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### Nova scientia vol.12 no.24 León may. 2020  Epub 02-Jul-2020

#### https://doi.org/10.21640/ns.v12i24.2371

Natural sciences and engineering

Importance of the class of harmonic sources in the identification of sources in the inverse electroencephalographic problema

Importancia de la clase de fuentes armónicas en la identificación de fuentes en el problema inverso electroencefalográfico

Abstract

Introduction:

In this work we discuss the relevance of the harmonic sources on the brain volume, which reproduce a given potential distribution on the scalp. These sources, apart from being a unicity class, they play a fundamental role in the resolution of the inverse problem of source identification with respect to any other sources class.

Method:

We make use of the volume conductor model for the head, in order to relate sources and reproduced measurements. The problem is rewritten as an operational formulation which allows to characterize the admissible measurements with respect to any considered sources class.

Results:

The admissible data set is characterized for the harmonic sources class on the brain volume. Also, the importance of this class in the context of the source estimation problem, with respect to any sources class, is shown. This is specifically illustrated considering the class of harmonic sources on a neighborhood of the cortex. Moreover, it is also shown the role the harmonic sources class on the brain plays when applying the Admissible Data Method (ADM) in order to get a general regularization scheme for the source estimation problem with respect to a unicity sources class.

Conclusion:

A general resolution methodology for the source estimation problem in the context of the inverse electroencephalographic problem is proposed, in which the harmonic sources class on the brain volume is crucial. Namely, given an arbitrary sources unicity class (for this inverse problem), a general method is developed for identifying the source in this class whose reproduced potential distribution best approximates a given potential measurement on the scalp. We consider sources classes in connection with the electrical activity near the cortex.

Keywords: volume conductor model; harmonic sources; equivalent sources; inverse electroencephalographic problema; admissible data method

Resumen

Introducción:

En este trabajo se estudiará la importancia de las fuentes armónicas en el volumen cerebral que reproducen una distribución de potencial en el cuero cabelludo determinada, las cuales además de ser clase de unicidad, tienen una importancia fundamental para la solución del problema inverso de identificación de fuentes en cualquier otra clase de fuentes, para reproducir la misma medición.

Método:

Se propone el modelo de medio conductor para relacionar las mediciones con las fuentes que las reproducen. Se reduce el problema a un planteamiento operacional que nos permitirá caracterizar las llamadas mediciones admisibles con respecto a la clase de fuentes que se considere.

Se caracteriza el conjunto de datos admisibles para la clase de fuentes armónicas en el volumen del cerebro y se ve la importancia de la clase de fuentes armónicas para la identificación de fuentes en otras clases de fuentes, lo cual se ejemplifica con la clase de fuentes armónicas en una vecindad de la corteza cerebral. También se muestra la importancia de la clase de fuentes armónicas en la aplicación del método de datos admisibles (MDA) a un esquema general de regularización para el problema de identificación de fuentes pertenecientes a clases de unicidad.

Conclusión:

Se propone una metodología general de resolución del problema inverso electroencefalográfico de identificación de fuentes, haciendo uso de la clase de fuentes armónicas en el volumen cerebral. Dada una clase arbitraria F de fuentes con la propiedad de unicidad para la solución del problema inverso, se desarrolla un método general para la identificación de la fuente en F que mejor aproxima una medición del potencial sobre el cuero cabelludo.

Palabras clave: modelo de medio conductor; fuentes armónicas; fuentes equivalentes; problema inverso electroencefalográfico; método de datos admisibles

Introduction

There are many works in the specialized literature dealing with the electrical activity source estimation problem in the brain, starting from potential distribution measurements on the scalp, corresponding to instantaneous electroencephalographic (EEG) measurements (Munck, Van Dik, & Spekreijse, 1988), (Amir, 1994), (El Badia & Ha Duong, 1998), (Fraguela Collar, Morín Castillo, & Oliveros Oliveros, 2008), (Morín Castillo, et al., 2013), (Fraguela Collar, Oliveros Oliveros, Morín Castillo, & Conde Mones, 2015) and references therein. Roughly speaking, this problem consists on identifying or estimate the source on the brain volume, including location and description, which yields the measured electric potential in form of electroencephalographic signal (EEG). These works make use of the volume conductor model, justified in (Sarvas, 1987), (Plonsey & Fleming, 1969). From a mathematical point of view, some drawbacks arise from the kind of models, partially but not completely solved in the literature.

This current work is mainly devoted to highlight the importance of the harmonic sources on the brain volume in the context of the previously mentioned source estimation problem. This kind of sources has no physiological meaning, since they are distributed over the whole brain volume and it is a well-known fact that EEG basically reflects the electrical activity close to the cortex at a macro spatial scale (regardless of whether or not it is influenced by any other inner sources) (Nunez, Nunez, & Srinivasan, 2019). However, from a mathematical perspective, this sources class plays a fundamental role in the sources characterization with respect to any other sources class, especially the ones relevant from a physiological point of view, as the multipoles, the sources concentrated in the cortex, or the spatially piecewise constant sources. It is worth to mention that these classes, commonly used in this context, have empty interior, which turns to be a serious drawback in the application of optimization methodologies.

We make emphasis on the unicity property for the sources class, an important requirement in the identification problem, which seems to be overlooked in previous related works. On the other hand, a big part of this manuscript is dedicated to show how the harmonic sources class on the brain volume is used to identify sources (belonging to a unicity class) which reproduce a potential measurement on the scalp.

