s2 s3 En ðs1 ÞGm 3, n

(3:96)

Enm ðb1 Þ m In ðb1 , s1 Þ Gm 3, n

(3:97)

fm fm bm Cfn = Cbn = Cbn +

Enm ðf1 ÞGm 1, n m In ðf1 , b1 Þ σ b Gm 3, n

(3:98)

fm cm cm Ccn = Cfn = Cfn +

Enm ðc1 ÞGm 2, n m In ðc1 , f1 Þ σ f Gm 3, n

(3:99)

bm bm sm Cbn = Csn = Csn +

where Inm ðx, yÞ = Inm ðxÞ − Inm ðyÞ m m and the constants Gm 1, n , G2, n , G3, n are given in Chapter 5.

(3:100)

Chapter 4 The spherical model The spherical model of the head–brain system was analyzed in Section 3.2. For the electroencephalogram (EEG) problem, the monopolic potential υs on the head surface  r = s is given by formula (3.8), where the sequence sn g∞ n = 1 of coefficients involves the radii c, f , b, s as well as the conductivity constants σc , σf , σb , σs . In this chapter, we calculate analytically the above coefficients. This is achieved by computing all electric potentials in the cerebrum and in all shells that surround the cerebrum. In fact in this chapter we also establish closed-form solutions for the spherical geometry. Related results for magnetoencephalography (MEG) are also included.

4.1 The interior electric potential In this chapter, we assume that the conductive brain is modeled by the sphere x12 + x22 + x32 = α2

(4:1)

which contain in its interior a dipolar current located at the point r0 , with jr0 j < α and dipole moment Q. In this case, the interior electric potential u − solves the Neumann problem σΔu − ðrÞ = Q · ∇r δðr − r0 Þ, ∂ − u ðrÞ = 0, ∂n

r r0 , and hence in a neighborhood of the boundary r = α. Therefore, the boundary condition (4.3) is written as   ∞ X n X m d n An Q · ∇r0 r Ynm ð^rÞ dr n=0 m= −n   ∞ X n X 1 r0n d 1 Y m ð^rÞYnm *ð^r0 Þ = − Q · ∇r0 2n + 1 dr rn + 1 n σ n=0 m= −n

(4:16)

which has to be evaluated at r = α. Utilizing the orthogonality of the surface spherical harmonics, we obtain A00 = arbitrary Am n =

(4:17)

1 n+1 rn Q · ∇r0 2n0+ 1 Ynm *ð^r0 Þ α σ nð2n + 1Þ

(4:18)

for every n ≥ 1 and jmj ≤ n. The above arguments are valid because expansion (4.15) converges in the spherical shell r0 < r < a. Inserting expressions (4.17) and (4.18) into expansion (4.11) and substituting the result, as well as eq. (4.15), in representations (4.8) and (4.6) we obtain   ∞ X n X 1 r0n n + 1 rn 1 − Y m *ð^r0 ÞYnm ð^rÞ, r 2 ðr0 , αÞ + u ðrÞ = c + Q · ∇r0 2n + 1 n α2n + 1 rn + 1 n σ n=1 m= −n (4:19) where c is the arbitrary constant of the Neumann problem. An analogous expression that holds for every r < α is given by 1 1 Q · ∇r0 4πσ jr − r0 j ∞ X n X 1 n+1 r0n rn m Y *ð^r0 ÞYnm ð^rÞ, + Q · ∇r0 σ nð2n + 1Þ α 2n + 1 n n=1 m= −n

u − ðrÞ = c +

(4:20) r α:

(4:29)

Note that the appearance of the monopolic term Q · r0 α 4πσr03 r in expression (4.29) for the exterior potential is due to our choice of having a vanishing interior potential at the center of the sphere. Hence, we see that the value of u − at the center of the sphere is connected to the position of the dipole.

4.3 Closed expressions of the electric potential Expressions (4.22) and (4.29) for the electric potential are given in the form of eigenexpansions of the Laplacian in spherical coordinates. In what follows, we will demonstrate how we can manipulate these expressions and obtain closed-form solutions for the corresponding potentials. First, we show how we can sum up the series f ð ρÞ =

∞ X ρn n=1

n

Pn ðcos θÞ

(4:30)

for ρ < 1. Since the Legendre polynomial is bounded by 1, it follows that the series converges absolutely and almost uniformly. Hence, within the region ρ < 1, we can d on reladifferentiate and integrate the series term by term. Applying the operator ρ dρ tion (4.30) and differentiating term by term we obtain

80

Chapter 4 The spherical model

ρf ′ð ρÞ =

∞ X

ρn Pn ðcos θÞ.

(4:31)

n=1

Then using expansion (4.12), we rewrite (4.31) in the form 1 ρf ′ð ρÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 1 − 2 ρ cos θ + ρ2

(4:32)

which is an ordinary differential equation of the first order that has to be solved with the initial condition f ð0Þ = 0

(4:33)

dictated via (4.30). This initial value problem is solved as follows: ð ð dρ dρ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − f ð ρÞ = +c ρ ρ 1 − 2ρ cos θ + ρ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2ρ cos θ + 2 1 − 2 ρ cos θ + ρ2 − ln ρ + c = − ln ρ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = − ln 1 − ρ cos θ + 1 − 2 ρ cos θ + ρ2 − ln 2 + c:

(4:34)

Condition (4.33) implies that c = ln 4

(4:35)

therefore, the function f ð ρÞ is given by f ð ρÞ = − ln

1 − ρ cos θ +

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 2 ρ cos θ + ρ2 : 2

(4:36)

Next we combine formulae (4.30) and (4.36) to write the series on the right-hand side of eq. (4.22) in the closed form: ∞ X n + 1 r0n rn Pn ð^r · ^r0 Þ n α2n + 1 n=1 ∞ n n ∞ 1X r0 r 1X 1 r0 n r n = Pn ð^r · ^r0 Þ + Pn ð^r · ^r0 Þ α n=1 α α α n=1 n α α

=

∞ ∞ 1X 1 1X ρn ρn Pn ð^r · ^r0 Þ − + Pn ð^r · ^r0 Þ α n=0 α α n=1 n

=

1 1 1 1 1 − ρð^r · ^r0 Þ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − − ln α 1 − 2ρð^r · ^r0 Þ + ρ2 α α

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 2ρð^r · ^r0 Þ + ρ2 (4:37) 2

81

4.3 Closed expressions of the electric potential

where ρ=

r r 0 0, that is, generated by a positive unit monopolic source at the point r0 = ðx01 , x02 , x03 Þ with x03 > 0, where this harmonic function vanishes on the plane x3 = 0. The mathematical formulation of this problem is given by ΔuðrÞ = δðr − r0 Þ,

x3 > 0,

x03 > 0

uðx1 , x2 , 0Þ = 0:

(4:49) (4:50)

The solution of this problem has the form uðrÞ = −

1 1 + wðrÞ 4π jr − r0 j

(4:51)

where the function wðrÞ is harmonic for x3 > 0 and satisfies the condition that for x3 = 0 1 1 has the value 4π jr − r0 j. The regular function w can be calculated via spectral methods, by separating variables for the Laplacian in Cartesian coordinates and using expansions of the corresponding eigenfunctions. However, we can use an alternative approach, which is based on the observation that the function w is the harmonic function, generated by a negative unit monopole located at the symmetric with respect ′ to the plane x3 = 0 point r0 of r0 . Then, the solution of eqs. (4.49) and (4.50) is given by the function 2 3 uðrÞ =

1 6 1 1 7

− 4

5 4π

r − r′

jr − r0 j

(4:52)

0

which holds for every r with x3 > 0. Because of symmetry we have



′

jr − r0 j = r − r0

(4:53)

for every r with x3 = 0. Hence, it is obvious that the introduction of an image with inten′ sity –1, at the symmetric to r0 position r0 , eliminates the need for calculating the boundary condition (4.50), since this condition is automatically satisfied by symmetry. Although the basic idea about the method of images appears in the monumental work of Green, published individually in 1828 under the title An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, it

84

Chapter 4 The spherical model

was Lord Kelvin who made a complete use of images around the middle of the nineteenth century. Example 4.2 A more interesting case was analyzed by Kelvin: the problem (4.49), (4.50) when the plane boundary x3 = 0 is replaced by a sphere of radius α having a monopole with intensity +1 in its interior. The harmonic function is specified to vanish on the surface r = α of the sphere. This problem is defined by ΔuðrÞ = δðr − r0 Þ,

r < α,

uðrÞ = 0,

r = α:

r0 < α

(4:54) (4:55)

The image that solves this interior Dirichlet problem is located at the position ′

r0 =

α2 r0 r02

(4:56)

α : r0

(4:57)

with intensity q′ = −

The proof of this result can be found in ref. [72]. The solution of the problem (4.54), (4.55) is written as 2 3 uðrÞ =

7 1 6 6 α 1 − 1 7: 4



4π r0 α2 jr − r0 j5

r − r2 r0

(4:58)

0

The structure of solution (4.58) motivates the form of the solution of the interior and exterior electric potentials in terms of images that lie on the complement of the fundamental domain. We start with the last term on the right-hand side of eq. (4.44), which we will rewrite in a form that can be recognized in terms of images. If we denote this term by L we have



α2 − r · r0 + r0 r − α2^r

1 Q · ∇r0 ln L= − 4πσα 2α2

2

2

α ^

α ^ r · + r r − r r − r



0 0 r r 1 =− Q · ∇r0 ln 2 4πσα 2α



α2 ^

r − r − r + r∇

r0 r 0

1

:

=− (4:59) Q· 2

2

4πσα r · αr ^r − r0 + r αr ^r − r0

4.4 Connection with the method of images

85

Using

2

α2 ^

α

r − r0

∇r0

^r − r0

= −

r2

α ^ r

r r − r0

(4:60)

it follows that relation (4.59) can be written as

2

2

α ^

α ^ r + r r − r r − r



0 0 r r Q

h

i :



·

2 L= 4πσα α ^r − r

r · α2 ^r − r + r

α2 ^r − r

0 0 0 r r r Expression (4.61) of L involves the function F given by eq. (4.41). Hence,

2

2

α ^

α ^ ^ + r r − r r − r



0 0 r r α

L= : Q· 2 4πσr F αr ^r ; r0

(4:61)

(4:62)

2

It is easy to identify the term αr ^r in formula (4.62) as the Kelvin image of the point r 2 with respect to the sphere of radius α. Obviously, the image point αr ^r lies outside the sphere r = a. The function jr − r0 j − 1 can be considered either as potential at point r due to a monopole at point r0 , or as the potential at point r0 due to a monopole at point r. Similarly, the vector potential of a dipole at point r0 is proportional to the field ∇r0 =

1 r − r0 = jr − r0 j jr − r0 j3

(4:63)

while the vector potential of a dipole at point r is proportional to the opposite field ∇r =

1 r − r0 : =− jr − r0 j jr − r0 j3

(4:64)

In other words, the differential operator ∇r + ∇r0 annihilates the monopolic potential 2 jr − r0 j − 1 . By distributing dipoles in the direction ^r all the way from the Kelvin point αr to infinity, we generated a potential that has the form ∞ ð

α2 r

∞  ð 1 t^r − r0 dt = ∇r0 dt 3 jt^r − r0 j jt^r − r0 j α2 r

= ^r

∞ ð

∞ ð

tdt jt^r − r0 j

3

α2 r

− r0

dt jt^r − r0 j

3

α2 r

:

(4:65)

86

Chapter 4 The spherical model

By employing the integral formulae ð xdx 2ðbx + 2cÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 = 2 ð − 4ac Þ ax2 + bx + c b ax2 + bx + c ð

dx − 2ð2ax + bÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 = 2 ðb − 4acÞ ax2 + bx + c ax2 + bx + c

(4:66)

a>0

(4:67)

we obtain the following: ∞ ð

α2

tdt jt^r − r0 j

3

=

1 cos θ r2 r − r0 α2 cos θ



+ 0 2 r0 sin θ r2 r sin2 θ

α2 ^r − r

0 0 r

(4:68)

1 r0 r cos θ − α2



+ 2 sin θ r2 r sin2 θ

α2 ^r − r0

0 r

(4:69)

r

and ∞ ð

dt jt^r − r0 j

3

α2 r

=

r02

where cos θ = ^r · ^r0 :

(4:70)

Finally, by inserting expressions (4.68) and (4.69) in (4.65) we find the following: 2 3 ∞  ð 2 1 r0 r − α cos θ 7 6 1 cos θ

