This textbook provides a comprehensive and didactic introduction from the basics to the current state of the art in the field of EEG/MEG source reconstruction. Reconstructing the generators or sources of electroencephalographic and magnetoencephalographic (EEG/MEG) signals is an important problem in basic neuroscience as well as clinical research and practice. Over the past few decades, an entire theory, together with a whole collection of algorithms and techniques, has developed.
In this textbook, the authors provide a unified perspective on a broad range of EEG/MEG source reconstruction methods, with particular emphasis on their respective assumptions about sources, data, head tissues, and sensor properties. An introductory chapter highlights the concept of brain imaging and the particular importance of the neuroelectromagnetic inverse problem. This is followed by an in-depth discussion of neural information processing and brain signal generation and an introduction to the practice of data acquisition. Next, the relevant mathematical models for the sources of EEG and MEG are discussed in detail, followed by the neuroelectromagnetic forward problem, that is, the prediction of EEG or MEG signals from those source models, using biophysical descriptions of the head tissues and the sensors. The main part of this textbook is dedicated to the source reconstruction methods. The authors present a theoretical framework of the neuroelectromagnetic inverse problem, centered on Bayes’ theorem, which then serves as the basis for a detailed description of a large variety of techniques, including dipole fit methods, distributed source reconstruction, spatial filters, and dynamic source reconstruction methods. The final two chapters address the important topic of assessment, including verification and validation of source reconstruction methods, and their actual application to real-world scientific and clinical questions.This book is intended as basic reading for anybody who is engaged with EEG/MEG source reconstruction, be it as a method developer or as a user, including advanced undergraduate students, PhD students, and postdocs in neuroscience, biomedical engineering, and related fields.