In spite of an EEG measurement is given on a finite number of electrodes, we consider the potential measurement is known on the scalp. The interpolation problem consisting in extend this kind of data to the whole scalp is out of the scope of the present article.

Here we focus on showing the general methodology. Therefore, we choose a simple geometric model for the head, consisting in two concentric spheres modeling the splitting surfaces between the brain and the rest of the head. In this case, this makes possible the use of explicit analytic expressions for the solutions of certain contour problems. In case of requiring more realistic models, more complex geometries (with more layers and more involved surfaces) are mandatory, and the contour problems need to be numerically solved, in general.

Finally, since the source estimation problem is not well posed, a regularization algorithm is required in order to minimize the error sensitivity on the potential measurement (the problem of lack of uniqueness is easily solved by choosing appropriate unicity sources classes) (Kirsch, 2013). Usually, iterative methods are used in this context. However, we decided to use a different approach: the “Admissible Data Method” (Hernandez Montero, Fraguela Collar, & Henry, 2019). This methodology allows to clarify a priori if a certain sources class is appropriate to identify a given measurement.

We apply the previously mentioned identification methodology to the specific case of sources supported and harmonic on a neighborhood of the cortex, which turn to be a more natural and convenient class than the above cited harmonic on the whole brain volume class. This should be understood in the following sense: by natural we mean an equivalent mathematical source which both reproduces the measurement and it is concentrated at the biological active zone. As it has been said above, the EEG reflects synaptic activity occurring near the cortex (Nunez, Nunez, & Srinivasan, 2019).

Method

The model and the Inverse Electroencephalographic Problem

In the simple volume conductor model (studied in (Sarvas, 1987) based on the results in (Geselowitz, 1967)), the brain is considered as conducting medium of electrical current, in which there is also a generating mechanism of other biological currents produced by neuronal activity, called impressed currents.

We will denote by Ω1 the region occupied by the brain, and by Ω1=S1 its border, corresponding to the cortex. σ1 denotes the Ohmic current conductivity (in a normal brain is considered to be constant and equal to the one corresponding to the saturated salt water (Nunez & Srinivasan, 2006)) and μ denotes the magnetic permeability. From the perspective of a macro spatial scale, the EEG measurements detect average synaptic source activity (Nunez, Nunez, & Srinivasan, 2019), so the constant conductivities reflect the average effect of microscopic spatial fluctuations. Additionally, we suppose that the electric field generated in the brain is due to their conductive properties as physical medium, and electrical sources originated by neuronal activity. Finally, u1 will be the electric potential in the brain volume Ω1. On the other hand, the rest of the head is considered also as a homogeneous conducting medium Ω2, with outer border Ω2 =S2 corresponding to the scalp. Its average conductivity, considered constant, is σ2. In this region there are no sources of electrical activity (see Fig. 1). Analogously, u2 will be the electric potential distribution in Ω2. We set Ω=Ω1Ω2.

In this way, other layers such as the skull or the spinal brain fluid, among others, are not considered. The outer medium (outside the head) is supposed to have a vanishing conductivity. Hence, the simple quasi-static model describes the behavior of potentials u1 and u2, in the following way:

-σ1u1=f in Ω1, (1)

u2=0  in Ω2, (2)

u1=u2 on S1, (3)

σ1u1n1 =σ2u2n1 on S1, (4)

u2n2xΩ2=0  on S2, (5)

Where uinj denotes the normal derivative of ui on Sj with respect to the normal unitary vector nj, outer to Ωj, i,j=1,2. Here, the forward problem consists on, given a source f, finding the potentials u1 and u2. Thus, the corresponding inverse problem consists on, given a measurement V, finding an equivalent source f satisfying ( 1 ) - ( 5 ) in such a way that the corresponding potential simultaneously fulfills the condition u2S2=V.

We consider the Hilbert spaces L2Ωi, L2Si and L2Ω of square summable functions defined on Ωi, Si and Ω; respectively i=1,2. Analogously, we will denote by H1Ωi and H1Ω the corresponding Sobolev spaces i=1,2. We will denote also by H1Si the subspaces of L2Si given by the traces in Si of functions in H1Ωi i=1,2.

The superindex (1) will indicate subspace corresponding to functions orthogonal to constants (with respect to the appropriate inner product). Namely, if W is a function Hilbert space with inner product ,W, then W1=wW:w,1W=0, where 1 is the indicator function of the corresponding domain. Also, the superindex denotes the orthogonal complement in W of a set F in W, that is, F=wW:w,fW=0, fF.

In the neuronal activity source estimation problem, via model ( 1 ) - ( 5 ), the operative formulation requires to find an operator A which associates to each neuronal activity source f (in a certain class F) the measurement V. Namely, the inverse source estimation problem reduces to solve the operational equation

Af=V, (6)

Where V=u2S2 is the potential distribution measurement on S2.

Hence, solving the inverse source estimation problem means that, starting from the instantaneous electroencephalographic measurement V=u2S2, a source f “reproducing this measurement” could be find, i.e., substituting f in the model ( 1 ) - ( 5 ), there exists a unique solution u1f,u2f  satisfying u2fS2=V.

