**Mathematical Foundation of Electroencephalography**

DOI: 10.5772/68021

Mathematical Foundation of Electroencephalography

Michael Doschoris and Foteini Kariotou

Additional information is available at the end of the chapter Michael Doschoris and Foteini Kariotou

http://dx.doi.org/10.5772/68021 Additional information is available at the end of the chapter

#### Abstract

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34 Electroencephalography

Electroencephalography (EEG) has evolved over the years to be one of the primary diagnostic technologies providing information concerning the dynamics of spontaneous and stimulated electrical brain activity. The core question of EEG is to acquire the precise location and strength of the sources inside the human brain by knowledge of an electrical potential measured on the scalp. But in what way is the source recovered? Leaving aside the biological mechanisms on the cellular level responsible for the recorded EEG signals, we pay attention to the mathematical aspects of the narrative. Our goal is to provide a brief and concise introduction of the mathematical terminology associated with the modality of EEG. We start from the very beginning, presenting step by step the mathematical formulation behind EEG in a simple and clear manner, keeping the mathematical notation to a minimum. Whilst we serve only the key relations for the described problems, we focus specifically on the limitations of each modelling approach. In this fashion, the reader can appreciate the beauty of the formulas presented and discover every single piece of information encoded within these formulas.

Keywords: EEG, mathematical analysis, forward problem, inverse problem, spherical conductor, ellipsoidal conductor

#### 1. Introduction

The human brain is a remarkable and fascinating organ exhibiting a tremendous complexity. It makes us unique and defines who we are. In spite of our scientific and technological progression, we do not know the particulars of its operating, and as we delve into its secrets, various surprises emerge, for example, nearly 100 previously unidentified brain areas have been recently discovered [1]. Consisting of an inconceivable network of interconnected nerve cells and fibres,

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

continuously transporting and processing information, the brain is extremely vulnerable and requires paramount protection. Several layers of safety are incorporated starting with three connective sheets of tissue, called the meninges containing the cerebrospinal fluid, followed by plates of bones, the skull. Above machinery safeguards the brain from mechanical damage. On the other hand, a sophisticated barrier within the brain provides a natural defence against toxic or infective agents.

Examining the anatomy of the brain, we recognize three distinct regions. The largest part of the brain is the cerebrum, divided into two hemispheres. The outermost layer of the cerebrum is the cerebral cortex, consisting of four lobes. Cognitive awareness has its origins here. The second largest part of the brain is the cerebellum, located underneath the cerebrum and responsible for motor control and learning. Last, but not least, an integral part of the brain connecting the cerebrum with the spinal cord is the brainstem, regulating reflexes and crucial, basic life functions. Detailed information can be found in [2].

The operational status of the brain is based on an alternating chain of electrical and chemical events. On the microscopic level, encoding and transmitting of information via electrochemical signals is achieved by the active participation of neuronal and non-neuronal constituents. Brain cells communicate through synaptic transmissions by controlling chemical transmitters or ionic currents which flow across their membranes. As a consequence, an electromagnetic field is generated. For a far-reaching introduction on the subject, see [3]. The question at hand now lies in the possibility exploiting these provoked fields. It seems only reasonable that if a substantial number of cells form a critical mass, which activates synchronously, the emerging electric and magnetic field should be detectable. This is indeed the case and a deeply rooted concept in electrophysiology [4, 5]. From an electrofunctional point of view, the ionic micro-flow within a single brain cell creates an opposite polarity between two point electrical charges very close together, leading to the notion of a dipole, a physical quantity one could say consists of the 'fundamental unit', which produces the observed fields. Dipoles are characterized by a vector called moment, the product of the charge and distance, visualized as an arrow pointing from a minimum (negative charge) to a maximum (positive charge), ergo featuring direction and magnitude. For that reason, it may be argued that the macroscopic description of the brain's activity is best achieved when simulated as an array of dipoles, that is, a non-uniform distribution of positive and negative charges. According to the latter, if a small neighbourhood is stimulated, an excessive number of dipoles concur and their electric fields would add or cancel one another depending on the direction. This complicated and difficult situation can be avoided by introducing the concept of the equivalent current dipole (ECD), namely a single dipole which generates the identical electric field as all of the individual dipoles together, hence summarizing the net effect of all microscopic currents located in the distinct region of the brain under consideration. This is a widely used approximation concept in the framework of neuroelectromagnetism [6, 7]. On the other hand, when the exertion is no longer confined to a focal region of the brain, then every one of these regions is simulated by an equivalent current dipole, leading to a distribution of sources.