Author(s): Thomas R. Knösche, Jens Haueisen
Publisher: Springer
Year: 2022
Language: English
Pages: 428
City: Cham
Preface
Mathematical Notation and Symbols
General Rules and Symbols
Differential Operators
Norms
Integrals
Special Functions
Indexing
General Quantities
Physical Quantities
Contents
About the Authors
1: Introduction
Definition 1.1
References
2: Neural Tissue and Its Signals
2.1 Overview
2.2 Basics of Neuroanatomy
2.3 Neural Information Processing
2.4 Apertures to Brain Activity
2.4.1 Overview
2.4.2 Electrophysiological Methods
2.4.2.1 Electrical Activity in Neural Tissue
2.4.2.2 Polarity of EEG and MEG
2.4.2.3 Primary and Secondary Currents
2.4.2.4 Far Field and Volume Conduction
2.4.3 Biochemical Methods
2.4.3.1 Processes of Interest
2.4.3.2 Using Radiotracers in PET and SPECT
2.4.3.3 Magnetic Resonance Spectroscopy (MRS)
2.4.4 Metabolic Methods
2.4.4.1 Energy Supply of the Brain
2.4.4.2 Functional MRI (fMRI)
2.4.4.3 Near-Infrared Spectroscopy (NIRS)
2.4.4.4 Fluorodeoxyglucose (FDG) PET
2.5 The Role of EEG and MEG in Functional Neuroimaging
References
3: Electro- and Magnetoencephalographic Measurements
3.1 Introduction
3.2 Electroencephalography
3.2.1 Introduction
3.2.2 Electrodes
3.2.3 Electrode Arrangements and Montages
3.2.4 Measurement Instrumentation
3.2.5 Spontaneous and Event-Related EEG
3.3 Magnetoencephalography
3.3.1 Introduction
3.3.2 Sensors
3.3.3 Magnetometers and Gradiometers
3.3.4 Shielding
3.3.5 Cryostats
3.3.6 Example Biomagnetometers
3.3.7 Spontaneous and Event-Related MEG
3.4 Intracranial Recordings
3.4.1 Introduction
3.4.2 Electrocorticography
3.4.3 Intracerebral Electrodes
3.4.4 Applications
3.5 Co-Registration of EEG and MEG Sensors with Head Models
3.6 Preprocessing of EEG and MEG Recordings
3.6.1 Introduction
3.6.2 Biosignal Characteristics
3.6.3 Artifact Detection
3.6.4 Interference Reduction
3.7 Comparison of Electric and Magnetic Recordings
3.7.1 Introduction
3.7.2 Vortex Currents
3.7.3 EEG/MEG Sensitivity Profiles of Source Orientation and Depth
3.7.4 Conclusions
References
4: Source Models
4.1 Introduction
4.2 Modeling the Spatial Distribution of the Primary Current Density
4.2.1 Introduction
4.2.2 Multipole Expansion
4.2.2.1 General Concept: Taylor Expansions of the Poisson’s Equations
4.2.2.2 The Multipole Terms for the Electric Scalar Potential
4.2.2.3 The Multipole Terms for the Magnetic Vector Potential
4.2.2.4 Interpretation of the Multipole Moments
4.2.2.5 The Choice of the Expansion Point
4.2.3 Line, Surface, and Volume Sources
4.2.4 Dipole Modeling
4.2.4.1 Overview
4.2.4.2 Dipole Constraints
4.2.4.3 Spatio-temporal Dipole Models
4.2.4.4 Current Density Modeling
4.3 Dynamic Source Models
4.3.1 Basic Concepts
4.3.2 Models of Single Neurons
4.3.2.1 Overview
4.3.2.2 Leaky Integrate-and-Fire (LIF) Models
4.3.2.3 Non-linear Integrate-and-Fire Models
4.3.2.4 Firing Rate Models
4.3.2.5 The Hodgkin-Huxley Model and Its Relatives
4.3.2.6 Cable Equations and Compartment Models
4.3.3 Neural Mass Models
4.3.3.1 Overview
4.3.3.2 Criteria for Neural Masses
4.3.3.3 General Framework
4.3.3.4 Linear NMM Approaches Based on the Mean Membrane Potential
Overview
The Models of Lopez da Silva et al. and Jansen and Rit
The Robinson Model
The Liley Model
4.3.3.5 The Montbrió Model
4.3.3.6 Neural Mass Models as Source Models for EEG and MEG
General Remarks
Neural Mass Models in Dynamic Causal Modeling
4.3.4 Abstract Dynamic Models
References
5: Forward Models
5.1 Basics
5.1.1 Introduction
5.1.2 The Neuroelectromagnetic Field
5.1.2.1 Overview
5.1.2.2 The Magnetic Field
5.1.2.3 The Electric Field
5.1.2.4 Links Between Electric and Magnetic Fields
5.1.2.5 Maxwell’s Equations in Biological Tissue
5.1.2.6 Poisson’s Equations
5.1.3 Sensor Models
5.1.3.1 Overview
5.1.3.2 MEG Sensors
5.1.3.3 EEG Electrodes
5.1.4 Conductivity Distribution
5.1.5 Classification of Forward Models
5.2 Analytic and Quasi-Analytic Methods
5.2.1 Overview
5.2.2 Infinite Space and Half-Space
5.2.2.1 Electric Potentials: EEG
5.2.2.2 Magnetic Flux Density: MEG
5.2.3 Spherical and Ellipsoidal Models
5.2.3.1 Electric Potentials: EEG
5.2.3.2 Magnetic Flux Density: MEG
5.2.4 Approximation of the Head by Spheres and Ellipsoids
5.2.4.1 Introduction
5.2.4.2 Single Sphere/Ellipsoid Models for EEG
5.2.4.3 Single Sphere/Ellipsoid Models for MEG
