The inverse EEG problem

Bachelor Thesis , 2007 , 64 Pages , Grade: 1.0

Computer Science - Miscellaneous

Abstract:
The inverse EEG problem is a well-studied, ill-posed problem in mathematics and neuroinformatics. Given a record of a limited number of electrodes (e.g. 21) that are placed on scalp, it is the task to estimate a three dimensional distribution of neural currents in the brain. The actual thesis deals with this problem and proposes a propabilistic Bayesian approach that assumes the distribution of neural currents to be heterogeneous - active and inactive regions in the brain are expected. This can be formalized with a mixture distribution. Furthermore, an expectation-maximization (EM) algorithm is presented that performs simultaneous classification and computation of neural currents, given an EEG measurement.


Zusammenfassung:
Diese Bachelorarbeit thematisiert das inverse EEG Problem. Dies ist ein umfangreich beschriebenes, schlecht gestelltes mathematisches Problem in der medizinischen Visualisierung. Bei einer EEG-Messung werden eine bestimmte Anzahl (z.B. 21) Elektroden an der Kopfhaut angebracht und elektrische Ströme dort detektiert. Das hier beschriebene Problem besteht darin, aus dieser Messung eine dreidimensionales Verteilung neuronaler Gehirnströme zu rekonstruieren. Es wird ein propabilistischer Bayesianischer Ansatz vorgestellt, um dieses Problem zu lösen. Dabei wird angenommen, dass die Verteilung neuronaler Ströme heterogen ist - es gibt aktive und nicht-aktive Bereiche im Gehirn. Dies wird mathematisch mit einer Mischverteilung formalisiert. Dieser Ansatz ermöglicht Gehirnströme sowohl zu berechnen, also auch zu klassifizieren. Ein entsprechender EM Algorithmus, der dies simultan durchführt, wird vorgestellt.

1. Introduction

Approximately 100,000,000,000 (100 billion) neurons ean be found in the brain. That/s the origin of thinking, feeling and conscious acting. That means that there are more than 74,000,000 (74 million) neurons in each eem. They are interacting with each other in various ways using excitatory and inhibitory synapses. As a result of this complex network, a human being has intelligence, feeling, emotions and personality traits.

In this network, interaction ean be measured and monitored by tracing neural electric currents, because neurons communicate with each other via electric signals/potentials, which are submitted to synapses that ean be seen as junctions between neurons. This tracing of neural currents is done with an electroencephalography (EEG) measurement, which is widely regarded as the physiological gold standard to monitor brain activity. Therefore, an accurate analysis of EEG data is expedient, because this technology en­ables a non-invasive, harmless and inexpensive neural examination. It is desirable to get detailed and high-resolution three-dimensional distributions of neural currents based on EEG measurements, because localizing brain activity in a three-dimensional space enables a highly specific diagnosis of neural diseases. This information could stand on its own or it could be corroborated by results of measurements based on distinct technologies like Functional Magnetic Resonance Imaging (fMRI) or Positron Emission Tomography (PET), These provide a much higher spatial resolution but a low temporal resolution. Determining a three dimensional distribution of neural currents is a very complex mathematical problem, thus the "traditional" EEG technology and the anal­ysis of its records is still an area of ongoing research. This is part of the relativtely new and multidisciplinary research held of functional brain mapping, that encompasses techniques devoted to a better understanding of the human brain through noninvasive imaging of electrophysiological, hemodynamic, metabolic and neurochemical processes that underlie normal and pathological brain function (Baillet et ah, 2001),

This current theses deals with the analysis of EEG data and presents a probabilistic approach that uses mixture distributions to identify active and inactive regions in the brain. In contrast to deterministric approaches like LORETA that is introduced in Chapter 6, the presented approach to solve the inverse EEC problem assumes the distri­bution of neural currents to be heterogeneous. Furthermore, the propabilistie approach presented makes it possible to compute and to classify the distribution of electric cur­rents simultanously.

During an EEC measurement, voltage fluctuations on the scalp are recorded across time and the discussed problem is how to calculate a three dimensional distribution of neural currents in the brain. This is an inverse problem that can be formulated mathematically with <f = KJ + e, (1,1)

where <f is a vector with electrode potentials on the scalp, J is a vector that describes the neural currents in a three-dimensional brain model, e is an error vector and K is a lead field matrix that formalizes the linear dependence between the spatial distribution of neural currents J and observed potentials 0, K and <f are known after an experiment and it is the task to find out a reasonable solution for J, which solves Equation 1,1, This problem is discussed in detail in Chapter 6 and Chapter 7, Generally, an inverse problem exists when the causes for an observed effect are yet to be determined. Chapter 4 concerns inverse problems in detail. Solving this inverse problem is not trivial, because it belongs to the class of ill-posed inverse problems (Hadamard, 1902) and there might be an unlimited number of solution vectors J, that correctly solve Equation 1,1, Thus, it can be stated that a perfect EEC tomography can not exist and determining a proper solution of this inverse problem is still a current area of research, A review of investigated methods in this field is given in Baillet et al, (2001); Koles (1998).