5 dt = ^r 4 + ∇r0 jt^r − r0 j r0 sin2 θ rr sin2 θ

α2 ^r − r

0 0 r 2 α r

2

3

r0 r cos θ − α 7 61 1

5 + − ^r0 4 r0 sin2 θ rr sin2 θ

α2 ^r − r

0 0 r 2

=

^r cos θ − ^r0 ^rðr0 r − α2 cos θÞ − ^r0 ðrr0 cos θ − α2 Þ



+

2

r0 sin2 θ rr0 sin2 θ α ^r − r0

r

2 ^r 6 = 4cos θ + r0 sin2 θ

3 2

2

^r0 6 r0 − αr cos θ7

5− 41 +

α2 ^

r0 sin2 θ

r r − r0

3 2

r0 cos θ − αr 7

5

α2 ^

r r − r0

2

2 2

3 2

3 α ^ α ^ ^ ^ · r r − r r · r − r 0 0 ^r ^r0 r 6^ ^ 0 r 6^ ^

7

7 = r ·r− 2 r · r −

2 5− 5: 2 4 0 2 4 0



α α r0 sin θ r0 sin θ

r ^r − r0

r ^r − r0

(4:71)

4.5 The inhomogeneous model

87

Introducing the symbol R=

α2 ^r − r0 r

(4:72)

with R = jRj

(4:73)

and using the identities

R−

α2 r

cos θ + r0

r0 sin2 θ

=

R+ R+

α2 r

α2 r

− r0 cos θ

(4:74)

and R−

α2 r

+ r0 cos θ

r0 sin2 θ

=

r0 R+

α2 r

− r0 cos θ

(4:75)

we can rewrite relation (4.71) in the form ∞ ð

α2

 1 α2 dt = ∇r0 r jt^r − r0 j





2

+ ^r αr ^r − r0



: 2 F αr ^r ; r0

α2 ^ r r − r0

(4:76)

r

Hence, by using expression (4.76) for the contribution of the distributed dipoles along the ray ^r in formula (4.62), we obtain Q · L= 4πσα

∞ ð

α2

 1 dt: ∇r0 jt^r − r0 j

(4:77)

r

4.5 The inhomogeneous model In the inhomogeneous spherical model of the head–brain system, the region Ωc is a homogeneous sphere, of radius c, with conductivity σc , and Ωf , Ωb , Ωs are concentric homogeneous shells with outer radii f , b, s, and conductivities σf , σb , σs , respectively. The region exterior to Ωs is denoted by Ωe . In each of the regions Ωi , i = c, f , b, s, e, we need to find harmonic functions υi , i = c, f , b, s, e supported in Ωi , which satisfy the transmission conditions (2.29), (2.30), (2.32), (2.33), (2.35), (2.36) and (2.38) and the Neumann condition (2.39). Because of eqs. (2.26) and (2.27), the potential υi , i = c, f , b, s, e, is due to a monopolic excitation. They admit the following expansions:

88

Chapter 4 The spherical model

υc ðr ; r0 Þ =

∞ X n X n=0 m= −n

^ m^ Cnm ðr0 Þrn Y m n *ðr0 ÞYn ðrÞ

  ∞ X n m X F2n ðr0 Þ m n ^ m^ Ym F1n ðr0 Þr + n + 1 υf ðr ; r0 Þ = n *ðr0 ÞYn ðrÞ r n=0 m= −n   ∞ X n X Bm m n 2n ðr0 Þ ^ m^ Ym B1n ðr0 Þr + n + 1 υb ðr ; r0 Þ = n *ðr0 ÞYn ðrÞ r n=0 m= −n   ∞ X n X Sm m n 2n ðr0 Þ ^ m^ Ym S1n ðr0 Þr + n + 1 υs ðr ; r0 Þ = n *ðr0 ÞYn ðrÞ r n=0 m= −n υe ðr ; r0 Þ =

∞ X n m X E2n ðr0 Þ m Y n *ð^r0 ÞYnm ð^rÞ: n+1 r n=0 m= −n

(4:78) (4:79) (4:80) (4:81) (4:82)

The n = 0 term specifies the arbitrary constant of the potentials and we assume that they are zero; all sums in eqs. (4.78)–(4.82) start with n = 1. The eight boundary conditions that determine the eight constants m m m m m m = F1 , F2n = F2 , Bm Cnm = C, F1n 1n = B1 , B2n = B2 , S1n = S1 , S2n = S2 , E2n = E

provide eight algebraic equations. We write these eight equations as follows: On the interface r = c 4π r0n 1 F2 + Ccn = F1 cn + n + 1 c σc ð2n + 1Þ cn + 1 σc −

n+1 rn n+1 4π n 0+ 2 + nCcn − 1 = nσf F1 cn − 1 − n + 2 σf F2 2n + 1 c c

(4:83) (4:84)

on the interface r = f F1 f n + σf nF1 f n − 1 − σf ðn + 1Þ

F2 n f +1

= B1 f n +

B2 fn+1

F2 B2 = σb nB1 f n − 1 − σb ðn + 1Þ n + 2 fn+2 f

(4:85) (4:86)

on the interface r = b B2 S2 = S1 bn + n + 1 bn + 1 b B2 S2 σb nB1 bn − 1 − σb ðn + 1Þ n + 2 = σs nS1 bn − 1 − σs ðn + 1Þ n + 2 b b B1 bn +

(4:87) (4:88)

and on the boundary r = s S1 sn +

S2 E2 = sn + 1 sn + 1

(4:89)

4.5 The inhomogeneous model

nS1 sn − 1 − ðn + 1Þ

S2 = 0: sn + 2

89

(4:90)

We set " ~ ð xÞ = W

x − ð n + 1Þ

nxn − 1

− ðn + 1Þx − ðn + 2Þ

" ~ ðx; σÞ = W

#

xn

xn

x − ðn + 1Þ

σnxn − 1

− σðn + 1Þx − ðn + 2Þ

(4:91) # :

(4:92)

Since ~ ðxÞ = − 2n + 1 ≠ 0 det W x2

(4:93)

and 2n + 1 ≠0 x2 both matrices are invertible. Then, eqs. (4.83) and (4.84) are written as ! ! C  F1 1 ~ − 1 ~ = W c; σf · Wðc; σc Þ 4πrn : 0 σc F2 2n + 1 ~ ðx; σÞ = − σ det W

Similarly, eqs. (4.85) and (4.86) are written as ! !   F1 B1 −1 ~ f ; σf ~ ðf ; σ b Þ · W =W B2 F2 eqs. (4.87) and (4.88) are written as ! ! B1 S1 −1 ~ ~ = W ðb; σs Þ · Wðb; σb Þ S2 B2 and eqs. (4.89) and (4.90) are written as ! s − ðn + 1Þ E2 0

~ ð sÞ =W

S1

(4:94)

(4:95)

(4:96)

(4:97)

!

S2

:

(4:98)

Furthermore, we write   ~ − 1 c; σf · W ~c=W ~ ðc; σc Þ M

(4:99)

  ~ − 1 ð f ; σb Þ · W ~ f ; σf ~f =W M

(4:100)

~ − 1 ðb; σs Þ · W ~ ðb; σb Þ ~b=W M

(4:101)

90

Chapter 4 The spherical model

~ ðsÞ ~s =W M

(4:102)

which lead to F1

!

F2 B1

1 ~ = M c σc ! ~f =M

B2 S1

! ~b =M

S2

s − ðn + 1Þ E2 0

!

C

(4:103)

n 4πr0

2n + 1

F1

!

(4:104)

F2 B1

! (4:105)

B2

! ~s =M

S1

!

S2

:

(4:106)

Then s − ð n + 1Þ E

!

0

~s ·M ~b·M ~f ·M ~c 1 =M σc

C

!

n 4πr0 2n + 1

~ =M

C n 4πr0 2n + 1

! :

(4:107)

If " ~= M

m11

m12

m21

m22

# ~b·M ~f ·M ~c ~s·M =M

1 σc

(4:108)

then eq. (4.107) gives m11 C +

4πr0n m12 = s − ðn + 1Þ E2 2n + 1

(4:109)

4πr0n m22 = 0: 2n + 1

(4:110)

4πr0n m22 2n + 1 m21

(4:111)

m21 C + Hence,

C= −

and then eq. (4.103) gives F1 , F2 , eq. (4.104) gives B1 , B2 , eq. (4.105) gives S1 , S2 and eq. (4.106) gives E. After performing the above calculations and obtain " # − ðn + 2Þ − ðn + 1Þ 2 x σ ð n + 1 Þx x −1 ~ ðx; σÞ = (4:112) W σð2n + 1Þ σnxn − 1 − xn

4.5 The inhomogeneous model

91

and 2

σ

2 1 6 ðn + 1Þ + n σ1 −1 ~ ~ W ðx; σ1 Þ · Wðx; σ2 Þ =

4 2n + 1 n 1 − σ2 x2n + 1 σ

ðn + 1Þ 1 −

σ2 σ1

x − ð2n + 1Þ

σ n + ðn + 1Þ σ2 1

1

3 7 5:

(4:113)

With this notation we find 2

σ

n + ðn + 1Þ σcf ~ c = 1 σc 6 M

4 2n + 1 σf n σf − 1 c2n + 1 σc 2

σ

n + ðn + 1Þ σb 1 σf 6 f ~ Mf =

4 2n + 1 σb n σb − 1 f 2n + 1 σ

3

− 1 c − ð2n + 1Þ 7 5 σf ðn + 1Þ + n σc

ðn + 1Þ

~b= M

σs

1 σb 6 n + ðn + 1Þ σb

4 2n + 1 σs n σs − 1 b2n + 1 σb

" ~s= M

f

σc

(4:114)

3 − 1 f − ð2n + 1Þ 7 5 σb ðn + 1Þ + n σ

(4:115)

3

ðn + 1Þ σσs − 1 b − ð2n + 1Þ b 7 5 σs ðn + 1Þ + n σ

(4:116)

ðn + 1Þ

f

2

σ



σb σf

f



b

sn

s − ðn + 1Þ

nsn − 1

− ðn + 1Þs − ðn + 2Þ

# :

(4:117)

~f , N ~ b, N ~ s via the relations ~ c, N We introduce N ~c ~ c = ð2n + 1Þ σf M N σc

(4:118)

~f ~ f = ð2n + 1Þ σb M N σf

(4:119)

~ b = ð2n + 1Þ σs M ~b N σb

(4:120)

~s ~s = M N

(4:121)

~ =N ~s · N ~b · N ~f · N ~ c: N

(4:122)

and

Then ~= M If we set

1 σs ð2n + 1Þ3

~ N:

(4:123)

92

Chapter 4 The spherical model

" ~= N

n11

n12

n21

n22

# (4:124)

then s − ð n + 1Þ E

! =

0

"

n11

n12

σs ð2n + 1Þ3 n21

n22

1

#

C n 4πr0 2n + 1

! :

(4:125)

Consequently C= −

4πr0n n22 2n + 1 n21

(4:126)

and E=

  ~ n22 4πr0n sn + 1 det N =− n − n : 12 11 4 4 n21 σs ð2n + 1Þ σs ð2n + 1Þ n21 4πr0n sn + 1

(4:127)

Inserting eq. (4.127) in (4.82) we obtain, for r = s, the formula υe ðs^r; r0 Þ = −

~ det N ^ m^ Ym n *ðr0 ÞYn ðrÞ n 21 n = 1 m = − n σs ð2n + 1Þ

∞ X n X

4πr0n

4

(4:128)

for the values of the exterior potential on the surface of the head, where EEG is measured. In view of the finite expansion Pn ð^r · ^r0 Þ =

n X

4π m ^ Yn ð^rÞY m n *ðr0 Þ 2n + 1 m= −n

(4:129)

expansion (4.128) is written as υe ðs^r; r0 Þ = −

∞ X

~ det N

3 n = 1 σs n21 ð2n + 1Þ

^ r0n Ynm ð^rÞY m n *ðr0 Þ

(4:130)

~ Comparing eq. (4.130) with (3.8) we obtain where n21 is the 21-entry of the matrix N. the coefficients sn in the form sn = −

~ det N σs n21 ð2n + 1Þ3

,

n = 1, 2, ...

