Importancia de la clase de fuentes armónicas en la identificación de fuentes en el problema inverso electroencefalográfico Importance of the class of harmonic sources in the identification of sources in the inverse electroencephalographic problem

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.


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) 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 Importancia de la clase de fuentes armónicas en la identificación de fuentes en el problema inverso electroencefalográfico Nº 24, Vol. 12 (1), 2020. ISSN 2007 -0705, pp.: 1 -32 -4 -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).

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 Ω 1 the region occupied by the brain, and by Ω 1 = 1 its border, corresponding to the cortex. 1 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 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, 1 will be the electric potential in the brain volume Ω 1 . On the other hand, the rest of the head is considered also as a homogeneous conducting medium Ω 2 , with outer border Ω 2 = 2 corresponding to the scalp. Its average conductivity, considered constant, is 2 . In this region there are no sources of electrical activity (see Fig. 1). Analogously, 2 will be the electric potential distribution in Ω 2 . We set Ω = Ω 1 ∪ Ω 2 .
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 1 and 2 , in the following way: Where denotes the normal derivative of on with respect to the normal unitary vector , outer to Ω , , = 1,2. Here, the forward problem consists on, given a source , finding the potentials 1 and 2 . Thus, the corresponding inverse problem consists on, given a measurement , finding an equivalent source satisfying ( 1 ) -( 5 ) in such a way that the corresponding potential simultaneously fulfills the condition 2 | 2 = .
In the neuronal activity source estimation problem, via model ( 1 ) -( 5 ), the operative formulation requires to find an operator which associates to each neuronal activity source (in a certain class ℱ) the measurement . Namely, the inverse source estimation problem reduces to solve the operational equation Where = 2 | 2 is the potential distribution measurement on 2 .
Hence, solving the inverse source estimation problem means that, starting from the instantaneous electroencephalographic measurement = 2 | 2 , a source "reproducing this Given a class ℱ of functions defined on the brain, the image of ℱ by the operator 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 restricted to ℱ is injective. In what follows the admissible data set associated to a sources class ℱ will be denoted by ℳ[ℱ]: In (Fraguela Collar, Oliveros Oliveros, Morín Castillo, & Conde Mones, 2015) was proved that operator is compact from 2 (1) (Ω 1 ) to 2 (1) ( 2 ). In addition, in Theorem 2.3 was shown that From this result yield the following conclusions, which will be important in what follows: a) If the desired source is required to be in ℋ (1) (Ω 1 ), then the unicity theorem for the solution of the inverse problem corresponding to the operational equation ( 6 ) is fulfilled.
b) If the source which reproduces a given potential distribution is required to be in a certain class ℱ which satisfies the unicity condition for the solution of the inverse problem, and can also be reproduced by ℎ ∈ ℋ (1) (Ω 1 ) (note that each one uniquely reproduces in their own class. Certainly, there is no unique representation in 2 (Ω 1 )), then ℎ is the orthogonal projection of on ℋ (1) (Ω 1 ), independently of the chosen class ℱ.
We note that another unicity class for the source estimation problem was obtained in (Fraguela

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 1 , of radius 1 > 0 corresponds to the brain volume Ω 1 , and the interior of the outer sphere 2 , of radius 2 > 1 , corresponds to the whole head Ω and outlines the region Ω 2 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 2 (1) (Ω 1 ). Note that, in = 0 on S 2 .
Nº 24, Vol. 12 (1), 2020. ISSN 2007 -0705, pp.: 1 -32 -11 -Where | | denotes the Lebesgue measure of . 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: By using these operators , , and , the solution of the inverse problem associated to the contour problem ( 1 ) -( 5 ) can be obtained by solving the following system of operational equations: = .
Once is known, it can be substituted in ( 26 ), obtaining the operational equation: Thus, equation ( 28 ) give us , so system ( 26 ) -( 27 ) is equivalent to inverse source estimation problem. Finally, the operator which relates the source with the measurement is given by: Importancia de la clase de fuentes armónicas en la identificación de fuentes en el problema inverso electroencefalográfico This theorem justifies remarks a) and b) above. An important corollary follows also from it: if ∈ ℱ reproduces a measurement , and the harmonic function ℎ 0 is the unique harmonic source on Ω 1 which also reproduces , then ℎ 0 is the orthogonal projection of on ℋ (1) (Ω 1 ), independently of the chosen class ℱ, and − ℎ 0 is the component of in Ker .

Admisible data method
The such a way that the minimum distance problem to ℳ 0 , would be well posed in 2 ( 2 ).
2. Assuming fixed a measurement error and a measurement ̃∈ 2 ( 2 ) (with error), find the admissible data 0 ∈ ℳ 0 which attains the minimum distance to ̃, and check that this distance has the magnitude order of the measurement error. 4. Finally, use ℎ 0 to characterize the source ∈ ℱ which best reproduces the measurement 0 in the class ℱ. In order to do that, one makes use of remark after Theorem 2: − ℎ 0 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.

