The inverse EEG problem

Inhaltsverzeichnis (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 multipier
    • 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 >
    • 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

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis examines the inverse EEG problem, a complex and ill-posed problem in mathematics and neuroinformatics. It aims to develop a probabilistic Bayesian approach to solve this problem, assuming a heterogeneous distribution of neural currents in the brain. The work focuses on the simultaneous classification and computation of neural currents based on EEG measurements.

• The nature and challenges of the inverse EEG problem
• Bayesian methods for solving ill-posed problems
• The assumption of heterogeneous neural current distribution
• Expectation-Maximization (EM) algorithms for classification and computation
• Application of the proposed approach to real EEG data

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter 1: Introduction Provides a general overview of the inverse EEG problem and its significance in neuroinformatics.
• Chapter 2: EEG Discusses the physiological basics of the brain, including cells, resting potential, and action potentials. It then delves into the technology of EEG measurements, including its history and the 10-20 system.
• Chapter 3: Mathematical theory and functions Introduces essential mathematical concepts and functions used in the thesis, including convex functions, Lagrange multipliers, and the Laplace operator.
• Chapter 4: Inverse problems Defines inverse problems and provides examples, discussing methods like Tikhonov regularization for solving them.
• Chapter 5: The inverse EEG problem Focuses on the spatial discretization of the brain using the Talairach space and analyzes data from a neurophysiological experiment.
• Chapter 6: LORETA Introduces the LORETA approach for solving the inverse EEG problem, detailing its methodology and classification techniques.
• Chapter 7: Bayesian approach to solve the inverse EEG problem Explores Bayesian statistics and its application to the inverse EEG problem, discussing both homogeneous and heterogeneous prior distributions. It presents an EM algorithm developed under the assumption of heterogeneity and demonstrates its application to clinical data.

Schlüsselwörter (Keywords)

The key terms and concepts in this thesis include the inverse EEG problem, Bayesian inference, heterogeneous neural currents, EM algorithm, LORETA, spatial discretization, Talairach space, EEG measurements, and neuroinformatics. These terms represent the main areas of research focus and the fundamental concepts explored in this work.