(4:131)

4.6 The exterior magnetic field As we have shown in Chapter 3, the magnetic field B in the exterior of the brain–head system is both solenoidal and irrotational:

93

4.6 The exterior magnetic field

∇ · B = 0,

∇ × B = 0:

(4:132)

Thus, it can be represented as the gradient of a harmonic function U, which we will call the magnetic potential B = μ0 ∇U,

r 2 Ωe :

(4:133)

Furthermore, because of the identity   r − r′ ^r · n ^ r′ ×

=0

r − r′ 3

(4:134)

  ^ r′ denoting the unit normal on the boundary of the sphere, it follows that with n for the spherical model of MEG there is no contribution from the interfaces between spherical shells of different conductivities. Consequently, the MEG problems for the homogeneous and the inhomogeneous models are identical. For a single dipole excitation at the point r0 with moment Q, the radial component of the exterior magnetic field is given by ^r · Bðr; r0 Þ = −

μ0 Q × r0 · ^r : 4π jr − r0 j3

(4:135)

∂ U ðr; r0 Þ ∂r

(4:136)

Using ^r · Bðr; r0 Þ = μ0 it follows that ∂ 1 Q × r0 · ^r : U ðr; r0 Þ = − ∂r 4π jr − r0 j3

(4:137)

Integrating along a ray in the direction ^r from r to infinity relation (4.134) gives +ð∞

U ðr; r0 Þ = − r

=

 ∂ ′ 1 U r ^r; r0 dr′ = 4π ∂r′

Q × r0 · ^r 4π

+ð∞

r

dt jt^r − r0 j

3

+ð∞

r

Q × r0 · ^r ′

dr

r′ − r0 3

:

(4:138)

After long but straightforward calculations we can show that +ð∞

r

where F is the function

dt jt^r − r0 j

3

=

r F ðr; r0 Þ

(4:139)

94

Chapter 4 The spherical model

  r − r0 ^ Fðr; r0 Þ = jr − r0 j½rjr − r0 j + r · ðr − r0 Þ = jr − r0 j r · r + jr − r0 j 2

(4:140)

which also appeared in expressions (4.41)–(4.45) in relation to the image representation of the electric potential. Consequently, U ðr; r0 Þ =

Q × r0 · r 4πF ðr; r0 Þ

(4:141)

and Bðr; r0 Þ = =

μ0 ðQ × r0 · rÞ ∇r 4π F ðr; r0 Þ μ0 ∇r ðQ × r0 · rÞ μ0 ∇r F ðr; r0 Þ ðQ × r0 · rÞ 2 − : 4π Fðr; r0 Þ 4π F ðr; r0 Þ

(4:142)

Straightforward calculations lead to ∇r ðQ × r0 · rÞ = Q × r0

(4:143)

and

∇r F ðr; r0 Þ = ∇r rjr − r0 j2 + ∇r ½jr − r0 jr · ðr − r0 Þ r − r0 r · ðr − r0 Þ + jr − r0 jð2r − r0 Þ jr − r0 j h i r−r 0 = jr − r0 jðr + jr − r0 jÞ^r + jr − r0 j2 + 2rjr − r0 j + r · ðr − r0 Þ : (4:144) jr − r0 j = ^rjr − r0 j2 + 2rðr − r0 Þ +

Hence, we finally obtain " μ0 Q × r0 μ0 Q × r0 · r Bðr; r0 Þ = − jr − r0 jðr + jr − r0 jÞ^r 4π F ðr; r0 Þ 4π F 2 ðr; r0 Þ #

r−r 0 + jr − r0 j + 2rjr − r0 j + r · ðr − r0 Þ jr − r0 j

2

(4:145)

for the exterior magnetic field in closed form. For the case of a distributed current, the magnetic field is given by the integral ð μ0 Bðr; r0 Þ dυðr0 Þ (4:146) BðrÞ = 4π Ωc

where Ωc denotes the cerebrum.

Chapter 5 The ellipsoidal model The ellipsoidal model of the head-brain system was analyzed in Section 3.3. In the present chapter, we will obtain analytic solutions for EEG and MEG in ellipsoidal geometry for both the homogeneous and the 3-shell inhomogeneous problems (just as was done in Chapter 4 for the spherical model). In this way analytical expressions for km that appear in expressions (3.96)–(3.100) will be derived. the coefficients Cin In the present chapter, as well as in all other cases where we will have ellipsoidal coordinates we will denote the magnetic parameter by μ^0 in order to distinguish it from the coordinate μ0 of the ellipsoidal system.

5.1 EEG for the homogeneous model We consider the reference ellipsoid (3.28) as the boundary S of the head Ω and we assume the simple homogeneous model where the shells surrounding the cerebrum are ignored. Then, for an isolated dipole excitation fr0 , Qg the interior electric potential u − solves the Neumann problem σ Δr u − ðr, r0 Þ = Q · ∇r δðr − r0 Þ,

r2Ω

(5:1)

∂ (5:2) u − ðr, r0 Þ = 0, r 2 S ∂nðrÞ   where r = ðρ, μ, νÞ and r0 = ρ0 , μ0 , ν0 . After we obtain the solution of the problems (5.1) and (5.2), we can define the Dirichlet problem, Δr u + ðr, r0 Þ = 0,

r 2 Ωc

u + ðr, r0 Þ = u − ðr, r0 Þ, r 2 S   1 u + ðr, r0 Þ = O 2 , r ! + ∞ r

(5:3) (5:4) (5:5)

for the exterior electric potential u + . Writing u − in the form u − ðr, r0 Þ =

 1 1  + wðr, r0 Þ Q · ∇r0 4πσ jr − r0 j

(5:6)

where wðr, r0 Þ is an interior harmonic function, we obtain the boundary condition  ∂ ∂ 1  1 wðr, r0 Þ = − Q · ∇r0 ∂ρ 4πσ ∂ρ jr − r0 j for ρ = a1 . https://doi.org/10.1515/9783110547535-006

(5:7)

96

Chapter 5 The ellipsoidal model

The expansion of the fundamental solution jr − r0 j−1 in the ellipsoidal system [26] is given by ∞ 2n +1 X X  1 4π 1 m  IEn ρ0 , μ0 , ν0 IFnm ðρ, μ, νÞ, = jr − r0 j n = 0 m = 1 2n + 1 γm n

ρ0 < ρ

(5:8)

Therefore, if we assume an expansion of w in terms of interior ellipsoidal harmonics in the form ∞ 2n +1 X   X   m Bm w ρ, μ, ν;ρ0 , μ0 , ν0 = n ρ0 , μ0 , ν0 IEn ðρ, μ, νÞ

(5:9)

n=0 m=1

with h2 ≤ ρ < a1 , then the Neumann condition (5.7) takes the form ∞ 2n +1 X X n=0 m=1

=−

 m   m ′ Bm n ρ0 , μ0 , ν0 En ða1 Þ Sn ðμ, νÞ

∞ 2X n+1     1 X 4π 1  Q · ∇r0 IEnm ρ0 , μ0 , ν0 Fnm ða1 Þ ′Sm n ðμ, νÞ. (5:10) m 4πσ n = 0 m = 1 2n + 1 γn

Utilizing the orthogonality relation (3.47), we resolve the relation (5.10) in its spectral components to obtain, Bm n



ρ0 , μ 0 , ν0





    Q · ∇r0 IEnm ρ0 , μ0 , ν0 Fnm ða1 Þ ′   . =− ð2n + 1Þσγm Enm ða1 Þ ′ n

(5:11)

However,  m   m    Fn ða1 Þ ′ En ða1 Þ ′Inm ða1 Þ + Enm ða1 Þ Inm ða1 Þ ′   = ð2n + 1Þ   Enm ða1 Þ ′ Enm ða1 Þ ′ " # 1 m   . = ð2n + 1Þ In ða1 Þ − a2 a3 Enm ða1 Þ Enm ða1 Þ ′

(5:12)

Therefore,

Bm n



ρ0 , μ0 , ν0



#   " Q · ∇r0 IEnm ρ0 , μ0 , ν0 1 m   − I ða1 Þ : = σγm a2 a3 Enm ða1 Þ Enm ða1 Þ ′ n n 

(5:13)

Inserting, (5.13) in (5.9) and the resulting equation in (5.6), we obtain the interior electric potential

5.1 EEG for the homogeneous model

u

−



ρ, μ, ν; ρ0 , μ0 , ν0



97

"  1  1 = Q · ∇r0 4πσ jr − r0 j

∞ 2n +1 X X 1 m γ n=1 m=1 n ! #   m 1 m m   − I ða1 Þ IEn ρ0 , μ0 , ν0 IEn ðρ, μ, νÞ : (5:14) a2 a3 Enm ða1 Þ Enm ða1 Þ ′ n

+ 4π

Keeping in (5.14) only the terms of degree 1 and 2 we obtain the following Cartesian approximation of the electric potential:  3  3 X 1 Qm xm I1m ðρÞ − I1m ða1 Þ + u − ðr, r0 Þ = 4πσ m = 1 a1 a2 a3      5 Λ − a21 Λ − a22 Λ − a23 1 1 1 − I2 ðρÞ − I2 ða1 Þ + 4πσ 2Λa1 a2 a3 ðΛ − Λ′Þ X  3  3 Qk x0k X x2m + 1 · Λ − a2k m = 1 Λ − a2m k=1      5 Λ′ − a21 Λ′ − a22 Λ′ − a23 1 2 2 I2 ðρÞ − I2 ða1 Þ + + 4πσ ðΛ − Λ′Þ 2Λ′a1 a2 a3 X    3 3 Qk x0k X x2m +1 · Λ′ − a2k m = 1 Λ′ − a2m k=1 " # 15 1 3 3   ðQ1 x02 + Q2 x01 Þx1 x2 + I ðρÞ − I2 ða1 Þ + 4πσ 2 a1 a2 a3 a21 + a22 " # 15 1 4 4   ðQ3 x01 + Q1 x03 Þx3 x1 + I ðρÞ − I2 ða1 Þ + 4πσ 2 a1 a2 a3 a23 + a21 " # 15 1 5 5   + ðQ2 x03 + Q3 x02 Þx2 x3 I ð ρÞ − I 2 ð a 1 Þ + 4πσ 2 a1 a2 a3 a22 + a23   + O el3 , ρ0 < ρ < a1

(5:15)

  where the symbol O el3 indicates ellipsoidal terms in ðρ, μ, νÞ of degree greater or equal to 3. For the exterior electric potential u + which solves the Dirichlet problem (5.3)–(5.5), we can use the expansions (5.8) and (5.14) to generate the Dirichlet data.

u

+



a1 , μ, ν; ρ0 , μ0 , ν0



    ∞ 2X n+1 Q · ∇r0 IEnm ρ0 , μ0 , ν0 m 1X   Sn ðμ, νÞ: = m ′ σ n=1 m=1 γm n a2 a3 En ða1 Þ

(5:16)

98

Chapter 5 The ellipsoidal model

Then, the solution of (5.3)–(5.5) is given by     ∞ 2X n+1   Q · ∇r0 IEnm ρ0 , μ0 , ν0 m 1 X +   u ρ, μ, ν; ρ0 , μ0 , ν0 = IFn ðρ, μ, νÞ, m ′ m σa2 a3 n = 1 m = 1 γm n En ða1 Þ Fn ða1 Þ

ρ > a1 . (5:17)

5.2 MEG for the homogeneous model We recall that the exterior magnetic field for the dipole excitation fr0 , Qg is given by the Geselowitz representation (1.72) which for the case of the ellipsoid is written as ð μ^ 1 μ^ σ 1 ^ ðr′Þ × ∇r′ Bðr; r0 Þ = 0 Q × ∇r0 u − ðr′; r0 Þn − 0 dsðr′Þ (5:18) 4π jr − r0 j 4π jr − r′j ρ = a1

where the interior electric potential (5.14), on the boundary ρ = a1 , takes the form

∞ 2X n+1  1

 1  1X 1 − u ða1 , μ′, ν′; r0 Þ = + Q · ∇r0 IEnm ðr0 Þ Q · ∇r0

m 4πσ σ n = 1 m = 1 γn jr′ − r0 j

ρ = a1

! 1 m m   − En ða1 ÞIn ða1 Þ Enm ðμ′ÞEnm ðν′Þ. a1 a2 a3 Enm ða1 Þ ′

(5:19)

Keeping in (5.15) only the dipole and quadrupole terms, we obtain the Cartesian representation u − ða1 , μ′, ν′; r0 Þ =

3 3 X Q m x ′m 4πσ m = 1 a1 a2 a3 #     3 " 3 2 5 Λ − a21 Λ − a22 Λ − a23 X Qk x0k X x ′m − +1 8πσ a1 a2 a3 ΛðΛ − Λ′Þ Λ − a2k m = 1 Λ − a2m k=1

#     3 " 3 2 5 Λ′ − a21 Λ′ − a22 Λ′ − a23 X Qk x0k X x ′m + +1 8πσ a1 a2 a3 Λ′ðΛ − Λ′Þ Λ′ − a2k m = 1 Λ′ − a2m k=1 +

15 Q1 x02 + Q2 x01 15 Q3 x01 + Q1 x03   x 1 ′x 2 ′ +   x1 ′x3 ′ 4πσ a1 a2 a3 a21 + a22 4πσ a1 a2 a3 a21 + a23

+

15 Q2 x03 + Q3 x02   x2 ′x3 ′ + Oðel3 Þ. 4πσ a1 a2 a3 a22 + a23

Similarly, for the ellipsoidal representation, 3 5 X X u − ða1 , μ′, ν′; r0 Þ = Jm E1m ðμ′ÞE1m ðν′Þ + θm E2m ðμ′ÞE2m ðν′Þ + Oðel3 Þ, m=1

m=1

(5:20)

(5:21)

5.2 MEG for the homogeneous model

99

where Jm =

am hm ^ m Þ, ðQ · x σVh1 h2 h3

θ1 = −

m = 1, 2, 3

  5 ~ Q  r0 : Λ ′ 6σV ðΛ − Λ Þ

(5:22) (5:23)

  5 ~′ Q  r0 :Λ 6σV ðΛ − Λ′Þ

(5:24)

θ3 =

^1  x ^ +x ^ x ^1 Þ 5a1 a2 a3 Q  r0 :ðx  2 2 2 2 σVh1 h2 h3 a 3 h3 a 1 + a 2

(5:25)

θ4 =

^1  x ^ +x ^ x ^1 Þ 5a1 a2 a3 Q  r0 :ðx  2 3 2 3 σVh1 h2 h3 a2 h2 a1 + a3

(5:26)

θ5 =

^2  x ^ +x ^ x ^2 Þ 5a1 a2 a3 Q  r0 :ðx  3 3 σVh1 h2 h3 a1 h1 a22 + a23

(5:27)

4π a1 a2 a3 3

(5:28)

θ2 =

V=

with V denoting the volume of the ellipsoid ρ = a1 ; also ~= Λ Λ~′ =

3 X ~m ~m  x x Λ − a2m m=1 3 X ~m ~m  x x m=1

Λ′ − a2m

(5:29) .