2. EEG

In this chapter, technology and methodology of an EEG measurement is introduced. This measurement and the resulting data represents the basis of following computations so that it is important to highlight this technology in detail. Furthermore, physiological basics that are connected with this technology are discussed,

2.1. Physiological basics

2.1.1. Cells of the central nervous system

The central nervous system is a complex network of cells in the brain, which represents the origin of thinking, feeling and conscious acting. The brain of a mammal is the most complex organ, consisting of hundreds of different neural cell types. These are different from each other, varying in their shape and biochemical composition. This complex interplay only conduce to an optimized signaling transduction between nerve cells, also called neurons, which are highly diversified on their own, A schematic illustration of an idealized neuron is given in Figure 2,1, A signal/action potential is received in synapse of a dendrite and can be forwarded to synapses of the axon terminal.

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2.1.2. Resting potential

Neurons are interacting with each other via excitatory and inhibitory synapses. When there is no stimulus, neurons hold a resting potential. This is a negative electrical potential between -90mV and -70mV on the neuron membrane, which is generated by excess negative charges in the inside compared to the outside of the cell. This poten­tial is conserved bv numerous ion pumps (Na+/K+-ATPase), which actively transport potassium-ions (K+) in the interior and sodium-ions (Na+) in the exterior. The mem­brane possesses highlv-selective ion channels for K+, which tend to be open. Ion channels permeable for Na+ are generally closed during resting potential. Thus, K + is able to diffuse back through the membrane, while Na+ is carried out of the cell so that the extracellular space gets positively charged,

2.1.3. Action potential

A neural signal or stimulus is coded with an action potential. If there is a stimulus that excesses the threshold (marked in Figure 2,2 with a blue line), voltage controlled Na+- channels open rapidly so that Na+-ions diffuse into the interior of neurons. They are driven by electrochemical repulsion, because complementary charges attract each other. This ion-flow induces a dramatic change in membrane potential, that is illustrated in Figure 2,2, After the short, but dramatic Na+ flow into the cell, Na+-channels close again and the Na+/K+-ATPase transports Na+-ions out of the cell. Thus, the resting potential is restored.

Signal transduction of neural cells is a very well-studied and complex area of research, an overview is given in Schmidt (1995),

2.2. Technology

As described above, action potentials represent neural activity. Thus, determining elec­tric currents in the brain is a useful approach to detect and localize neural activity. Because of the configuration of skull and skullcap, it is not possible to measure voltage fluctuations directly in the brain without endangering a patient. But organic matters in brain conduct electrical currents, so that it is reasonable to detect and to analyze voltage fluctuations on the scalp. This is done in an EEG measurement.

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2.2.1. History

There are many modern technologies to measure, localize and visualize activity in the brain. Electroencephalography (EEG) is widely regarded as the physiological gold stan­dard to monitor and quantify brain activity. This long-standing technology was first applied in the early 20th century, when the Russian physiologist Vladimir Vladimirovich Pravdich-Xeminsky measured evoked potential and voltage fluctuations across time us­ing electrodes placed on dogs’ scalps, German physiologist Hans Berger (1873 - 1941) began his studies of human EEG in 1920 and his work was later expanded by Edgar Douglas Adrian (1889 - 1977), In 1936, the first EEG laboratory was set up for clinical application with the main focus on analyzing epileptic patients. There have been con­tinual improvements to the technology regarding mathematical and mechanical issues.

2.2.2. EEG measurement

Electrodes are placed on specific positions on the scalp. These positions are predeter­mined e.g. through the 10-20 system, that is described in Section 2,3, These electrodes detect electric signals, which are induced by neural currents in their surroundings. Each electrode record is amplified between 1000 and 100,000 times by a connected differential amplifier. The resulting voltage signal is filtered by a low-pass filter which filters out signals at 35-70 Hz and a high-pass filter, which filters out signals at 0,5 Hz, One of the advantages of measuring brain aetivity with EEG is that the time reso­lution is very high. Unlike modern methods that measure blood flow or metabolism in the brain to suggest neural aetivity, reeording voltage fluetuations on the sealp has a temporal resolution down to sub-milliseeonds. Furthermore, this method is non-invasive, harmless, inexpensive and easy to apply.

Beside these advantages, there are also several limitations to this technology. First of all, only summations of postsynaptie potentials of cortical neurons are recognized, single action potentials don’t cause evident voltage fluctuations on the sealp.

It is also not possible to separate between inhibitory and excitatory potentials, both are recorded in the same way as neural generators and neural aetivity respectively. The analysis of EEG data is also problematic due to the fact that several neural regions can be active simultaneously and potentials of these sources interfere with each other on single detectors. This problem is called blind source separation and is described in more detail by Knuth (1997), Likewise, interpreting EEG signals as spatial three-dimensional images is still very difficult and represents an ongoing area of research in mathematics and neuroinformatics. This work contributes to this ongoing research.