Given a class F of functions defined on the brain, the image of F by the operator A will be called admissible data set. This is exactly the set of measurements which could be reproduced by sources in the class F. Furthermore, the class F will be said to be a unicity class if operator A restricted to F is injective. In what follows the admissible data set associated to a sources class F will be denoted by MF:

AF=MF. (7)

In (Fraguela Collar, Oliveros Oliveros, Morín Castillo, & Conde Mones, 2015) was proved that operator A is compact from L21Ω1 to L21S2. In addition, in Theorem 2.3 was shown that KerA=H1Ω1, where HΩ1 is the closed subspace in L2Ω1 of harmonic functions. From this result yield the following conclusions, which will be important in what follows:

1. If the desired source is required to be in H1Ω1, then the unicity theorem for the solution of the inverse problem corresponding to the operational equation ( 6 ) is fulfilled.

2. If the source f which reproduces a given potential distribution V is required to be in a certain class F which satisfies the unicity condition for the solution of the inverse problem, and V can also be reproduced by hH1Ω1 (note that each one uniquely reproduces V in their own class. Certainly, there is no unique representation in L2(Ω1)), then h is the orthogonal projection of f on H1Ω1, independently of the chosen class F.

We note that another unicity class for the source estimation problem was obtained in (Fraguela Collar, Oliveros Oliveros, Morín Castillo, & Conde Mones, 2015) consisting on certain kind of piecewise constant sources.

Operational formulation of the inverse problem

In order to solve the source estimation problem for certain unicity classes in the context of the volume conductor model ( 1 ) - ( 5 ), we need to reduce the inverse problem to an equivalent operational formulation. In general, solving this inverse problem in a realistic geometry is a quite involved mathematical problem. For the sake of simplicity, in order to explain our methodology more easily, we will consider a simple geometric model for the head.

We consider two concentric spheres. The interior of the inner sphere S1, of radius R1>0 corresponds to the brain volume Ω1, and the interior of the outer sphere S2, of radius R2>R1, corresponds to the whole head Ω and outlines the region Ω2 corresponding to the spherical crown (see Fig. 1 and Fig. 2).

This simple spherical model allows us to build explicit analytical expressions for the solutions of the volume conductor model ( 1 ) - ( 5 ) in terms of the Fourier series with respect to classical orthonormales bases, which eases the qualitative analysis, and also constitutes the basis of an algorithm for the numerical resolution of the forward and inverse problems.

We start from assuming that the source is in a unicity class F in L21Ω1. Note that, in L21Ω1, the forward problem ( 1 ) - ( 5 ) has an unique solution in the Sobolev space H11Ω, and hence its restriction to the border S2 could be considered in the sense of Sobolev traces on S2, (Mijailov, 1978).

Next, we introduce some contour problems, which are auxiliary in solving the main inverse problem of source estimation, and associated to the corresponding solution. Note that these contour problems have no physical meaning; we use them for analyzing the operational formulation of the inverse problem. Their solutions will be interpreted in a weak sense. Fix a source fL21Ω1 and ψL21S1.

σ1Δw1=f in 1, (8)

w1n1=0 on S1,

Δv1=0 in 1, (9)

σ1v1n1=ψ on S1,

Δv2=0 in 2, (10)

σ2v2n1=ψ on S1,

v2n2=0 on S2,

Definition 1. The functions w1,v1H1Ω1, v2H1Ω2, satisfying

σ1Ω1w1ω1dx=Ω1fω1dx, ω1H1Ω1, (11)

σ1Ω1v1ω1dx=S1ψω1dx, ω1H1Ω1, (12)

σ2Ω2v2ω2dx=-S1ψω2ds, ω2H1Ω2, (13)

will be called weak solutions of problems ( 8 ), ( 9 ) and ( 10 ), respectively.

The following result holds.

Theorem 1. Problems ( 8 ), ( 9 ) and ( 10 ) have weak solution if and only if fL21Ω1 and ψL21S1 . In this case, the weak solutions are unique in their respective spaces H11Ω1 and H11Ω2, and the following inequalities hold:

w1H1Ω1C1fL2Ω1, (14)

v1H1Ω1K1ψL2S1, (15)

v2H1Ω2C2ψL2S1, (16)

Where constants C1, K1 and C2 do not depend on f and ψ, (see Mijailov, 1978).

Consequently, the following operators are well defined by using the weak solutions of problems ( 8 ), ( 9 ) and ( 10 ):

A0:L21Ω1L2S1,  A0f=w1S1, (17)

B0:L21S1L2S1,  B0ψ=v1S1, (18)

C0:L21S1L2S1,   C0ψ=v2S1, (19)

D0:L21S1L2S2,   D0ψ=v2S2. (20)

Making use of previous results and the compactness of the trace operators from H1Ω1 to L2S1, and from H1Ω2 to L2S1 and to L2S2, we conclude the compactness of the above operators, A0, B0, C0 and D0. It is convenient to consider the projection operators Pi:L2SiL21Si, defined by:

Piφi=φi-1SiSiφids, i=1,2, (21)

Where Si denotes the Lebesgue measure of Si. These averages could be considered as reference potentials, and these substractions are required in order to assure existence of the corresponding solutions. Thus we define the following operators:

A=P1A0:L21Ω1L21S1, (22)

B=P1B0:L21S1L21S1, (23)

C=P1C0:L21S1L21S1, (24)

D=P2D0:L21S1L21S2. (25)

By using these operators A,B,C and D, the solution of the inverse problem associated to the contour problem ( 1 ) - ( 5 ) can be obtained by solving the following system of operational equations:

Af+Bψ=Cψ, (26)

Dψ=V. (27)

Actually, equation ( 27 ) is equivalent to the Cauchy problem in region Ω2 for the Laplace operator with respect to the Cauchy data u2n2=0 on S2 and u2S2=V , where VL21S2. Once ψ is known, it can be substituted in ( 26 ), obtaining the operational equation:

Af=-B-Cψ. (28)

Thus, equation ( 28 ) give us f, so system ( 26 ) - ( 27 ) is equivalent to inverse source estimation problem. Finally, the operator which relates the source f with the measurement V is given by:

A=D-B-C-1A=-DB-C-1A. (29)

The following result can be found in (Fraguela Collar, Oliveros Oliveros, Morín Castillo, & Conde Mones, 2015).