(5:30)

The expression (5.21) provides the factor u − of the integrand in (5.18). The corre~ ðr′Þ × r − r′ 3 can also be calculated as follows: sponding expression of the factor n jr − r′j first, by utilizing the identity ~ ðr′Þ = x ^ ρ′ = n

3V ~ −1 ða1 Þ · r′ la ðμ′, ν′ÞM 4π 1

(5:31)

where la1 is the weighting function defined in (3.93) and ~ ð ρÞ = M

3  X m=1

 ~m . ~m  x ρ2 − a21 + a2m x

(5:32)

This implies ~ −1 ða1 Þ = M

3 X ~m ~m  x x m=1

a2m

.

(5:33)

100

Chapter 5 The ellipsoidal model

Second, employing the identity r − r′ ~ 1 ð ρÞ + F ~ ðrÞ · r′ + Oðel2 ′Þ = 3r · H 3 jr − r′j

(5:34)

with ~ 1 ð ρÞ = H

3 X

~m  x ~m I1m ðρÞx

(5:35)

~i  x ~j  x ~i  x ~j I2i + j ðρÞx

(5:36)

m=1

~ 2 ðρÞ = H

3 X i, j = 1 i≠j

and 1 2 ~~ ðρÞ. ~ ðrÞ = − IF2 ðrÞ Λ ~ + IF2 ðrÞ Λ ~ ′ + 15r  r:H F 2 Λ − Λ′ Λ − Λ′

(5:37)

Then, ~ ðr′Þ × ∇r′ n



1

r − r′

′ ^ = ρ × 3

jr − r′j ρ = a1 jr − r′j ρ = a1 = a1 a2 a3 la1 ðμ′, ν′Þ

 X

1 3 2 ~ −1 ða1 Þ × F ~ ðrÞ · x ^m ^m · M am x 3 m=1

+

3

X 1 ~ −1 ða1 Þ × H ~ 1 ðρÞ · r Em ðμ′ÞEm ðν′Þ ^m · M a m hm x 1 1 3h1 h2 h3 m = 1

−

3

X 1 a2m ~ −1 ða1 Þ × F ~ ðrÞ · x ^ m E21 ðμ′ÞE21 ðν′Þ ^m · M x 2 3ðΛ − Λ′Þ m = 1 Λ − am

3

X 1 a2m ~ −1 ða1 Þ × F ~ ðrÞ · x ^ m E21 ðμ′ÞE21 ðν′Þ ^m · M x 2 3ðΛ − Λ′Þ m = 1 Λ′ − am  3

ai aj 1 X −1 i+j i+j ~ ~ ′ ′ ^ ^ + 2 2 2 xi · M ða1 Þ × FðrÞ · xj E2 ðμ ÞE2 ðν Þ h1 h2 h3 i, j = 1 h6 − ði + jÞ

+

i≠j

+ Oðel3 ′Þ:

(5:38)

The expression (5.38) involves three dipole terms and five quadrupole terms, but as we will show below, the three dipole terms vanish. Indeed, 3 X m=1

3

X ~ −1 ða1 Þ × F ~ ðrÞ · x ~ ðrÞ · x ^m = ^m ^m · M ^m × F a2m x x m=1

=−

3 3 ~ m IF22 ðrÞ X ~m ~m × x ~m × x IF21 ðrÞ X x x + Λ − Λ′ m = 1 Λ − a2m Λ − Λ′ m = 1 Λ′ − a2m

5.2 MEG for the homogeneous model

+ 15

3 X

101

~~ ^m ^ m × r  r:H x 2 ðρÞÞ · x

m=1

= 15

3 X

^j × x ^ j = 0: xi xj I2i + j ðρÞx

(5:39)

i, j = 1 i≠j

The relation (5.39) is consistent with the laws of magnetostatics and in particular with the nonexistence of magnetic monopoles. Inserting (5.21) and (5.39) in the integral on the right hand side of (5.18) and using orthogonally, we obtain ð 3 3 X X 1 ^ ðr′Þ × ∇r′ u − ðr′; r0 Þn Jm γm β + θm γm (5:40) dsðr′Þ = 1 m 2 δm + Oðel3 Þ jr − r′j m=1 m=1 ρ = a1

where βm = 3

a1 a2 a3 hm ~ 1 ðρÞ, ^ m  r ×. H x h1 h2 h3 a m

δ1 = −

m = 1, 2, 3

a1 a2 a3 ~ ~ Λ × FðrÞ 3ðΛ − Λ′Þ .

a1 a2 a3 ~ ~ Λ × FðrÞ 3ðΛ − Λ′Þ .   a1 a2 a3 a2 a1 ~ ðrÞ ^ ^ ^ ^ ×. F  x +  x x x δ3 = 1 2 2 1 a2 h1 h2 h23 a1   a1 a2 a3 a3 a1 ~ ^ ^ ^ ^  x +  x δ4 = x x 1 3 3 1 × . FðrÞ a3 h1 h22 h3 a1   a1 a2 a3 a3 a ~ ðrÞ. ^3 + 2 x ^ 2 ×. F ^2  x ^3  x δ5 = 2 x a3 h1 h2 h3 a2 δ2 =

(5:41) (5:42) (5:43) (5:44)

(5:45)

(5:46)

But 3 X m=1

μ

Jm γ1 βm =

3 3X ~ 1 ðρÞ = 3 Q  r × H ~ ^m Þ x ^ m  r ×. H ðQ · x . 1 ðρÞ: σ m=1 σ

(5:47)

After extended manipulations involving the ellipsoidal harmonics, which are given in [27], we arrive at the expression Bðr′; r0 Þ =

3 3 ^0 IF21 ðρ, μ, νÞ X ^0 IF22 ðρ, μ, νÞ X di di μ μ ^ ^i − x x i 4π Λ − Λ′ i = 1 Λ − a2i 4π Λ − Λ′ i = 1 Λ′ − a2i

−

3   ^0 X 15 μ ^ j + O ρ−4 di xi xj I2i + j ðρÞx 4π i, j = 1 i≠j

where the dipole dependent vector d is given by

(5:48)

102

Chapter 5 The ellipsoidal model

d=

a22 Q2 x03 − a23 Q3 x02 a23 Q3 x01 − a21 Q1 x03 a21 Q1 x02 − a22 Q2 x01 ^ ^ ^3 . + + x x x 1 2 a22 + a23 a21 + a23 a21 + a22

(5:49)

Very elaborate manipulations are needed to calculate the octapolic term n = 2 for the field B. The complete analysis can be found in reference [35].

5.3 The complete magnetic field In this section, we demonstrate an analytic technique to obtain the general expression of B in ellipsoidal geometry. The idea is as follows: we know that in order to obtain the magnetic potential U for the spherical model of the brain we need to integrate the radial component ^r · BðrÞ along a ray from the exterior observation point r to infinity in the direction ^r. This ray is the intersection of the cone with the direction ^r as its generating line and the azimouthal plane that contains ^r. This ray is the intersection of the two coordinate surfaces which specify the orientation of points in space. For the ellipsoidal system we can follow the corresponding curve in space specified by the intersection of the hyperboloid of one sheet and the hyperboloid of two sheets that pass through the point ðρ, μ, νÞ, that is, the μ = constant and the ν = constant coordinate surfaces, (since the coordinates μ, ν are those that specify the spatial orientation in terms of ellipsoidal coordinates). This particular curve is defined by  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ^ 1 + h2 ρ2 − h23 μ20 − h23 h23 − ν20 x ^2 h1 ρμ0 ν0 x CðρÞ = h 1 h2 h3 qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^3  (5:50) + h3 ρ2 − h22 h22 − μ20 h22 − ν20 x     where ρ 2 ρ0 , + ∞ is the parameter of the curve and r0 = ρ0 , μ0 , ν0 is the point from where the ellipsoidal ray emanates. If the point where the magnetic potential is evaluated belongs to the surface ρ = a1 , then the parameterization of (5.50) is ρ 2 ½a1 , + ∞Þ. We write the electric potential on the ellipsoid ρ = a1 as uða1 , μ′, ν′; r0 Þ =

∞ 2X n+1 X n=1 m=1

m ′ m ′ Am n ðr0 ÞEn ðμ ÞEn ðν Þ

(5:51)

where Am n ðr0 Þ =

Q · ∇r0 IEnm ðr0 Þ   . m ′ σa2 a3 γm n En ða1 Þ

(5:52)

5.3 The complete magnetic field

103

Also



∞ 2X n+1 X  m

r − r′

1

4π 1  m ′ = − ∇r =− ∇r IFn ðrÞ IEn ðr Þ

3

2n + 1 γm jr − r′j ρ′ = a1 jr − r′j ρ′ = a1 n ρ′ = a1 n=0 m=1 =−

∞ 2X n+1 X n=0 m=1

m ′ m ′ Bm n ðrÞEn ðμ ÞEn ðν Þ

(5:53)

with Bm n ðrÞ =

  4π ∇r IFnm ðrÞ Enm ða1 Þ m ð2n + 1Þγn

(5:54)

and

3 3 X xi ′

aaa X hi ^i : ^i = 1 2 3 ^′ = a1 a2 a3 la1 ðμ′, ν′Þ la1 ðμ′, ν′ÞE1i ðμ′ÞE1i ðν′Þx x ρ

2 h h h a a 1 2 3 i ′ i = 1 i ρ = a1 i=1 Therefore, ð ∞ 2X n+1 X ∞ 2X k+1 X r − r′ m ′ ^′ × u − ðr′; r0 Þρ ds ð r Þ = Aλk ðr0 ÞCmλ − nk × Bn ðrÞ 3 jr − r′j n=0 m=1 k=1 λ=1

(5:55)

(5:56)

ρ′ = a1

where Bm n are given in (4.97) and Cmλ nk = ð

3 a1 a2 a3 X hi h1 h2 h3 i = 1 a i

^i : Enm ðμ′ÞEnm ðν′ÞEkλ ðμ′ÞEkλ ðν′ÞE1i ðμ′ÞE1i ðν′Þla1 ðμ′, ν′Þdsðμ′, ν′Þx

(5:57)

ρ′ = a1

Expanding the first term on the right hand side of (5.18), we obtain Q×

r − r0 jr − r0 j3

= − Q × ∇r

∞ 2n +1 X X 1 4π 1 m IE ðr ÞQ × ∇r IFnm ðrÞ: =− m n 0 jr − r0 j 2n + 1 γ n n=0 m=1

(5:58)

Then, in view of (5.56) and (5.58), formula (5.18) becomes ^0 Bðr; r0 Þ = μ

−

 ∞ 2X n+1 X ∇r IFnm ðrÞ × QIEnm ðr0 Þ m ð 2n + 1 Þγ n n=1 m=1

 ∞ 2k +1 X 1 X Enm ða1 Þ Q · ∇r0 IEkλ ðr0 Þ   Cmλ a2 a3 k = 1 λ = 1 nk Ekλ ða1 Þ ′ γλk

(5:59)

104

Chapter 5 The ellipsoidal model

where Cmλ nk are given in (5.57). In the expansion (5.59), we have used the fact that the n = 0 term vanishes. Indeed, if we denote the n = 0 term by B0 ðr; r0 Þ, we have " # ∞ 2k +1 X Q · ∇r0 IEkλ ðr0 Þ ∇r I01 ðρÞ 1 X 1 1λ ^0  B0 ðr; r0 Þ = μ × Q− C  (5:60) a2 a3 k = 1 λ = 1 0k Ekλ ða1 Þ ′ γ10 γλk where C1λ 0k =

= =

3 a1 a2 a3 X hi ^i x h1 h2 h3 i = 1 a i

a1 a2 a3 h1 h2 h3

3 X i=1

ð

Ekλ ðμ′ÞEkλ ðν′ÞE1i ðμ′ÞE1i ðν′Þla1 ðμ′, ν′Þds′

ρ′ = a1

3 X hi i 4π δk1 δiλ ^i = ^i γ1 δk1 δiλ x x a1 a2 a3 h1 h2 h3 ai a i hi 3 i=1

4π δk1 ^λ . x a1 a2 a3 h1 h2 h3 aλ hλ 3

(5:61)

Inserting (5.61) in (5.60), we find     3 X ^0 hλ h1 h2 h3 μ ^λ Q · ∇r0 x0λ x ∇r I01 ðrÞ × Q − 4π hhh hλ λ=1 1 2 3   3 X ^ μ ^λ  x ^ λ = 0. Q·x = 0 ∇r I01 ðrÞ × Q − 4π λ=1

B0 ðr; r0 Þ =

(5:62)

The above calculations verify that the dipole term contribution from the source fr0 , Qg and from the interface at ρ = a1 cancel each other. Therefore, the exterior magnetic field starts with the quadrupolic contribution   1 (5:63) ∇r IF1i ðrÞ = O 3 , r ! ∞. r λ The term Cmλ 1k with k ≥ 3 has to vanish by orthogonality, since Sk should have the m i same degree as S1 · S1 . Also, note that the surface ellipsoidal harmonic i m m i i Sm n ðμ, νÞS1 ðμ, νÞ = En ðμÞEn ðνÞE1 ðμÞE1 ðνÞ

(5:64)

lives in the subspace generated by the surface harmonics of degree less or equal to n + 1. So, by orthogonality Cmλ nk = 0,

k ≥ n + 2.