The main task and problem is to correlate active regions with associated generated electric fields. This essential step is closely connected with the installation of physical structures, namely a boundary or number of boundaries enclosing distinct regions with specific physical characteristics, such as conductivity. The head model obtained is termed the volume conductor model. Clearly, the level of details incorporated into the head model provides an analogous

degree of operational freedom when it comes to investigate how the fields generated by brain cells are transmitted through various biological tissues towards measurement apparatus. As a result, the volume conductor model consists of the physical foundation for source analysis, which is categorized into two major problems. The first one is associated with the calculation of the electric potential, generated by known electrochemical sources within the brain, at precise points at the surface of the scalp. This is the forward electroencephalographic (EEG) problem [8–11]. The forward EEG problem has been extensively scrutinized for over 60 years since Wilson and Bayley [12] attempted to quantify the interplay between neuronal activity and the potentials they generate at the scalp. The reconstruction of the sources responsible for the recorded values is called the corresponding inverse EEG problem [13].

continuously transporting and processing information, the brain is extremely vulnerable and requires paramount protection. Several layers of safety are incorporated starting with three connective sheets of tissue, called the meninges containing the cerebrospinal fluid, followed by plates of bones, the skull. Above machinery safeguards the brain from mechanical damage. On the other hand, a sophisticated barrier within the brain provides a natural defence against toxic

Examining the anatomy of the brain, we recognize three distinct regions. The largest part of the brain is the cerebrum, divided into two hemispheres. The outermost layer of the cerebrum is the cerebral cortex, consisting of four lobes. Cognitive awareness has its origins here. The second largest part of the brain is the cerebellum, located underneath the cerebrum and responsible for motor control and learning. Last, but not least, an integral part of the brain connecting the cerebrum with the spinal cord is the brainstem, regulating reflexes and crucial,

The operational status of the brain is based on an alternating chain of electrical and chemical events. On the microscopic level, encoding and transmitting of information via electrochemical signals is achieved by the active participation of neuronal and non-neuronal constituents. Brain cells communicate through synaptic transmissions by controlling chemical transmitters or ionic currents which flow across their membranes. As a consequence, an electromagnetic field is generated. For a far-reaching introduction on the subject, see [3]. The question at hand now lies in the possibility exploiting these provoked fields. It seems only reasonable that if a substantial number of cells form a critical mass, which activates synchronously, the emerging electric and magnetic field should be detectable. This is indeed the case and a deeply rooted concept in electrophysiology [4, 5]. From an electrofunctional point of view, the ionic micro-flow within a single brain cell creates an opposite polarity between two point electrical charges very close together, leading to the notion of a dipole, a physical quantity one could say consists of the 'fundamental unit', which produces the observed fields. Dipoles are characterized by a vector called moment, the product of the charge and distance, visualized as an arrow pointing from a minimum (negative charge) to a maximum (positive charge), ergo featuring direction and magnitude. For that reason, it may be argued that the macroscopic description of the brain's activity is best achieved when simulated as an array of dipoles, that is, a non-uniform distribution of positive and negative charges. According to the latter, if a small neighbourhood is stimulated, an excessive number of dipoles concur and their electric fields would add or cancel one another depending on the direction. This complicated and difficult situation can be avoided by introducing the concept of the equivalent current dipole (ECD), namely a single dipole which generates the identical electric field as all of the individual dipoles together, hence summarizing the net effect of all microscopic currents located in the distinct region of the brain under consideration. This is a widely used approximation concept in the framework of neuroelectromagnetism [6, 7]. On the other hand, when the exertion is no longer confined to a focal region of the brain, then every one of these regions is simulated by an

The main task and problem is to correlate active regions with associated generated electric fields. This essential step is closely connected with the installation of physical structures, namely a boundary or number of boundaries enclosing distinct regions with specific physical characteristics, such as conductivity. The head model obtained is termed the volume conductor model. Clearly, the level of details incorporated into the head model provides an analogous

basic life functions. Detailed information can be found in [2].

equivalent current dipole, leading to a distribution of sources.

or infective agents.

36 Electroencephalography

As of today, a high level of details can only be achieved with the aid of numerical models, which are generally categorized into boundary element models (BEMs) and finite element models (FEMs). Whereas boundary element models are adequate to portray major tissue compartments, such as the cerebrum and skull, they fail to represent detailed anatomical information within the compartments, such as the cerebral folding [14, 15]. Finite element models, on the other hand, are efficient in capturing these details, but are labour intensive and computationally demanding [16, 17].

Nonetheless, in order to gain a deeper comprehension of the problem a rigorous mathematical analysis is essential in providing a vital step towards the recognition of the underlying phenomena as well as identifying the limitations of the developed algorithms. The importance of mathematical analysis cannot be emphasized enough, since (i) it allows testing the impact of modifications regarding various variables upon the output of the system and provides further insight into underlying physical behaviour. (ii) It serves as validation tools for the numerical models.
