The inverse EEG problem

Table of Contents

1. Introduction

2. EEG

2.1. Physiological basics

2.1.1. Cells of the central nervous system

2.1.2. Resting potential

2.1.3. Action potential

2.2. Technology

2.2.1. History

2.2.2. EEG measurement

2.3. 10-20 system

2.4. Evoked potentials

3. Mathematical theory and functions

3.1. Functions

3.1.1. Convex functions

3.1.2. Lagrange multiplier

3.1.3. Laplace operator

4. Inverse problems

4.1. Definition

4.2. Example of ill-posed inverse problems

4.2.1. Inverse problem of a Hilbert matrix equation system

4.3. Solving inverse problems

4.3.1. Tikhonov regularization

4.4. Methods for estimating regularization parameter λ

4.4.1. L-curve method

5. The inverse EEG problem

5.1. Spatial discretization in Talairach space

5.2. Data from a neurophysiological experiment

6. LORETA

6.1. The LORETA approach

6.2. Classification of J in a two-step-algorithm.

7. Bayesian approach to solve the inverse EEG problem

7.1. Bayes statistics

7.2. Bayesian solution of the inverse EEG problem

7.2.1. Homogeneous prior distribution

7.2.2. Heterogeneous prior distribution

7.3. EM algorithm

7.4. Developed algorithm under the assumption of heterogeneity

7.5. Applying presented approach on clinical data

8. Discussion

A. Mathematical background and terminology

A.1. Matrix algebra

A.2. Norm of a matrix

A.2.1. l1-norm

A.2.2. l2-norm

A.2.3. l∞-norm

A.2.4. Frobenius norm

A.3. Conjugate transpose

A.4. Trace of a matrix

A.5. The condition number of a matrix

A.6. Derivatives of vectors

Research Objectives and Core Themes

The primary objective of this thesis is to address the ill-posed inverse EEG problem by proposing a probabilistic Bayesian framework. The study aims to estimate the three-dimensional distribution of neural currents from scalp electrode measurements, assuming a heterogeneous distribution of neural activity to improve source localization and classification.

• Mathematical formulation of the inverse EEG problem and lead field matrices.
• Application of Bayesian statistics and mixture distributions to model neural current heterogeneity.
• Implementation of an Expectation-Maximization (EM) algorithm for simultaneous computation and classification of neural currents.
• Comparison of the proposed probabilistic approach with deterministic methods like LORETA using clinical test data.

Excerpt from the Thesis

Summary of Chapters

1. Introduction: Outlines the physiological basis of brain activity and the challenges involved in non-invasive EEG source localization.

2. EEG: Introduces the technology and methodology of EEG measurements, including physiological basics and the 10-20 electrode system.

3. Mathematical theory and functions: Details fundamental mathematical concepts used in the thesis, including convex functions, Lagrange multipliers, and the Laplace operator.

4. Inverse problems: Explains the nature of ill-posed inverse problems and provides methods for solving them, specifically focusing on Tikhonov regularization.

5. The inverse EEG problem: Discusses the spatial discretization of the brain into voxels and the use of the Talairach coordinate system.

6. LORETA: Presents the deterministic LORETA approach and its application to the inverse EEG problem.

7. Bayesian approach to solve the inverse EEG problem: Introduces the core probabilistic framework, including Bayes statistics, the EM algorithm, and the proposed heterogeneous model.

8. Discussion: Evaluates the experimental results, discusses limitations of the proposed approach, and offers perspectives for future research.

Keywords

EEG, inverse problem, neural currents, LORETA, Bayesian approach, EM algorithm, source localization, Talairach space, mixture distribution, Tikhonov regularization, brain mapping, neuroinformatics, clinical data, electroencephalography, probability.

Frequently Asked Questions

What is the core problem addressed in this thesis?

The thesis addresses the inverse EEG problem, which involves reconstructing the three-dimensional distribution of neural currents within the brain based on voltage fluctuations recorded by scalp electrodes.

What are the central thematic fields?

The central fields include electrophysiology, mathematical optimization (inverse problems), Bayesian statistical modeling, and computational neuroinformatics.

What is the primary research goal?

The primary goal is to develop a probabilistic Bayesian framework that allows for the simultaneous computation and classification of neural currents by assuming a heterogeneous distribution of activity.

Which scientific methods are utilized?

The work employs Bayesian inference, mixture distribution modeling, and the Expectation-Maximization (EM) algorithm, while comparing results against the standard LORETA method.

What does the main body cover?

It covers the mathematical foundations (convex functions, regularization), the description of the inverse problem, the implementation of the Bayesian approach, and the application of this method to clinical test data.

What are the characterizing keywords?

Key terms include EEG, inverse problem, LORETA, Bayesian approach, EM algorithm, and source localization.

How does the Bayesian approach differ from LORETA?

While LORETA is a deterministic approach, the Bayesian approach introduced here is probabilistic, allowing for the explicit modeling of heterogeneous neural regions through mixture distributions.

What role does the EM algorithm play in this study?

The EM algorithm is used as an iterative framework to handle the latent variables (the classification of voxels into different neural activity groups) while simultaneously estimating the current distributions.

Why is Talairach space relevant to this work?

Talairach space provides a standardized coordinate system that allows for the spatial discretization of the brain into voxels, which is necessary to define the discrete model of neural generators.

How is the issue of "overfitting" addressed in the algorithm?

Overfitting is mitigated by introducing a lower bound on the variance of the neural current distribution, specifically setting a threshold for the regularization parameter during iterations.