(5:65)

Hence, in the expansion (5.59) the k − summation goes from 1 to n + 1. In fact, introducing the source dependent dyadic

105

5.3 The complete magnetic field

~ m ðr0 Þ = D n

1 ð2n + 1Þγm n  − IEnm ðr0 Þ~I +

n + 1 2X k+1  1 X 1 Enm ða1 Þ    ∇r0 IEkλ ðr0 Þ  Cmλ nk a2 a3 k = 1 λ = 1 γλk Ekλ ða1 Þ ′

 (5:66)

we can write the complete exterior magnetic field as ^0 Bðr; r0 Þ = μ

∞ 2n +1 X X n=1 m=1

~ ðr0 Þ × ∇r IFm ðrÞ. Q·D n n m

(5:67)

In view of the differential representation ^0 ∇r U ðr; r0 Þ Bðr; r0 Þ = μ

(5:68)

an integration along the ellipsoidal coordinate curve (5.50) gives +ð∞

U ðr; r0 Þ = − ρ

=−

=−

1 ^0 μ 1 ^0 μ

∂ U ðρ′, μ, ν; r0 Þdρ′ ∂ρ′ +ð∞

  hρ′ ^ρ′ · ∇ρ′ μ0 U ðρ′, μ, ν; r0 Þ dρ′

ρ +ð∞

^′ · Bðρ′, μ, ν; r0 Þdρ′ hρ′ ρ

(5:69)

ρ

with ρ ≥ a1 and the contour of integration is CðρÞ. Some further calculations lead to the formula

~ m × ∇ ′ IFm ðρ′, μ, ν; r0 Þ ^′ · Q · D ρ ρ n n  = Fnm ðρ′Þ



^ ∂ ^ ∂ m ν μ ~ m ðr0 Þ . En ðμÞEnm ðνÞ · Q · D − n hμ ∂μ hν ∂ν

(5:70)

Utilizing formulae (3.52) and (3.53), we can perform a series of calculations that lead to   3 X ^ ∂ ^ ∂ m E1i ðρ′Þ ν μ ^i ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi fnim ðμ, νÞx qffiffiffiffiffiffiffiffiffiffiffiffiffiffi En ðμÞEnm ðνÞ = (5:71) − hμ ∂μ hν ∂ν 2 2 i=1 ρ′ − μ2 ρ′ − ν2 with fnim ðμ, νÞ =

E12 ðμÞE12 ðνÞE13 ðμÞE13 ðνÞ hi E1i ðμÞE1i ðνÞ h1 h2 h3 ðμ2 − ν2 Þ

106

Chapter 5 The ellipsoidal model

    m  m ν En ðμÞ ′En ðνÞ μEnm ðμÞ Enm ðνÞ ′ · − ν2 − a21 + a2i μ2 − a21 + a2i =

hi E25 ðμÞE25 ðνÞ  i  i   m  m E ðμÞ E1 ðνÞ ′ En ðμÞ ′En ðνÞ h1 h2 h3 ðμ2 − ν2 Þ 1     − E1i ðμÞ ′E1i ðνÞEnm ðμÞ Enm ðνÞ ′.

(5:72)

Hence, (5.69) reads U ðr; r0 Þ = −

∞ 2n +1 X 3 X X n=1 m=1 i=1

fnim ðμ, νÞ



+ð∞ Fnm ðρ′ÞE1i ðρ′Þ m ~ ^i ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dρ′ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q · Dn ðr0 Þ · x 2 2 ρ′ − h23 ρ′ − h22 ρ

(5:73)

where the coefficients fnim ðμ, νÞ depend only on the orientation point ðμ, νÞ of the point r, while the dependence on the distance ρ enters via the integral factor. The quantity ~m · x ^ i depends solely on the dipole source fr0 , Qg. The expression (5.73) provides Q·D n the magnetic potential in a separable form.

5.4 EEG for the 3-shell model In the present section, we follow the steps that are described in Section 4.5, for the solution of the spherical model, in the case where the 3-shell model is based on the system of confocal ellipsoids. In order to keep the presentation as closely related as possible to the spherical geometry, we will use the same notation as in the spherical case, introducing a cup “˄” on the top of the corresponding coefficients. For the notation of the ellipsoidal har  monics, we refer to Section 3.3. We write ðρ, μ, νÞ and ρ0 , μ0 , ν0 for the ellipsoidal coordinates of the observation point and the location of the monopole, respectively. ^i due to a dipolic excitation Furthermore, we assume that the electric potentials u ^i due to a monopolic exciare obtained from the corresponding electric potentials υ tation, via the action of the source depended directional derivative 1 Q · ∇r0 4π as in the expressions (2.26) and (2.27). Then, in view of (5.8), we obtain ^c ðr; r0 Þ = u

   1 1  ^c ðr; r0 Þ Q · ∇r0 +υ 4πσc jr, r0 j

(5:74)

 1  ^i ðr; r0 Þ Q · ∇r0 υ 4π

(5:75)

^s ðr; r0 Þ = u

107

5.4 EEG for the 3-shell model

for j = f , b, s, e where the monopolic potentials admit the following expansions: ^c ðr; r0 Þ = υ

∞ 2X n+1 X n=1 m=1

^f ðr; r0 Þ = υ

    ^ m ρ Em ðρÞSm μ , ν0 Sm ðμ, νÞ C 0 1n 0 n n n

(5:76)

∞ 2n +1 h i  X X      ^ m ρ Fm ðρÞ Sm μ , ν0 Sm ðμ, νÞ ^ m ρ E m ð ρÞ + F F 0 1n 0 n 2n 0 n n n

(5:77)

∞ 2n +1 h i  X X      ^ m ρ F m ðρÞ Sm μ , ν0 Sm ðμ, νÞ ^ m ρ Em ðρÞ + B B 0 1n 0 n 2n 0 n n n

(5:78)

∞ 2n +1 h i  X X      ^Sm ρ Em ðρÞ + ^Sm ρ F m ðρÞ Sm μ , ν0 Sm ðμ, νÞ 0 1n 0 n 2n 0 n n n

(5:79)

n=1 m=1

^b ðr; r0 Þ = υ

n=1 m=1

^s ðr, r0 Þ = υ

n=1 m=1

^e ðr, r0 Þ = υ

∞ 2n +1 X X n=1 m=1

    ^ m ρ F m ðρÞSm μ , ν0 Sm ðμ, νÞ. E 0 2n 0 n n n

(5:80)

Comparing the expansions (4.78)–(4.82) for the sphere with the corresponding expansions (5.76)–(5.80) for the ellipsoid we see that all we need to do is to replace: – the interior radial function rn with the interior Lamé function Enm ðρÞ – the exterior radial function r−ðn + 1Þ with the exterior Lamé function Fnm ðρÞ – the spherical surface harmonic Ynm ð^rÞ with the ellipsoidal surface harmonic Sm n ðμ, νÞ and insert the hat “˄” on the top of every coefficient of the ellipsoidal expansions. In the ellipsoidal system we have ∂ 1 ∂ = ^r · ∇r = ∂n hρ ∂ρ

(5:81)

∂ appears in both and since in the relative transmission conditions the operator ∂n  sides of the equations, the factors 1 hρ are eliminated; thus we only need to replace ∂ ∂ ∂r by ∂ρ. Using (5.74) and (5.75) and the ellipsoidal addition theorem

Pn ð^r · ^r0 Þ =

2n +1 X

 4π 1 m  Sn μ0 , ν0 Sm n ðμ, νÞ m 2n + 1 γn m=1

(5:82)

where γm n are the normalizing constants of the surface ellipsoidal harmonics, we obtain the conditions below. On the interface ρ = c1 we have the conditions     1 4π Fnm ðc1 Þ m   m m ^ ρ ð c Þ C ρ + E E 1 1n 0 0 n n σ c γm n 2n + 1     ^ m ρ + F m ðc1 ÞF ^m ρ = Enm ðc1 ÞF (5:83) 1n 0 n 2n 0 and

108

Chapter 5 The ellipsoidal model

    m   ′ 1 4π Fm m n ðc1 Þ m ^ En ρ0 + En ′ðc1 ÞC1n ρ0 σf γm n 2n + 1   m   ^m ^m ′ ′ = Em n ðc1 ÞF1n ρ0 + F n ðc1 ÞF2n ρ0 . On the interface ρ = f1 we have         ^ m ρ + F m ð f1 Þ F ^ m ρ = Em ðf1 ÞB ^ m ρ + F m ð f1 Þ B ^m ρ Enm ðf1 ÞF 1n 0 n 2n 0 n 1n 0 n 2n 0

(5:84)

(5:85)

and h h   m  i   m  i ′ ^m ′ ^m ′ ^m ′ ^m σf Em = σb E m n ðf1 ÞF1n ρ0 + F n ðf1 ÞF2n ρ0 n ðf1 ÞB1n ρ0 + F n ðf1 ÞB2n ρ0 . On the interface ρ = b1 we have         ^ m ρ + Fm ðb1 ÞB ^ m ρ = Em ðb1 Þ^Sm ρ + F m ðb1 ÞS^m ρ Enm ðb1 ÞB 1n 0 n 2n 0 n 1n 0 n 2n 0

(5:86)

(5:87)

and h h    i    i m′ ^Sm ρ + F m ′ðb1 Þ^Sm ρ . ^ m ρ + F m ′ðb1 ÞB ^m ρ ′ b b ð Þ B E ð Þ σ b Em = σ 1 s 1 n 1n 0 n 2n 0 n 1n 0 n 2n 0 Finally, on the interface ρ = s1 we obtain the transmission condition   m   m   ^m ^m Enm ðs1 Þ^Sm 1n ρ0 + Fn ðs1 ÞS2n ρ0 = Fn ðs1 ÞE2n ρ0

(5:88)

(5:89)

and the Neumann condition     m′ ^m ^m ′ Em n ðs1 ÞS1n ρ0 + F n ðs1 ÞS2n ρ0 = 0.