Reduction of the instantaneous source estimation problem to a Cauchy data problem on cortex
In order to find the restriction to 1 of the solution 2 on Ω 2 , we will study the following contour problem, which is a Cauchy problem for the Laplace equation on Ω 2 , with vanishing Neumann contour condition and Dirichlet data : Where is the potential distribution electroencephalographic measurement in 2 (the whole scalp) and 2 is the electric potential in Ω 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): Importancia de la clase de fuentes armónicas en la identificación de fuentes en el problema inverso electroencefalográfico Where are the Fourier coefficients of . It can be checked that the corresponding solution to problem ( 30 ) -( 32 ) is given by 2 ( , , ) = ∑ ∑ ( With convergence in 2 (Ω 2 ). The solution of the Cauchy problem corresponds to restrict 2 to 1 , which means to evaluate ( 35 ) In order to assure that the above expression for 2 ( 1 , , ) to be the trace in 1 of a function in 1 ( 1 ), the series composed with the squares of Fourier coefficients of ( 36 ), multiplied by , must converge (Mijailov, 1978, p. 221). Then, by using an elementary result on convergence of numerical series 1 we obtain the following condition on the Fourier coefficients of : 1 If and are nonnegative term sequences, and lim →∞ = ∈ (0, ∞), then these sequences are said to be "equivalent", and then ∑ ∞ =1 < +∞ iff ∑ ∞ =1 < +∞ Consequently, ( 38 ) must converge in 2 ( 1 ). Proceeding as before, we get the following result: Theorem 3. One has ∈ 2 (1) ( 1 ) if and only if the Fourier coefficients satisfy 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 of satisfy condition ( 39 ). From an operational point of view, it makes sense to solve equation ( 28 ) from given by ( 38 ). That is Which could be viewed as the identification of the source , which reproduces the potential on the cortex 1 , as it would be artificially measured.
In order to obtain an explicit expression for = − , we make use of ( 38 ). Since operators and linear and continuous, it is enough to compute ( ( , )) and ( ( , )). For this purpose, in view of ( 18 ) and ( 19 ), we solve problems ( 9 ) and ( 10 ) with contour data ( , ).
In this way we get In summary, we conclude that, if the Fourier coefficients of measurement satisfy condition ( 39 ), then the source estimation problem turns to solve equation = , where 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" 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 on the scalp, and we study under what conditions there exists an (unique) harmonic source in the brain which reproduces . By the way, we will find an explicit expression for .
We start from the fact that any harmonic source in Ω 1 should take the following form: Which is a development with respect to the orthonormal basis of spherical harmonics √ 2 +3 1 2 +3 ( , ). Hence, the convergence of this series (in the 2 (Ω 1 ) sense) is equivalent to the condition Since operator is linear and continuous, we get In order to compute In view of ( 17 ), it is required to solve the associated problem Thus we look for a solution of the form The general solution of this equation takes the form Where 0 is solution for the associated homogeneous equation and is a particular solution. We propose a particular solution of the form From ( 53 ) Thus we get the condition = + 2, so And = − 1 2 1 (2 + 3) .
The function 0 satisfies the homogeneous equation And must be bounded at = 0, so takes the form 0 ( ) = .
Theorem 4. There exists a biunivocal correspondence between the set of harmonic sources in the brain Ω 1 and the set of measurements on the scalp (in 2 ( 2 )) whose Fourier coefficients with respect to the orthonormalized spherical harmonics ( , ) satisfy: Furthermore, given a measurement satisfying condition ( 65 ), the unique harmonic source in Ω 1 which reproduces it is given by ( 64 ).
We recall that from the previous theorem it follows that ℋ (1) (Ω 1 ) is a sources unicity class, and the admissible data set corresponding to this class of harmonic sources orthogonal to the constants, denoted by ℳ[ℋ (1) (Ω 1 )] following notation ( 7 ), turns to be the set of measurements satisfying ( 65 ).
Note that, from Theorem 2 and 4 it follows that for any sources class ℱ in 2 (Ω 1 ) the corresponding admissible data set ℳ[ℱ] is contained into the set of measurements in 2 ( 2 ) which satisfy condition ( 65 ). Note also that this condition ( 65 ) is a quite strong smooth requirement on the spatial distribution of the measurements , 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 on the scalp, and we study the existence of a source in the brain reproducing it. For this source we assume it is supported and harmonic in a neigbourhood of the cortex. We stablish conditions on for its existence, and we will find the admissible data set for this class.
Finally, have the sum for any positive integer and from − to . If we take into account that is harmonic in Ω 1 , we have the following conclusions: converges, since are the Fourier coefficients of .
3. Then the series with general term the right side of ( 96 ) also converges, whose terms are equivalent to 3 ( 2 1 ) 2 | | 2 .
Then, we get (1) = (2) , but nothing can be said about (1) and (2) . From this it can be deduced that an unicity subclass contained in ℋ (1) (Ω 1 (1) ) is obtained considering harmonic functions given by series with respect only the first terms; i.e., those of the following form: Starting from this development, and computing their coefficients via the equation = as before, we get From now, this subclass of harmonic sources will be denoted by ℋ 0 (1) (Ω 1 (1) ).
Note that sources class ( 107 ) precisely matches the set of sources that equal harmonic sources in 1 when restricted at 1 (1) , and vanish at 1 (0) .
The following result summarizes these results.

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 in 2 can be reproduced by a harmonic source concentrated as close as wanted to the cortex 1 . Furthermore, although these sources extend in a unique way to Ω 1 , these specific extensions need not to reproduce the same measurement , when the corresponding support is contained into a different neighborhood (of the cortex) from that corresponding to the original source.