2.3. 10-20 system

In 1958, the Canadian neuroscientist Herbert Henri Jasper (1906 - 1999) developed a system of placing electrodes on the sealp that takes variable head size and head shape into account (see Klem et ah, 1999; Jasper, 1958), The skull is sized with the distance from Xasion to Inion, This distance represents the reference, or 100%, The system is called 10-20 system, because the electrode positioning is done by steps of 10% or 20%, In the end, 21 electrodes are placed on the sealp, see Figure 2,3, To get a three-dimensional orientation, Figure 2,4 shows the electrode positioning with regard to the isolated and standardized brain.

Because of its easy-to-apply and universally valid properties, this electrode positioning method is still in practice.

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Evoked potentials measure changes in brain electrical activity that are associated with a sensory or psychological process or stimulus (Pieton and Stuss, 1984), Thus, electrical potentials are recorded as a response to an artificial stimulus. In contrast to a conventional EEG measurement, where spontaneous potentials are recorded, these evoked potentials tend to be hidden in the larger background EEG activity. That is why these potentials (also named event-related potentials, ERP) are determined by averaging repeated responses so that it is possible to distinguish between evoked potentials, spon­taneous potentials and background noise. Averaging is necessary in order to increase the signal-to-noise ratio for the brain activity and the events not of interest. An image of a measurement of evoked potentials is given in Figure 5,2 in page 27, where the neural answer to a visual stimulus is marked with a red bar.

Example 2.1. A typical experiment might be to give an acoustic stimulus to a subject: As a non-target stimulus, 175 consecutive clicks are defined. The target stimulus is defined with tones of 1000Hz, Identifying a target tone (acoustic stimulus), the subjects have the task to press a button.

As a response of the acoustic stimulus described above, a characteristic potential distri­bution at the scalp is measurable. There is a delay of 90 - 100 ms, which can be seen in Figure 2,6, where the record of three electrodes is plotted and the stimulus is marked with the vertical magenta line.

Another typical experiment might be to give a visual stimulus, showing a specific pic­ture on a computer screen for a preassigned time followed by a black screen. This visual stimulus is repeated many (e.g, 50) times and the resulting neural response is recorded and analyzed.

3. Mathematical theory and functions

In this chapter, mathematical theory and functions that are used in the following text are introduced. More general background information and terminology can be found in the Appendix,

3.1. Functions

3.1.1. Convex functions

Definition 3.1. A function f (x) is called convex on interval I, if for any points a, b G I and any z G [0,1]

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A function g(x) is called concave, if -g(x) is a convex function.

You can identify a concave function by analyzing its first derivative. If and only if the first derivative f(x) is a monotone increasing function, f(x) is convex. The first derivative of a strictly concave function is a strict monotone increasing function and cannot have more than one root. Thus, strictly convex functions have not more one minimum! Analogously, if f'(x) is a monotone decreasing function, f (x) is concave.

Example 3.1. As an example for a convex function, f(x) = y is illustrated in Figure

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Figure 3.1.: Example of a convex function: f(x) = where a = —25,6 = 15,2 = 0.7.

The green dashed line reflects the proportionality between f (za + (1 — z)b) and zf(a) + (1 — z)f (b)

3.1.2. Lagrange multipier

The Lagrange multiplier is a method to formulate and calculate optimization problems with constraints. Considering two functions f (x) and g(x), a local extremum x of f (x) is to be determined that satisfies the constraint g(x).

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Thus, the problem of finding the extremum of f (x) that satisfies the constraints g1..n(x) is reduced to finding the extremum of h(x, A), defined in Equation 3.4. This can be done bv determining the root of the first derivative of h(x, A)

The unknown multiplicators Ak can be determined applying their related constraint gk(x)

Example 3.2. The lazy tomcat Elvis lies on coordinate A (6, 6). He wants to reach coordinate B (8, 2) to get some food. On his wav from A to B, he wants to meet his girlfriend Jenny. He knows the path that Jenny walks every day and thinks about the best position to meet her. The path that Jenny walks can be put in the mathematical formulation

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A delineation of this situation is given in Figure 3.2.

Thus, Elvis wants to find out the coordinate C (x, y) for which the distance d(A, C) plus the distance d(C, B) is minimal under the constraint that C is a point on the path of Jenny, thus C = (x, y), where y = p(x). This can be formulated mathematically with

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where the square root of the euclidean distance function is neglected for simplicity. For better calculating with this constraint, it should be a function g(C), where the constraint is satisfied if g(C) = 0, In this context, the constraint could be defined with

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• College: Free University of Berlin (Institute of Mathematics and Informatics, Department Bioinfomatics)
• Grade: 1.0
• Author: Johannes Höhne (Author)
• Publication Year: 2007
• Pages: 64
• Catalog Number: V89576
• ISBN (eBook): 9783638068888
• ISBN (Book): 9783640185238
• Language: English