(5:90)

We rewrite the conditions (5.83)–(5.90) in matrix form on each of the interfaces ρ = c1 , f1 , b1 , s1 as follows: we define the Wronskian matrix " # Enm ðxÞ Fnm ðxÞ m ^ (5:91) Wn ð x Þ = m ′ En ′ðxÞ F m n ð xÞ which is invertible since the Lamé functions are linearly independent. Its inverse is given by qffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi " # m −1 x2 − h23 x2 − h22 − F m ′ Fnm ðxÞ n ð xÞ ^ Wn ð x Þ = . (5:92) 2n + 1 ′ Em − Enm ðxÞ n ð xÞ

5.4 EEG for the 3-shell model

We also define the parametric Wronskian matrix   " E m ð xÞ m σ n 1 ^ = σ1 m W n x; ′ E σ2 n ð xÞ σ 2

# Fnm ðxÞ . σ1 m ′ σ F n ð xÞ

109

(5:93)

2

Furthermore, for simplicity of notation we suppress for the moment the indices n ^m , F ^1 = C ^1 = F ^m , F ^2 = F ^m , B ^1 = B ^m , B ^2 = B ^ m , ^S1 = ^Sm , ^S2 = S^m , E ^2 = E ^m , and m, setting C 1n 1n 2n 1n 2n 1n 2n 2n m m σ σ 1 1 ^ ^ ^ ð xÞ = W ^ ðxÞ, Wðx; W n σ2 Þ = Wn ðx; σ2 Þ. Then, the conditions (5.83)–(5.90) takee the form 2   3  # ^1 ρ   C ^1 ρ 0 F 1 σ 0 c ^ c1 ; ^ ðc1 Þ 4 5 W   = W Enm ðρ0 Þ 4π ^ σ σ c f F 2 ρ0 m 2n + 1 γ "

(5:94)

n

"

 #  " ^   # ^1 ρ F 1 ρ0 B σ 0 f ^ ðf1 Þ ^ f1 ; W   =W   ^2 ρ ^2 ρ σb B F 0 0 "  # "  #   ^ ^S1 ρ B 1 ρ0 σb 0 ^ ^ Wðb1 Þ     = W b1 ; ^S2 ρ ^2 ρ σs B 0 0

(5:95)

(5:96)

and " ^ ðs1 Þ W

#  # " 0 S^1 ρ0   .   = ^2 ρ ^S2 ρ F ðs1 ÞE 0 0

(5:97)

The above notation leads to the expressions "

2   3 # ^1 ρ C ^1 0 F 1 ^ 5 = Mc ðn, mÞ4 m ρ E 4π n ð 0 Þ ^2 σc F m 2n + 1

"

"

"

#

^1 B ^ f ðn, mÞ =M ^2 B

^1 F ^2 F

γn

#

# " # ^S1 ^ ^ b ðn, mÞ B1 =M ^S2 ^2 B

^2 Fnm ðs1 ÞE 0

where

"

#

(5:98)

" # ^S1 ^ = Ms ðn, mÞ S^2

(5:99)

(5:100)

(5:101)

110

Chapter 5 The ellipsoidal model

2 σc m ′ m ′ − Enm ðc1 ÞF m n ðc1 Þ + σf En ðc1 ÞFn ðc1 Þ c c 2 3 ^ c ðn, mÞ = 4

M 2n + 1 1 − σc Em ðc1 ÞEm ′ðc1 Þ σf

− 1−

σc σf

n

n

′ Fnm ðc1 ÞF m n ðc1 Þ

m ′ Em n ðc1 ÞFn ðc1 Þ −

σc σf

3

′ Enm ðc1 ÞF m n ðc1 Þ

2

5

(5:102)

σ

f m m m′ m ′ f2 f3 6 − En ðf1 ÞF n ðf1 Þ + σb En ðf1 ÞFn ðf1 Þ ^

Mf ðn, mÞ = 4 σ 2n + 1 1 − f Em ðf ÞEm ′ðf Þ

σb

− 1−

σf σb

1

n

n

1

3

′ Fnm ðf1 ÞF m n ðf1 Þ

σf m ′ Em n ðf1 ÞFn ðf1 Þ − σb

′ Enm ðf1 ÞF m n ðf1 Þ

7 5

(5:103)

2 σb m m ′ ′ − Enm ðb1 ÞF m n ðb1 Þ + σs En ðb1 ÞFn ðb1 Þ b b 2 3 ^ 4

Mb ðn, mÞ = σ 2n + 1 1 − b Em ðb1 ÞEm ′ðb1 Þ σs

− 1−

σb σs

n

n

′ Fnm ðb1 ÞF m n ðb1 Þ σ

m m′ b m ′ Em n ðb1 ÞFn ðb1 Þ − σs En ðb1 ÞF n ðb1 Þ " # Enm ðs1 Þ Fnm ðs1 Þ ^ Ms ðn, mÞ = m ′ En ′ðs1 Þ F m n ðs1 Þ

3 5

(5:104)

(5:105)

and "

^ 11 ðn, mÞ m ^ 21 ðn, mÞ m

^ 12 ðn, mÞ m

#

^b ·M ^f ·M ^ c. ^s·M ^ ðn, mÞ = 1 M =M σc ^ 22 ðn, mÞ m

(5:106)

^2 T from (5.100), the vector In eq. (5.101), we substitute successively, the vector ½^S1 , S T T ^ 2  from (5.99) and the vector ½F ^1 , F ^ 2  from (5.98), to obtain the vector equation ^1, B ½B "

^2 Fnm ðs1 ÞE 0

#

2

  3 ^1 ρ C 0 ^ ðn, mÞ4 5 = Mg m ρ E 4π n ð 0 Þ 2n + 1

(5:107)

γm n

^ is given in (5.106). The above gives rise to two scalar equations: where M     4π Enm ρ0 m ^ ^ ^ ^ Fn ðs1 ÞE2 = m11 ðn, mÞC1 ρ0 + m12 ðn, mÞ 2n + 1 γm n

(5:108)

111

5.4 EEG for the 3-shell model

    4π Enm ρ0 ^ ^ 21 ðn, mÞC1 ρ0 + m ^ 22 ðn, mÞ 0=m . 2n + 1 γm n

(5:109)

  ^ m ρ coefficients are given by Hence, the C 1n 0

    ^ 22 ðn, mÞ 4π Enm ρ0 m m ^ . C1n ρ0 = − ^ 21 ðn, mÞ 2n + 1 γm m n

(5:110)

    ^ m ρ and F ^ m ρ in the Substituting (5.110) in (5.98) we obtain the coefficients F 1n 0 2n 0 form " # "  #   ^ ðn, mÞ m ^m ρ F 4π Enm ρ0 ^ − m^ 22 ðn, mÞ 1n 0 21 . (5:111) Mc ðn, mÞ   = ^m ρ 2n + 1 σc γm 1 F n 2n 0     ^m ρ ^ m ρ and B Next, we can insert (5.111) in (5.99) to obtain the coefficients B 0 1n 2n 0     ^m and finally eq. (5.100) gives the coefficients ^Sm 1n ρ0 and S2n ρ0 . The coefficients   ^ m ρ are obtained from eq. (5.108): E 2n 0       ^ 22 ðn, mÞ ^ 11 ðn, mÞm 4π Enm ρ0 m m ^ ^ − m12 ðn, mÞ E2n ρ0 = − m ^ 21 ðn, mÞ 2n + 1 γm m n Fn ð s 1 Þ   ^ ðn, mÞ 4π Enm ρ0 det M . (5:112) =− m m ^ 21 ðn, mÞ 2n + 1 Fn ðs1 Þ γn m Therefore, the values of the electric potential on the boundary ∂Ωs is given by   ^s s1 , μ, ν;ρ0 , μ0 , ν0 ^s ðr; r0 Þj∂Ωs = υ υ =−

∞ 2X n+1 X ^ ðn, mÞ 4π det M IEnm ðr0 ÞSm n ðμ, νÞ. ^ 2n + 1 γm n m21 ðn, mÞ n=1 m=1

(5:113)

Using the reduction Enm ðxÞ ! xn e!s

Fnm ðxÞ !

1

e!s xn + 1

n−1 ′ Em n ð xÞ ! nx e!s

′ Fm n ð xÞ ! − e!s

n+1 xn + 2

(5:114) (5:115) (5:116) (5:117)

where the rotation e ! s means: ‘as the ellipsoid degenerates to the sphere’, we see ^ f, M ^ b, M ^ s reduce to the corresponding matrices Mc , ^ c, M that all the matrices M Mf , Mb , Ms for the sphere. The same is true for the RHS of (5.98) and (4.103):

112

Chapter 5 The ellipsoidal model

"

^1 F ^2 F

#

" !

e!s

F1 F2

# .

(5:118)

Therefore, the forward problem for the ellipsoid is reduced to the forward problem for the sphere. In particular, the solution for the ellipsoid follows from the matrix " # ^ 12 ðn, mÞ ^ 11 ðn, mÞ m m 1 ^ ^ ^ ^ ^ ðn, mÞ = (5:119) = M M s · Mb · Mf · Mc σc ^ 22 ðn, mÞ ^ 21 ðn, mÞ m m ^ 22 =m ^ 21 appearing in (5.111). Then we can obtain the ratio m ^m , F ^ m ^ m ^ m ^ m ^m ^m Finally, having the coefficients C 1n 1n , F2n , B1n , B2n , S1n , S2n , we obtain the mo^f , υ ^b , υ ^s and from these we obtain the dipolic potentials ^c , υ nopolic potentials υ ^f , u ^b , u ^s . ^c , u u

5.5 Direct solution of the EEG problem We can also solve the EEG problem in ellipsoidal geometry for the 3-shell model via a direct solution of the 8 × 8 algebraic system that is generated by the transmission and boundary conditions. In fact, using the expansion (5.8) we express the electric potential in Ωc as ∞ 2n +1 X X   1 1 Q · ∇r0 tnm ρ0 , μ0 , ν0 IEnm ðρ, μ, νÞ + 4πσc jr − r0 j n = 1 m = 1   ∞ 2 n + 1 XX    m 1 m tnm ρ0 , μ0 , ν0 + Q · ∇ IE ρ , μ , ν ð ρ Þ IEnm ðρ, μ, νÞ, ρ < c1 = I r 0 0 0 n 0 n m γ σ c n n=1 m=1

  ue ρ, μ, ν;ρ0 , μ0 , ν0 =

(5:120) where the n = 0 term is annihilated by the operator Q · ∇r0 . Furthermore, the electric potentials within the shells take the form ∞ 2n +1   X   X  rnm ρ0 , μ0 , ν0 IEnm ðρ, μ, νÞ uf ρ, μ, ν;ρ0 , μ0 , ν0 = n=1 m = 1  m + qm c1 < ρ < f1 n ρ0 , μ0 , ν0 IFn ðρ, μ, νÞ,

(5:121)

∞ 2X n+1    X   knm ρ0 , μ0 , ν0 IEnm ðρ, μ, νÞ ub ρ, μ, ν;ρ0 , μ0 , ν0 = n = 1 m= 1  m + mm f1 < ρ < b 1 n ρ0 , μ0 , ν0 IFn ðρ, μ, νÞ,

(5:122)

∞ 2n +1  X   X   gnm ρ0 , μ0 , ν0 IEnm ðρ, μ, νÞ us ρ, μ, ν;ρ0 , μ0 , ν0 = n = 1 m= 1  m + hm b1 < ρ < s1 n ρ0 , μ0 , ν0 IFn ðρ, μ, νÞ,

(5:123)

5.5 Direct solution of the EEG problem

113

Finally, the exterior electric potential has the expansion ∞ 2m +1 X   X   fnm ρ0 , μ0 , ν0 IFnm ðρ, μ, νÞ, ue ρ, μ, ν;ρ0 , μ0 , ν0 =

ρ > s1 .

(5:124)

n=1 m=1

Introducing (5.76)–(5.80) in the transmition conditions uc ðρ, μ, ν; r0 Þ = uf ðρ, μ, ν; r0 Þ, σc

∂ ∂ uc ðρ, μ, ν; r0 Þ = σf uf ðρ, μ, ν; r0 Þ, ∂n ∂n uf ðρ, μ, ν; r0 Þ = ub ðρ, μ, ν; r0 Þ,

σf

ρ = c1

(5:126) (5:127)

ρ = f1

ρ = b1

∂ ∂ ub ðρ, μ, ν; r0 Þ = σs us ðρ, μ, ν; r0 Þ, ∂n ∂n us ðρ, μ, ν; r0 Þ = ue ðρ, μ, ν; r0 Þ,

(5:125)

ρ = f1

∂ ∂ uf ðρ, μ, ν; r0 Þ = σb ub ðρ, μ, ν; r0 Þ, ∂n ∂n ub ðρ, μ, ν; r0 Þ = us ðρ, μ, ν; r0 Þ,

σb

ρ = c1

(5:128) (5:129)

ρ = b1

ρ = s1

(5:130) (5:131)

as well as in the boundary condition ∂ us ðρ, μ, ν; r0 Þ = 0, ∂n

ρ = s1

(5:132)

and utilizing the orthogonality conditions of the surface harmonics Sm n ðμ, νÞ we end up with a linear system of order 8 for the 8 unknown coefficients tnm ðr0 Þ, rnm ðr0 Þ, m m m m m qm n ðr0 Þ, kn ðr0 Þ, mn ðr0 Þ, gn ðr0 Þ, hn ðr0 Þ, f n ðr0 Þ. Setting Dm n ðr0 Þ =

 1  Q · ∇r0 IEnm ðr0 Þ m γn

we can write this 8 × 8 system in the following form:   1 m m m m m m ð r ÞI ð c Þ tnm + Dm 0 n 1 En ðc1 Þ = rn En ðc1 Þ + qn Fn ðc1 Þ σc n        m  1 m 1 m ′ Dn ðr0 Þ Inm ðc1 Þ ′Enm ðc1 Þ + tnm + Dm ð r ÞI ð c Þ E ð c Þ σc 0 n 1 1 n σc σc n     m  ′ = σf rnm Enm ðc1 Þ ′ + qm n Fn ðc1 Þ m m m m m rnm Enm ðf1 Þ + qm n Fn ðf1 Þ = kn En ðf1 Þ + mn Fn ðf1 Þ     m      m  ′ = σb knm Enm ðf1 Þ ′ + mm ′ σf rnm Enm ðf1 Þ ′ + qm n Fn ðf1 Þ n Fn ðf1 Þ

(5:133)

(5:134)

(5:135) (5:136) (5:137)

114

Chapter 5 The ellipsoidal model

m m m m m knm Enm ðb1 Þ + mm n Fn ðb1 Þ = gn En ðb1 Þ + hn Fn ðb1 Þ     m      m  ′ = σs gnm Enm ðb1 Þ ′ + hm ′ σb knm Enm ðb1 Þ ′ + mm n Fn ðb1 Þ n Fn ðb1 Þ m m m gnm Enm ðs1 Þ + hm n Fn ðs1 Þ = fn Fn ðs1 Þ    m  gnm Enm ðs1 Þ ′ + hm n Fn ðs1 Þ = 0.