Theorem 2. One has

Ker A=H1Ω1.

This theorem justifies remarks a) and b) above. An important corollary follows also from it: if fF reproduces a measurement V, and the harmonic function h0 is the unique harmonic source on Ω1 which also reproduces V, then h0 is the orthogonal projection of f on H1Ω1, independently of the chosen class F, and f-h0 is the component of f in Ker A.

Hence, for the class F to be an unicity class is necessary and sufficient that

F-FKer A=0,

Where F-F=f1-f2 : f1,f2F and 0 is the vanishing function. Note that the class F need not to be a linear space. Indeed, F is a unicity class if A (or A) restricted to F is injective. This happens if for any f1,f2F with Af1=Af2 one gets f1=f2, but Af1-f2=Af1-Af2, so Af1=Af2 is equivalent to f1-f2KerA, which has to be trivial.

The “Admissible Data Method” (ADM), as a regularization strategy (Hernandez Montero, Fraguela Collar, & Henry, 2019), can be applied to any sources unicity class F. Concretely, in the context of the inverse source estimation problem linked to the instantaneous contour problem ( 1 ) - ( 5 ), when the admissible data set MF is contained into the admissible data set corresponding to the harmonic sources H1Ω1, consists of the following steps:

1. Choose, if possible, a closed vector subspace or compact convex subset M0 of MF in such a way that the minimum distance problem to M0, would be well posed in L2S2.

2. Assuming fixed a measurement error and a measurement V~L2S2 (with error), find the admissible data V0M0 which attains the minimum distance to V~, and check that this distance has the magnitude order of the measurement error.

3. Compute the harmonic source h0 which reproduces the measurement V0.

4. Finally, use h0 to characterize the source fF which best reproduces the measurement V0 in the class F. In order to do that, one makes use of remark after Theorem 2: f-h0 is orthogonal to any harmonic function.

This methodology will be illustrated in the last section, in the case of the sources supported and harmonic in a neighborhood of the cortex.

Results

Reduction of the instantaneous source estimation problem to a Cauchy data problem on cortex

In order to find the restriction to S1 of the solution u2 on Ω2, we will study the following contour problem, which is a Cauchy problem for the Laplace equation on Ω2, with vanishing Neumann contour condition and Dirichlet data V:

Δu2=0 in 2 , (30)

u2n2=0 on S2, (31)

u2S2=V, (32)

Where V is the potential distribution electroencephalographic measurement V in S2 (the whole scalp) and u2 is the electric potential in 2, at a given time instant. We proceed by computing formally the solution and studying the convergence of the obtained series (Mijailov, 1978).

We consider the normalized spherical harmonics (Tijonov & Samarsky, 1980, pp. 765-778):

Ynmθ,ϕ=2n+14πn-m!n+m!Pnmcosθeimϕ, (33)

Where Pnmθ,ϕ are the associated Legendre polynomials, and θ0,π, ϕ0,2π,1n<,-n mn. Thus, V is given by

Vθ,ϕ=n=0m=-nnVnmYnmθ,ϕ, (34)

Where Vnm are the Fourier coefficients of V. It can be checked that the corresponding solution to problem ( 30 ) - ( 32 ) is given by

u2r,θ,ϕ=n=1m=-nnn+12n+1rR2n+n2n+1R2rn+1VnmYnmθ,ϕ, (35)

With convergence in L22. The solution of the Cauchy problem corresponds to restrict u2 to S1, which means to evaluate ( 35 ) at r=R1, obtaining

u2R1,θ,ϕ=n=1m=-nnn+12n+1R1R2n+n2n+1R2R1n+1VnmYnmθ,ϕ. (36)

In order to assure that the above expression for u2R1,θ,ϕ to be the trace in S1 of a function in H1Ω1, the series composed with the squares of Fourier coefficients of ( 36 ), multiplied by n, must converge (Mijailov, 1978, p. 221). Then, by using an elementary result on convergence of numerical series1 we obtain the following condition on the Fourier coefficients of V:

n=1nR2R12nm=-nnVnm2<. (37)

Next we find out what conditions on V must be imposed in order for ψ=σ2σ1u2rr=R1 to be L21S1. 2 It is easy to see that

ψ=σ2σ1u2rr=R1=σ2σ1n=0m=-nnnn+12n+1R1n-1R2n-R2n+1R1n+2VnmYnmθ,ϕ. (38)

Consequently, ( 38 ) must converge in L2(S1). Proceeding as before, we get the following result:

Theorem 3. One has ψL21S1 if and only if the Fourier coefficients Vnm satisfy

n=1n2R2R12nm=-nnVnm2<. (39)

All of this machinery allows us to convert the source estimation problem in the head into a source estimation problem in the brain from Cauchy data on the cortex. From now on we will suppose that the Fourier coefficients Vnm of V satisfy condition ( 39 ). From an operational point of view, it makes sense to solve equation ( 28 ) from ψ given by ( 38 ). That is

Af=Cψ-Bψ=g, (40)

Which could be viewed as the identification of the source f, which reproduces the potential g on the cortex S1, as it would be artificially measured.