(5:138) (5:139) (5:140) (5:141)

Solving the system (5.134)–(5.141) in the way described in Section 5.4 or in some other suitable manner, we arrive at the potentials uc ðρ, μ, ν; r0 Þ = uf ðc1 , μ, ν; r0 Þ +

∞ 2X n+1 1 X Dm ðr0 ÞInm ðρ, c1 ÞIEnm ðρ, μ, νÞ, σc n = 1 m = 1 n

ρ 2 ½h2 , c1 Þ

(5:142)

uf ðρ, μ, ν; r0 Þ = ub ðf1 , μ, ν; r0 Þ +

∞ 2n +1 m X G2, n m 1 X Dn ðr0 ÞInm ðρ, f1 ÞIEnm ðρ, μ, νÞ, σf n = 1 m = 1 Gm 3, n

c1 < ρ < f1

(5:143)

ub ðρ, μ, ν; r0 Þ = us ðb1 , μ, ν; r0 Þ +

∞ 2n +1 m X G1, n m 1 X Dn ðr0 ÞInm ðρ, b1 ÞIEnm ðρ, μ, νÞ, σ b n = 1 m = 1 Gm 3, n

f1 < ρ < b 1

(5:144)

us ðρ, μ, ν; r0 Þ = ue ðs1 , μ, ν; r0 Þ +

∞ 2X n+1 X 1 m D ðr0 ÞInm ðρ, s1 ÞIEnm ðρ, μ, νÞ, Gm n n = 1 m = 1 3, n

b1 < ρ < s1

(5:145)

and ue ðρ, μ, ν; r0 Þ =

∞ 2n +1 X X 1 1 m Inm ðρÞ m IE ðρ, μ, νÞ, m Dn ðr0 Þ m m S G3, n In ðs1 Þ n n=1 m=1 n

ρ > s1 .

In the above expressions, the various constants are defined as follows:   Cnm = Enm ðc1 Þ Enm ðc1 Þ ′c2 c3   ^ m = Em ðf1 Þ Em ðf1 Þ ′f2 f3 F n n n  m  m ′ Bm n = En ðb1 Þ En ðb1 Þ b2 b3   m m ′ Sm n = En ðs1 Þ En ðs1 Þ s2 s3

(5:146)

(5:147) (5:148) (5:149) (5:150)

^ m introduced in order to be distinguished from the Lamé funcwith the notation F n tion Fnm

5.5 Direct solution of the EEG problem

ðy Inm ðx, yÞ = Inm ðxÞ − Inm ðyÞ =

 x

dt 2 qffiffiffiffiffiffiffiffiffiffiffiffi2ffipffiffiffiffiffiffiffiffiffiffiffiffi2ffi. Enm ðtÞ t 2 − h3 t 2 − h 2

115

(5:151)

The constants Gm i, n , i = 1, 2, 3, are defined as follows:   1 1 m m G1, n = σb + ðσb − σs Þ In ðb1 , s1 Þ + m − m Bm (5:152) n Sn Bn !   m 1 1 ^m m G2, n = σf + σf − σb In ðf1 , s1 Þ + m − F ^m n Sn F n   1 1 + ðσb − σs Þ Inm ðb1 , s1 Þ + m − m Bm n Sn Bn     σ f − σ b ðσ b − σ s Þ m 1 1 m ^m + In ðf1 , b1 Þ In ðb1 , s1 Þ + m − m Bm n Fn (5:153) σb Sn Bn     m 1 1 Cm = σ + σ − σ ð c , s Þ + − I Gm c c 1 1 f 3, n n Sm Cnm n n !   m 1 1 ^m + σf − σb In ðf1 , s1 Þ + m − F ^m n Sn F n   1 1 m + ðσb − σs Þ In ðb1 , s1 Þ + m − m Bm n Sn Bn !    σ c − σf σf − σb m 1 1 ^m m m + In ðc1 , f1 Þ In ðf1 , s1 Þ + m − F C ^m n n σf Sn F n 

   σc − σf ðσb − σs Þ m 1 1 m In ðc1 , b1 Þ Inm ðb1 , s1 Þ + m − m Bm n Cn σb S n Bn     σf − σb ðσb − σs Þ m 1 1 m ^m + In ðf1 , b1 Þ In ðb1 , s1 Þ + m − m Bm n Fn σb Sn Bn !    σc − σf σf − σb ðσb − σs Þ m 1 m + In ðc1 , f1 Þ In ðf1 , b1 Þ − ^m σf σb F n +

  1 1 ^m m Inm ðb1 , s1 Þ + m − m Bm n Fn C n . Sn Bn

(5:154)

The parenthesis on the right hand side of eq. (5.152) represents the expression "   m #   Fnm ðb1 Þ ′ Fn ðs1 Þ ′ 1 1 1 m   −  In ðb1 , s1 Þ + m − m = (5:155) Sn Bn 2n + 1 Enm ðb1 Þ ′ Enm ðs1 Þ ′

116

Chapter 5 The ellipsoidal model

and similarly for the corresponding parentheses in the formulae (5.153) and (5.154). Note the effect of the different conductivities in the expressions (5.152)–(5.154) m m besides the leading terms of the conductivity σb in Gm 1, n , σf in G2, n and σc in G3, n , all other terms vanish when the conductivities become equal. In fact, since the conductivities of the cerebrum, the cerebrospinal fluid and the scalp are close to each other, while the conductivity of the skull is rather different, we can assume the following relations already mentioned at the end of Chapter 1: 1 1 σc ≈ σf ≈ σ b ≈ σs . 3 12

(5:156)

Thus, the only safe assumption is the vanishing of the terms that are proportional to σc − σs but this difference does not appear in any term since the conductivity differences concern only neighboring shells. The terms of degree 1 and 2 of the exterior electric potential in Cartesian form are given by ue ðr; r0 Þ =

3 1 X Qm Inm ðρÞ xm m Vs m = 1 Gm 3, 1 In ðs1 Þ

" 3 X 5 1 I 1 ðρÞ IE21 ðrÞ   − Qm x0m 1 12 ′ 6Vs ðΛs − Λ s Þ m = 1 G3, 2 I2 ðs1 Þ Λs Λs − s2m 1 I 2 ð ρÞ IE2 ðrÞ  2  − 2 22 G3, 2 I2 ðs1 Þ Λ′s Λ′s − s2m +

#

3   5 X Qi x0j I i + j ð ρÞ

i2+ j xi xj + O el3 Vs i, j = 1 G6 − i − j s2 + s2 I2 ðs1 Þ i≠j

3, 2

i

(5:157)

j

where Vs =

4π s1 s2 s3 3

and Vs is the volume of the ellipsoidal ρ = s1 ; also   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λs 1 2 = s21 − h2 + h23 ± h41 + h22 h23 3 Λ′s

(5:158)

(5:159)

are the roots of the equation 3 X

1 = 0. − s2m Λ s m=1

(5:160)

5.6 MEG for the 3-shell model

117

5.6 MEG for the 3-shell model The exterior magnetic field for the 3-shell model with an isolated dipole excitation located at r0 with dipole moment Q is obtained from (2.55) in the form ^0 1 μ Q × ∇r0 4π jr − r0 j ð   ^ μ σc uc ðρ, r0 Þ − σf uf ðρ, r0 Þ Dðρ, rÞdsðρÞ − 0 4π

BðrÞ =

Sc

−

ð

^0 μ 4π



 σf uf ðρ, r0 Þ − σb ub ðρ, r0 Þ Dðρ, rÞdsðρÞ

Sf

−

ð

^0 μ 4π

ðσb ub ðρ, r0 Þ − σs us ðρ, r0 ÞÞDðρ, rÞdsðρÞ Sb

ð

^ μ − 0 4π

σs us ðρ, r0 ÞDðρ, rÞdsðρÞ

(5:161)

Ss

where ^ ðρÞ × ∇ρ Dðρ, rÞ = n

1 jr − ρj

(5:162)

is the dipole field generated at the surface point ρ with the normal to the surface ^ ðρÞ. unit moment n On the interfaces, we have the following continuity conditions: uc ðρ, r0 Þ = uf ðρ, r0 Þ,

ρ 2 Sc

(5:163)

uf ðρ, r0 Þ = ub ðρ, r0 Þ,

ρ 2 Sb

(5:164)

ub ðρ, r0 Þ = us ðρ, r0 Þ,

ρ 2 Ss

(5:165)

which we can insert in the expression (5.161) to rewrite this expression in the form BðrÞ =

 ^0 ^  r − r0 μ μ + 0 σf − σc Ic ðrÞ Q× 3 4π 4π jr − r0 j

+

 ^0  ^ ^ μ μ μ σb − σf If ðrÞ + 0 ðσs − σb ÞIb ðrÞ − 0 σs Is ðrÞ 4π 4π 4π

(5:166)

118

Chapter 5 The ellipsoidal model

where ð Ii ðrÞ =

^ ðρ′Þ × ui ðρ′, r0 Þn

Si

r − ρ′ dsðρ′Þ 3 j r − ρ′ j

(5:167)

for i = c, f , b, s. The explicit calculations for the above expressions are very complicated and for this reason we restrict our interest to the terms of order 1 and 2, that is, to the dipole and quadrupole terms, of the expansion of B in ellipsoidal harmonics. Long and tedious calculations lead to the expansion



1

r − ρ′

^ ð ρ′ Þ × ^ ðρ′Þ × ∇ρ′ =n n 3

jr − ρ′j ρ′ = c1 jr − ρ′j ρ′ = c1 X  3 5 X c m ′ ′ ′ ′ = lc1 ðμ′, ν′Þ βcm ðrÞSm ð μ , ν Þ + δ ð r ÞS ð μ , ν Þ + Oðel′3 Þ m 1 2 m=1

(5:168)

m=1

where we recall that the notation Oðel3 Þ represents ellipsoidal terms of order greater or equal to 3 and ρ′ = c1 defines the surface Sc . The coefficients in (5.168) are given by the following formulae: βcm = 3

c1 c2 c3 hm ~ 1 ðρÞ, ^ m  rÞ ×. H ðx h1 h2 h3 c m δc1 = −

m = 1, 2, 3

c1 c2 c3 ~ ~ c ðrÞ Λc ×. F 3ðΛc − Λ′c Þ

c1 c2 c3 ~ ~ c ðrÞ Λ′c ×. F 3ðΛc − Λ′c Þ   c1 c2 c3 c2 c1 c ~ c ðrÞ ^2 + x ^ 1 ×. F ^1  x ^2  x δ3 = x c2 h1 h2 h23 c1   c1 c2 c3 c3 c1 ~ c ðrÞ ^ ^ ^ ^ ×. F δc4 =  x +  x x x 1 3 3 1 c3 h1 h22 h3 c1   c1 c2 c3 c3 c ~ c ðrÞ ^3 + 2 x ^ 2 ×. F ^2  x ^3  x δc5 = 2 x c3 h1 h2 h3 c2 δc2 =

(5:169) (5:170) (5:171) (5:172)

(5:173)

(5:174)

where the cross-dot product is taken in the sense ðα  bÞ ×. ðc  dÞ = ðα × cÞðb · dÞ

(5:175)

5.6 MEG for the 3-shell model

119

and the related polyadics are defined below: 3 X ^m ^m  x x

~c = Λ

m=1

Λ~′c =

Λc − c2m

3 X ^m ^m  x x m=1

Λ′c − c2m

1 2 ~~ ~ c ðrÞ = − IF2 ðrÞ Λ ~ c + IF2 ðrÞ Λ~′c + 15r  r:H F 2 ðρÞ Λc − Λ′c Λc − Λ′c