In order to obtain an explicit expression for g=Cψ-Bψ, we make use of ( 38 ). Since operators B and C linear and continuous, it is enough to compute B(Ynmθ,ϕ) and C(Ynmθ,ϕ). For this purpose, in view of ( 18 ) and ( 19 ), we solve problems ( 9 ) and ( 10 ) with contour data Ynmθ,ϕ.

For B(Ynmθ,ϕ), we look for the solution v1 of the contour problem ( 9 ) of the form

v1r,θ,φ=ArnYnmθ,φ, (41)

And for C(Ynmθ,ϕ), we seek the solution v2 of the contour problem ( 10 ) of the form

v2r,θ,ϕ=Anrn+Bnr-n+1Ynmθ,ϕ. (42)

In this way we get

Bψ=σ2σ12n=1n+12n+1R2R1n+1R1R22n+1-1m=-nnVnmYnmθ,ϕ, (43)

Cψ=1σ2n=1nn+12n+11nR1R2n+1n+1R2R1n+1m=-nnVnmYnmθ,ϕ. (44)

From this expressions one easily gets

g=1σ1n=1n+12n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1m=-nnVnmYnmθ,ϕ. (45)

In summary, we conclude that, if the Fourier coefficients Vnm of measurement V satisfy condition ( 39 ), then the source estimation problem turns to solve equation Af = g, where g is given by ( 45 ). This problem can be interpreted as a source estimation problem in the brain, assuming it is isolated, and starting from the “measurement” g on the cortex. In what follows, we will apply this conclusion.

Identification of harmonic sources in the brain

In this section we consider a given instantaneous potential distribution V on the scalp, and we study under what conditions there exists an (unique) harmonic source f in the brain which reproduces V. By the way, we will find an explicit expression for f.

We start from the fact that any harmonic source f in 1 should take the following form:

fr,θ,ϕ=n=1m=-nnFnm2n+3R12n+3rnYnmθ,ϕ, (46)

Which is a development with respect to the orthonormal basis of spherical harmonics 2n+3R12n+3rnYnmθ,ϕ. Hence, the convergence of this series (in the L21 sense) is equivalent to the condition

n=1m=-nnFnm2<. (47)

Since operator A is linear and continuous, we get

Af=n=1m=-nnFnm2n+3R12n+3ArnYnmθ,ϕ. (48)

In order to compute

ArnYnmθ,ϕ, (49)

In view of ( 17 ), it is required to solve the associated problem

-σ1Δw1r,θ,ϕ=rnYnmθ,ϕ en Ω1, (50)

w1n1S1=0 sobre S1. (51)

Thus we look for a solution of the form

w1=vrYnmθ,ϕ, (52)

Where vr satisfies the equation

ddrr2dvrdr-vrnn+1=-rn+2σ1. (53)

The general solution of this equation takes the form

vr=v0r+vpr, (54)

Where v0 is solution for the associated homogeneous equation and vp is a particular solution. We propose a particular solution of the form

vpr=Lrk. (55)

From ( 53 ) we obtain

rk Lkk+1-nn+1=-1σ1rn+2. (56)

Thus we get the condition k=n+2, so

Ln2+5n+6-n2-n=-1σ1, (57)

And

L=-12σ12n+3. (58)

The function v0 satisfies the homogeneous equation

ddrr2dv0rdr-v0rnn+1=0, (59)

And must be bounded at 𝑟=0, so takes the form

v0r=Brn. (60)

From the contour condition ( 51 ) we get

dv0R1dr+dvpR1dr=0, (61)

So

B=n+2σ12n2n+3R12. (62)

In this way we have finally found that

Afr,θ,ϕ=n=1m=-nnFnm2n+3R1σ1n2n+3Ynmθ,ϕ. (63)

From ( 40 ) we must equate expressions ( 63 ) and ( 45 ). From the unicity of the coefficients, we obtain the Fourier coefficients Fnm in terms of the corresponding Vnm. Concretely, the final expression for f is:

f=n=1m=-nn12n+3R1nn+12n+32n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1Vnm2n+3R12n+3rnYnmθ,ϕ, (64)

Where the convergence must be in L21. Comparing expressions ( 48 ) and ( 64 ) we conclude the following result.

Theorem 4. There exists a biunivocal correspondence between the set of harmonic sources f in the brain 1 and the set of measurements V on the scalp (in L2S2) whose Fourier coefficients Vnm with respect to the orthonormalized spherical harmonics Ynmθ,ϕ satisfy:

n=1n3R2R12nm=-nnVnm2<. (65)

Furthermore, given a measurement V satisfying condition ( 65 ), the unique harmonic source in 1 which reproduces it is given by ( 64 ).

We recall that from the previous theorem it follows that H1Ω1 is a sources unicity class, and the admissible data set corresponding to this class of harmonic sources orthogonal to the constants, denoted by MH1Ω1 following notation ( 7 ), turns to be the set of measurements V satisfying ( 65 ).

Note that, from Theorem 2 and 4 it follows that for any sources class F in L21 the corresponding admissible data set MF is contained into the set of measurements V in L2S2 which satisfy condition ( 65 ). Note also that this condition ( 65 ) is a quite strong smooth requirement on the spatial distribution of the measurements V, for each time instant.

Importance of the class of harmonic sources in the sources identification methodology on arbitrary unicity classes. The particular case of harmonic sources in a neighbourhood of the cortex

In this section we assume as before a given instantaneous potential distribution V on the scalp, and we study the existence of a source in the brain reproducing it. For this source f we assume it is supported and harmonic in a neigbourhood of the cortex. We stablish conditions on V for its existence, and we will find the admissible data set for this class.

Let us first clarify what will be understood for “a neighbourhood of the cortex” (see Fig. 3): Fix R 0 satisfying 0<R0<R1. Ω10=0<rR0 represents the subcortical part of the brain, Ω11=R0<rR1 stands for a certain spherical neighbourhood of the cortex, and Ω2=R1<rR2 is the rest of the head as before. Analogously, the sphere S0=r=R0 is the border of Ω10, the sphere S1={r=R1} stands for the cortex and is the common border of Ω10 and Ω11, and S2={r=R2} stands for the scalp, where the measurements are taken, which is the outer border of Ω2.

It will be convenient to denote

r^=0  if 0<rR0 r   if R0<rR1. (66)

We consider the class H1Ω11 of sources in the brain which are supported and harmonic in Ω1 (vanishing outside). It is easy to see that this sources f can be represented in the form (Tijonov & Samarsky, 1980, p. 772):

fr,θ,ϕ=n=1m=-nnanm2n+3R12n+3-R02n+3r^n+bnm2n-1R0-2n+1-R1-2n+1r^-n+1Ynmθ,ϕ, (67)

Where the corresponding series of Fourier coefficients

n=1m=-nnanm2, n=1m=-nnbnm2, (68)

should converge.

Next, we solve the auxiliary contour problem ( 8 ), which in view of ( 17 ) means compute A( f ). Since operator A is linear and continuous, it is enough to compute Ar^nYnmθ,ϕ and Ar^-n+1Ynmθ,ϕ. In order to find:

Ar^nYnmθ,ϕ, (69)

it is necessary to solve the associated contour problem

-σ1Δw1r,θ,ϕ=r^nYnmθ,ϕ en Ω1, (70)

w1n1S1=0 on S1 (71)

whose solution is imposed to take the form

w1=arYnmθ,ϕ. (72)

It is easy to see that a (r) satisfies the equation

ddrr2dardr-arnn+1=-r^n+2σ1. (73)

If we denote by a0r, w10 and a1r, w11 the corresponding restrictions of a (r) and w 1 to Ω10 and Ω11, respectively, and consider the compatibility conditions on S 0,

w10r=R0=w11r=R0 on  S0, (74)

w10rr=R0=w11rr=R0 on S0, (75)

we arrive to functions a0r and a1r satisfy

ddrr2da0rdr-a0rnn+1=0, for 0<rR0, (76)

ddrr2da1rdr-a1rnn+1=-rn+2σ1, for R0<rR1, (77)

and the contour conditions at r=R0

a0R0=a1R0, (78)

da0drR0=da1drR0. (79)

a00<, (80)

da1drR1=0. (81)

Solving this problem for a0r and a1r gives us

a0r=1σ1-n+1n2n+12n+3R02n+3R12n+3+n+2R122n2n+3-R0222n+1rn, (82)

a1r=-n+1σ1n2n+12n+3R02n+3R12n+3+n+2R122σ1n2n+3rn-R02n+3σ12n+12n+3r-n+1-1σ122n+3rn+2. (83)

Thus, we finally get

Ar^nYnmθ,ϕ=-1+n1+R12R02n+3σ1n2n+12n+3R1n+3+R1n+2σ1n2n+3Ynmθ,ϕ. (84)

In a similar fashion one obtains

Ar^-n+1Ynmθ,ϕ=R1-n+1-R02R1-n-12σ1nYnmθ,ϕ. (85)

Equipped with ( 84 ) and ( 85 ), we can easily write the resulting formula for A ( f ):

Af=n=1m=-nnanm2n+3R12n+3-R02n+3-R02n+31+n1+R12σ1n2n+32n+1R1n+3+R1n+2σ1n2n+3+bnm2n-1R0-2n+1-R1-2n+1R1-n+1-R02R1-n-12σ1nYnmθ,ϕ. (86)

Finally, we must equate (86) and ( 45 ) and apply the coefficients unicity in order to obtain

anm2n+3R12n+3-R02n+3-1+n1+R12R02n+3n2n+12n+3R1n+3+R1n+2n2n+3+bnm2n-1R0-2n+1-R1-2n+1R1-n+1-R02R1-n-12σ1n=n+12n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1Vnm. (87)

Let us define αn, βn, and γn by

αn=2n+3R12n+3-R02n+3R1n+2n2n+3-1+n1+R12R02n+3n2n+32n+1R1n+3, (88)

βn=R12-R022-1n2nR1R1R1R02n-1-1, (89)

γn=n+12n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1. (90)

It is easy to see that

limnαnn-32=R12, (91)

limnβnn-12R0R1n=R12-R02R12R0, (92)

limnγnR2R1n=12R2R11+σ2σ1. (93)

Let us write ( 87 ) in the form

αnanm+βnbnm=γnVnm, (94)

or equivalently

anm+βnαnbnm=γnαnVnm. (95)

This yields

anm2+bnm2βn2αn2+2anmbnmβnαn=γn2αn2Vnm2. (96)

Finally, have the sum for any positive integer n and m from −n to n. If we take into account that f is harmonic in Ω1, we have the following conclusions:

1. n=1m=-nnanm2 converges, since anm are the Fourier coefficients of f.