~ 1 ð ρÞ = H

3 X

(5:176)

(5:177)

(5:178)

^m  x ^m I1m ðρÞx

(5:179)

^i  x ^j  x ^i  x ^j . I2i + j ðρÞx

(5:180)

m=1

~~ ðρÞ = H 2

3 X i, j = 1 i≠j

As it was mentioned in Section 4.1, the constants  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λc h2 + h23 2 = c21 − 2 h41 + h22 h23 ± 3 3 Λ′c

(5:181)

are the roots of the equation 3 X

1 =0 Λ − c2m m=1 c

(5:182)

and correspond to the ellipsoid Sc . Furthermore, because of confocality the constants Λf , Λ′f , Λb , Λ′b , Λs , Λ′s corresponding to the surfaces Sf , Sb , Sc , respectively, satisfy the relations Λc − Λ′c = Λf − Λ′f = Λb − Λ′b = Λs − Λ′s

(5:183)

Λc − c2m = Λf − fm2 = Λb − b2m = Λs − s2m

(5:184)

Λ′c − c2m = Λ′f − fm2 = Λ′b − b2m = Λ′s − s2m

(5:185)

for m = 1, 2, 3. ~ f for the surface ~ f , Λ~′f , F Because of the relations (5.183)–(5.185), the polyadics Λ ~ ~ ~ ~ ~ ~ Sf , the polyadics Λb , Λ′b , Fb for the surface Sb and Λs , Λ′s , Fs for the surface Ss satisfy the following degeneracies: ~f = Λ ~b = Λ ~s = Λ ~ ~c = Λ Λ

(5:186)

Λ~′c = Λ~′f = Λ~′b = Λ~′s = Λ~′

(5:187)

120

Chapter 5 The ellipsoidal model

~f = F ~b = F ~s = F ~ ~c = F F

(5:188)

where for simplicity of notation we have dropped the subindices and use the sym~ ~ Λ~′, F. bols Λ, Keeping only ellipsoidal harmonics of degree n = 1 and n = 2, it is possible to obtain the following expansion, derived analytically in [27] uc ðc1 , μ′, ν′Þ =

3 X m=1

5 X

c m ′ m ′ Jm E1 ðμ ÞE1 ðν Þ +

m=1

Θcm E2m ðμ′ÞE2m ðν′Þ + Oðel′3 Þ

(5:189)

where c Jm =

3hm ^m , Am Q · x 4πh1 h2 h3 1

Θc1 = − Θc2 = Θci+ j =

m = 1, 2, 3

5 ~ A1 Q  r0 :Λ 4πðΛc − Λ′c Þ 2

5 A2 Q  r0 :Λ~′ 4πðΛc − Λ′c Þ 2 5hi hj

4πðh1 h2 h3 Þ

2

  ^ j + x0j x ^i Ai2+ j Q · x0i x

(5:190) (5:191) (5:192) (5:193)

with i, j 2 f1, 2, 3g and i≠j. Furthermore, " ! Enm ðc1 Þ m 1 σ f − σb m 1 1 ^m m m In ðc1 , s1 Þ + m + In ðc1 , f1 Þ In ðf1 , s1 Þ + m − F An = ^m n σf Gm Sn Sn F 3, n n   σb − σs m 1 1 m I ðc1 , b1 Þ In ðb1 , s1 Þ + m − m Bm + n σb n Sn Bn     σf − σb ðσb − σs Þ m 1 1 + In ðc1 , f1 Þ Inm ðb1 , s1 Þ + m − m σb σf Sn Bn    1 ^m (5:194) · Inm ðf1 , b1 Þ − m Bm n Fn Fn ^ m , Bm , Sm are given in for n = 1, 2 and m = 1, 2, ..., 2n + 1, where the definitions of Inm , F n n n ′ ′ (5.148)–(5.151) and Gm 3, n is given in (5.154). Inserting in (5.167), uc ðc1 , μ , ν Þ and the expansion for the dipole field, and using orthogonality, we obtain Ic ðrÞ =

3 X m=1

c m c Jm γ1 βm ðrÞ +

5 X m=1

c Θcm γm 2 δm ðrÞ + Oðel3 Þ.

(5:195)

121

5.6 MEG for the 3-shell model

In expanded form, this equation becomes Ic ðrÞ =

3 X 3c1 c2 c3 m=1

cm

^ ^ Am 1 ðQ · xm Þðxm  rÞ × . H1 ðρÞ

    2 Λc − c21 Λc − c22 Λc − c23 ~ ~ Λ ~ ×F − c1 c2 c3 A12 Q  r0 :Λ . ðrÞ 3ðΛc − Λ′c Þ     2 Λ′c − c21 Λ′c − c22 Λ′c − c23 ~ ðrÞ c1 c2 c3 A22 Q  r0 :Λ~′  Λ~′ ×. F + 3ðΛc − Λ′c Þ 2 3

  3 ^j + x ^j  x ^ i  c2i x ^ i + c2j x ^j ^i  x ^j  x ^i  x x X 6 7 c1 c2 c3 Ai2+ j 5 + Q  r0 :4 c c i j i, j = 1 i≠j

+ Oðel3 Þ.

(5:196)

The previous calculations summarize the steps that lead to the expression (5.196) for Ic ðrÞ. Following similar arguments we can evaluate the integrals If ðrÞ, Ib ðrÞ and Is ðrÞ. After these four integrals are evaluated and substituted in the expression (5.166), we then perform some further calculations and reductions that can be found analytically in reference [27]. In this way, we finally arrive at the expression BðrÞ =

 ^0  μ dc + d2f − d2c + d1b − d1f + d0s − d0b 4π 3 X IF21 ðρ, μ, νÞ ~ IF22 ðρ, μ, νÞ ~ ^j ^i  x I2i + j ðρÞxi xj x Λ′ − 15 · Λ− Λc − Λ′c Λc − Λ′c i, j = 1

! + Oðel3 Þ

(5:197)

where   ~ ðc1 Þ × r0 · N ~ ðc1 Þ dc = Q · M   ~ ðc1 Þ × r0 · N ~ 2 ðc1 Þ d2c = Q · M   ~ ðf1 Þ × r0 · N ~ i ðf1 Þ, i = 1, 2 dif = Q · M   ~ ðb1 Þ × r0 · N ~ i ðb1 Þ, i = 0, 1 dib = Q · M   ~ ðs1 Þ × r0 · N ~ 0 ðs1 Þ d0s = Q · M ~ ðc1 Þ = M

3 X i=1

^i ^i  x c2i x

(5:198) (5:199) (5:200) (5:201) (5:202) (5:203)

122

Chapter 5 The ellipsoidal model

~ ðf1 Þ, M ~ ðb1 Þ and M ~ ðs1 Þ. and similarly with M Furthermore, ~ ðc1 Þ = N

3 X i=1

~ 2 ðc1 Þ = N

3 X G62,−2 i i=1

~ 1 ðc1 Þ = N ~ 0 ðc1 Þ = N

^i ^i  x x c21 + c22 + c23 − c2i

^i ^i  x x G63,−2 i c21 + c22 + c23 − c2i

3 X G61, −2 i

(5:204)

(5:205)

^i ^i  x x 2 + c2 + c23 − c2i

(5:206)

3 X ^i ^i  x σs x 6 − i c2 + c2 + c2 − c2 G 2 3 i i = 1 3, 2 1

(5:207)

G6 − i c2 i = 1 3, 2 1

~ when the c’s are replaced by the f ’s, the b’s and similarly for the corresponding N’s and the s’s. m m If we insert the values of Gm 1, n , G2, n , G3, n in the expressions (3.148)–(3.151) we obkm appearing in the expression (3.96)–(3.99). tain the values of the coefficients Cin

5.7 Problems Problem 1

Solve the Dirichlet problem (5.3)–(5.5) to obtain equation (5.17).

Problem 2 Define the Cartesian expression (5.15) starting with the expression (5.14). Problem 3

Prove the asymptotic expansion (5.34)–(5.37).

Problem 4 Prove the expansion (5.40)–(5.46). Problem 5

Starting from the asymptotic condition   1 ue ðr, r0 Þ = O 2 , r ! + ∞ r

prove that all potentials uc , uf , ub , us , ue have ellipsoidal expansions with leading terms a dipole field (that is they start with n = 1). Problem 6

Provide the details that lead to the expansions (5.57) and (5.58).

5.7 Problems

123

Problem 7 Assuming the representation BðrÞ = ∇ × AðrÞ for the solenoidal field BðrÞ, prove that ð ^′ μ0 Q ρ u − ðr′Þ −σ AðrÞ = dsðr′Þ 4π jr − r0 j jr − r′j ρ′ = a1

where A has the ellipsoidal expansion. ^0 AðrÞ = − μ

∞ 2X n+1 X n=1 m=1

~ m ðr0 Þ IF m ðrÞ: Q·D n n

Problem 8 Prove that the vector field A defined in Problem 7 is solenoidal, that is, ∇ · A = 0. Problem 9 Problem 10

Prove the identity (5.70). Prove the form (5.72) for the orientation depended coefficients fnim ðμ, νÞ.

Problem 11 Verify that the electric potentials uc , uf , ub , us , ue given by (5.142), (5.143), (5.144), (5.145), (5.146), respectively, together with notations (5.133) and (5.147)–(5.154) satisfy the transmition conditions (5.125)–(5.131) and the boundary condition (5.132). Problem 12 Prove the following identifies:     3    X  Λ − a21 Λ − a22 Λ − a23 2 = Λ − a Λ − a2m , − i)   i 2 Λ − a2i m=1

i = 1, 2, 3.

ii) The same as in i) with Λ replaced by Λ′.         Λ − a21 Λ − a22 Λ − a23 ~ ~ Λ′ − a21 Λ′ − a22 Λ′ − a23 ~ iii) Λ  Λ− Λ′  Λ~′ 3ðΛ − Λ′Þ 3ðΛ − Λ′Þ 3 X 1 ^i  x ^i  x ^i . ^i  x x = I  I− 3 i=1

  ^j + x ^j  x ^ xi  a2j x ^ j + a2i x ^i ^i  x ^i  x ^j  x x iv) a2i + a2j ^i  x ^j  x ^i  x ^j + x ^j  x ^i  x ^j  x ^i =x

  ^j − a2j x ^i  x ^i − x ^i  x ^j ^j  x ^i  x ^j  x a2i x + a2i + a2j for i ≠ j.

124

Chapter 5 The ellipsoidal model

Problem 13 Starting from the integral term Ic ðrÞ as given by (5.167), utilize the relations given in the text to provide the details that lead to the expression (5.195). Problem 14 Establish the expansion (5.196). Problem 15 Write the expansions (5.195) and (5.196) for the integrals If ðrÞ, Ib ðrÞ, Is ðrÞ.

Chapter 6 Determination of localized current via EEG or MEG data In this chapter and the following chapter, we discuss algorithms that invert the recorded electroencephalographic (EEG) and magnetoencephalographic (MEG) data for the purpose of identifying the localized neuronal current that has generated these data. We consider the spherical and the ellipsoidal models of the brain–head system, and data that are obtained from EEG or MEG. The general idea is that we expand the data in the geometrically appropriate system of complete orthogonal eigenfunctions and compare the coefficients of this expansion with the knowing structure of the corresponding coefficients obtained from the solution of the related forward problem. This algorithm leads to a set of nonlinear, in general, equations for the structure of the unknown current. It is important to mention here the fundamental work of Albanese and Monk [1], which proved that it is impossible to identify completely a current within a conducting medium when the support of the current is three-dimensional. On the other hand, they demonstrated that it is possible to identify the support of a current that is a subset of two-, one- or zero-dimensional space. Indeed, as we have shown in [28] if we assume that the current is localized in a small sphere we can find the center of this sphere but it is impossible to calculate its radius. The inversion procedure identifies the sphere as an equivalent dipole located at its center. For this reason, we propose two different inversion techniques. The first concerns the case where the current consists of a number of isolated dipoles. The second assumes a continuously distributed current within the whole brain tissue, which, as we have demonstrated, is not uniquely identifiable. In this case, we propose an optimization method that seeks the current source that has minimum L2 - norm, frequently referred to as minimum energy-norm. Cases where the support of the current has low dimensionality and lies on a curve, or on a surface patch, are reported in refs. [30, 31, 33, 34, 38]. Furthermore, refs. [39, 41] discuss the conditions under which we can discriminate among single, double or multiple dipolic excitations, while refs. [36, 40] concern the influence of the boundary perturbations of the geometrical model on the inversion algorithms. The inversion methods that identify localized dipoles are presented in this chapter while the continuously distributed currents are discussed in Chapter 7.

https://doi.org/10.1515/9783110547535-007

126

Chapter 6 Determination of localized current via EEG or MEG data

6.1 EEG for isolated dipoles in a sphere We consider the three-shell spherical model with radii 0