2. Consider the series with general term the second term of the left side of ( 96 ), which is equivalent to n2R0R12nbnm2. Since n2R0R12nn0 and bnm are Fourier coefficients, this series converges.

3. Then the series with general term the right side of ( 96 ) also converges, whose terms are equivalent to n3R2R12nVnm2.

In this way, we arrive to the equivalent condition:

n=1n3R2R12nm=-nnVnm2<+, (97)

which turns to be the necessary and sufficient condition previously obtained for V to be reproducible by an harmonic source defined in Ω1. That is, if there is an harmonic source fH1Ω11 which reproduces V, then ( 97 ) is fulfilled. In other words, ( 97 ) is a necessary condition.

Next, we will characterize the admissible data set corresponding to harmonic sources in a neighborhood of the cortex. By Theorem 2 we have that, if fH1Ω11 and h0H1Ω1 both reproduce V, then f-h0H1Ω1, so

f-h0,rnYnmθ,ϕ=0, (98)

for n1, -nmn, where , denotes the inner product in L2Ω1. Now, from ( 67 ) and since r^nYnmθ,ϕ and r^-n+1Ynmθ,ϕ are orthogonal in L2Ω11, it follows

f,rnYnmθ,ϕ=anmR12n+3-R02n+32n+3. (99)

On the other hand, we have

h0=n=1m=-nn12n+3R1nn+12n+32n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1Vnm2n+3R12n+3rnYnmθ,ϕ, (100)

So

h0,rnYnmθ,ϕ=R1n+1nn+12n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1Vnm. (101)

Thus, from ( 98 ), ( 99 ) and ( 101 ) we get

anm=2n+3R12n+3-R02n+3nn+12n+1R1n+1R1R2n1-σ2σ1+R2R1n+1nn+1+σ2σ1Vnm. (102)

also, from ( 94 ) we get

bnm=γnβnVnm-αnβnanm. (103)

Making the computations, from ( 102 ) and ( 103 ) we get that anm and bnm are equivalent to n32R2R1nVnm and nR0R1nR2R1nVnm, respectively, and then anm2+bnm2 is equivalent to n3R2R12n1+1n2R0R12nVnm2. Consequently, ( 97 ) is necessary and sufficient for V to be reproducible by a source in H1Ω11.

However, these classes H1Ω11 are not unicity classes. In fact, if f 1 and f 2 are sources in this class generating the same measurement V in MH1Ω11, thus f1-f2KerA, so

f1-f2,rnYnm=0, 1n<,-nmn. (104)

Then, we get anm1=anm2, but nothing can be said about bnm1 and bnm2. From this it can be deduced that an unicity subclass contained in H1Ω11 is obtained considering harmonic functions given by series with respect only the first terms; i.e., those of the following form:

f=n=1m=-nnanm2n+3R12n+3-R02n+3r^nYnmθ,ϕ. (105)

Starting from this development, and computing their coefficients via the equation Af = g as before, we get

f=n=1m=-nn1+1n2+3nR11-R0R12n+32+1n-1n1R121-n-1R0R12n+3n32R1R2n1-σ2σ1+R2R12n+1nn+1+σ2σ12n+3R12n+3-R02n+3r-nVnmYnmθ,ϕ. (106)

From now, this subclass of harmonic sources will be denoted by

H01Ω11 (107)

Note that sources class (107) precisely matches the set of sources that equal harmonic sources in Ω1 when restricted at Ω11, and vanish at Ω10.

The following result summarizes these results.

Theorem 5. If 0<R0<R1, then the admissible data sets MH1Ω11 and MH01Ω11 coincide with the admissible data set MH1Ω1. These sets are the potential distributions measurements V on the scalp whose Fourier coefficients satisfy

n=1n3R2R12nm=-nnVnm2<. (108)

In addition, the class H01Ω11 is an unicity class, while H1Ω11 is not.

This last statement could result a little bit confusing; let us clarify this point. By making use of the unicity of extension of harmonic functions, one can identify a harmonic source in H01Ω11 with its harmonic extension to Ω1. In this sense, the classes H1Ω1 and H01Ω11 coincide as sets, while that H1Ω11 strictly contains H1Ω1.

Conclusions

In this article, a methodology for solving the inverse bioelectric source estimation problem in the brain, starting from instantaneous electroencephalographic measurements on the whole scalp, is proposed. We make use of a volume conductor model, in which the head and the brain are represented by means of concentric spheres outlining conducting layers with different but constant conductivities.

The main product of this work is the proposal of a general methodology for solving the inverse electroencephalographic source estimation problem. Given an arbitrary sources class F satisfying unicity condition for the inverse problem, a general method for determining the source in F which best approximates a potential measurement on the scalp is developed.

The process is basically as follows. First of all, one obtains the harmonic source which reproduces approximately the measurement, via the Admissible Data Method. Later, the specific source in a given class which best approximates the measurement is determined by using additional information provided by the harmonic source. The representing functions sets generating series development solutions and the computational approach depend on the geometric head model.

All this methodology can be extended to the case of time-dependent measurements (EEG), and provides algorithms for directly identifying time-dependent sources which do not depend on previous time discretizations. For the sake of clarity, this is out of the scope of the current work.

Finally, we want to make emphasis on the importance of Theorem 5. As a consequence, we find that any measurement V in S2 can be reproduced by a harmonic source concentrated as close as wanted to the cortex S1 . Furthermore, although these sources extend in a unique way to Ω1, these specific extensions need not to reproduce the same measurement V, when the corresponding support is contained into a different neighborhood (of the cortex) from that corresponding to the original source.

Acknowledgements

Authors acknowledge CONACYT and Posgrado en Matemáticas de la BUAP financial support of Javier Mozqueda Lafarga through a PhD grant. Authors acknowledge CONACYT financial support of Moisés Soto-Bajo through "Cátedras CONACYT-CEMMAC-BUAP para Jóvenes Investigadores 2016" program. The author Javier Herrera-Vega acknowledges funding from the Mexican research council CONACYT under the posdoctoral scholarship number 741147.

This article was prepared in the context of the research project N° A1-S-36879, titled “Estudio teórico de las soluciones periódicas y de problemas inversos en sistemas de reacción difusión y elípticos que aparecen en los modelos matemáticos de generación y propagación de la actividad eléctrica en el corazón y el cerebro”.

The authors thankfully acknowledge the computer resources, technical expertise and support provided by the Laboratorio Nacional de Supercómputo del Sureste de México, CONACYT Network of National Laboratories.

References

Amir, A. (1994). Uniqueness of the Generators of Brain Evoked Potential Maps. IEEE Transactions on Biomedical Engineering, 41(1), 1-11. DOI:10.1109/10.277265 [ Links ]

El Badia, A., & Ha Duong, T. (1998). Some remarks on the problem of source identification from boundary measurements. Inverse Problems, 14(4), 883-891. [ Links ]

Fraguela Collar, A., Morín Castillo, M., & Oliveros Oliveros, J. (2008). Inverse electroencephalography for volumetric sources. Mathematics and Computers in Simulation, 78(4), 481-492. DOI:10.1016/j.matcom.2007.06.010 [ Links ]

Fraguela Collar, A., Oliveros Oliveros, J. J., Morín Castillo, M. M., & Conde Mones, J. J. (2015). Identification of piecewise constant sources in non-homogeneous media based on boundary measurements. Applied Mathematical Modelling, 39, 7697-7717. [ Links ]

Geselowitz, D. B. (1967). On bioelectric potentials in an inhomogeneous volume conductor. Biophysical journal, 7(1), 1-11. [ Links ]

Hernandez Montero, E., Fraguela Collar, A., & Henry, J. (2019, February 16). An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG. Mathematical Modelling of Natural Phenomena, 14(2), 1-22. [ Links ]

Kirsch, A. (2013). An Introduction to the Mathematical Theory of Inverse Problems (Vol. 120). Karlsruhe, Alemania: Springer Science & Business Media. [ Links ]

Mijailov, V. P. (1978). Ecuaciones diferenciales en derivadas parciales. Moscú: MIR. [ Links ]

Morín Castillo, M. M., Oliveros Oliveros, J. J., Conde Mones, J. J., Fraguela Collar, A., Gutiérrez Arias, F. F., & Flores Mena, E. (2013, April 13). Simplificación del Problema Inverso Electroencefalográfico a una Sola Región Homogénea con Condición de Neumann Nula. Revista Mexicana de Ingeniería Biomédica, 34(1), 41-51. [ Links ]

Munck, J. C., Van Dik, B. W., & Spekreijse, H. (1988). Mathematical Dipoles are Adequate to Describe Realistic Generators of Human Brain Activity. IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, 35(11), 960-966. [ Links ]

Nunez, P. L., & Srinivasan, R. (2006). Electric Fields of the Brain: The Neurophysics of EEG (2nd ed.). Oxford University Press. [ Links ]

Nunez, P. L., Nunez, M. D., & Srinivasan, R. (2019). Multi-Scale Neural Sources of EEG: Genuine, Equivalent, and Representative. A Tutorial Review. Brain Topography(32), 193-214. [ Links ]

Plonsey, R., & Fleming, D. G. (1969). Bioelectric Phenomena. New York: McGraw-Hill. [ Links ]

Sarvas, J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Physics in Medicine and Biology, 32(1), 11-22. [ Links ]

Speckmann, E. J., Elger, C. E., & Gorji, A. (2011). Niedermeyer's Electroencephalography. In L. d. Schomer Donald L. (Ed.), Basic Principles, Clinical Applications, and Related Fields (6th ed., pp. 17-24). Philadelphia, Pensilvania, United States: Lippincott Williams & Wilkins. [ Links ]

Tijonov, A. N., & Samarsky, A. A. (1980). Ecuaciones de la física matemática. Moscu, Rusia: MIR. [ Links ]

1If xn and yn are nonnegative term sequences, and limnxnyn= α0,, then these sequences are said to be “equivalent”, and then n=1xn<+ iff n=1yn<+

2This is a consequence of the necessary and sufficient condition for the existence of solution Neumann problem for the Laplace equation on Ω1Ω2 and on Ω1.

Received: January 29, 2020; Accepted: February 21, 2020

Autor para correspondencia: Javier Mozqueda Lafarga E-mail: javiermozqueda@gmail.com