5.2.4.4 Multiple Local Sphere/Ellipsoid Models for EEG and MEG
5.3 Numerical Methods
5.3.1 Introduction
5.3.2 Boundary Element Methods
5.3.2.1 Introduction
5.3.2.2 Method of Weighted Residuals
Overview
Approximation of the Potentials
Formulation of Goal Functions for Parameter Optimization
Solution of the Equation System
Interpolation of Potentials at Electrodes
5.3.2.3 “Center of Mass” Approach
5.3.2.4 Vertex Approach
5.3.2.5 Linear Galerkin Approach
5.3.2.6 Solution for the Magnetic Flux Density
5.3.2.7 Single Layer and Symmetric BEM
5.3.2.8 Construction of Models
Compartment Definition
Boundary Definition
Conductivity Values
5.3.3 Finite Difference Methods
5.3.3.1 Problem Description
5.3.3.2 The Finite Difference Formulation of the Laplace Operator
5.3.3.3 The Description of the Source Current Density
5.3.3.4 Solving the Linear Equation System
5.3.3.5 Computation of the Magnetic Field
5.3.4 Finite Element Methods
5.3.4.1 Introduction
5.3.4.2 General Approach
5.3.4.3 Discretization and Base Functions
5.3.4.4 Modeling the Sources
Direct Approaches
Subtraction Approach
5.3.4.5 Computation of Potentials
Overview
Iterative Solvers for FEM Problems
Parallelization
Transfer Matrix Approach
5.3.4.6 Computation of the Magnetic Flux Density
5.3.4.7 Construction of FEM Models
Overview
Segmentation
Tetrahedral Meshes
Hexahedral Meshes
5.4 Semi-analytic Solutions
5.4.1 Introduction
5.4.2 Multiple Multipole Method
5.4.3 Perturbative Analytical Solutions
5.5 Comparison of Forward Models
5.5.1 Overview
5.5.2 Accuracy
5.5.3 Information Availability
5.5.4 Computational Costs
5.5.5 Which Method to Choose?
5.6 Specifying Electric Conductivity
5.6.1 Overview
5.6.2 Local Biophysical Measurement
5.6.3 Global Fitting of Conductivities
5.6.4 Conductivity Anisotropy
5.7 Lead Field Concept
References
6: Inverse Methods
6.1 Introduction
6.2 Working with Probabilities: The Bayesian Approach
6.2.1 Overview
6.2.2 Definition of Priors
6.2.2.1 General Remarks
6.2.2.2 Prior Moments
6.2.2.3 Hyperpriors
6.2.3 Computation of Likelihood
6.2.4 Model Evidence
6.3 Overview on Inverse Methods
6.4 Assuming Focality: Single and Multiple Dipole Fit Methods
6.4.1 Overview
6.4.2 Dipole Fit as Bayesian Optimization
6.4.2.1 Model Specification
6.4.2.2 Likelihood
6.4.2.3 Priors
6.4.2.4 Model Evidence and Model Selection
6.4.2.5 Optimization (Sampling of the Posterior)
Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Solutions
Determination of the Full Posterior Distribution
6.4.2.6 Bayesian Dipole Fit Approaches in the Literature
6.4.3 Classical Spatio-temporal Dipole Fit
6.4.3.1 Model Specification
6.4.3.2 Goal Function
6.4.3.3 Goal Function Modifications: Constraints, Weightings, and Penalties
Constraints Encoded into the FO
Penalty Functions
6.4.3.4 Model Fitting
Overview
Goal Function or Deviation Scan
Local Optimization
Population Methods
Stochastic Methods
6.4.3.5 Model Selection: The Number of Dipoles
Overview
Model Selection Based on Goodness-of-Fit
Pairwise Model Comparison Based on Maximum Likelihood
Wald Tests
Information Criteria
6.5 Reconstruction of Distributed Sources: Minimum Norm Approaches
6.5.1 Introduction
6.5.2 Overview
6.5.2.1 Gaussian Noise and Gaussian Priors
6.5.2.2 Generalizations
6.5.3 Minimum ℓ2 Norm Solutions
6.5.3.1 Overview
6.5.3.2 The Resolution Matrix
6.5.3.3 (Weighted) Minimum Norm Estimates
Overview
Weighting Matrix
Regularization
6.5.3.4 Exact Low-Resolution Brain Electromagnetic Tomography (eLORETA)
6.5.3.5 Low-Resolution Electromagnetic Tomography (LORETA)
6.5.3.6 Encoding Prior Knowledge into the Prior Source Covariance Matrix
Overview
Diagonal Elements of Prior Covariance Matrix
Off-Diagonal Elements of Prior Covariance Matrix
Functional Area Constrained Estimator (FACE)
PatchLORETA
Combination of MN and LORETA
Use of Hyperpriors: VARETA and HIGGS
6.5.3.7 Anatomically Informed Basis Functions
6.5.4 Minimum ℓ1 Norm Solutions
6.5.4.1 Overview
6.5.4.2 LASSO Solutions
6.5.4.3 Solutions Using the ℓ1 Norm for the Data Term
6.5.4.4 Linear Programming Approaches
6.5.4.5 The Direction Bias Problem
6.5.4.6 Spatial Instability
6.5.4.7 Transformed ℓ1 Norms: Patchy Sources and Edge Preservation
6.5.5 Minimum Mixed Norm Solutions
6.5.5.1 Overview
6.5.5.2 Nested Norm Solutions
6.5.5.3 Summed Norm Solutions
6.5.6 Minimum ℓp Norm Solutions with 0 < p < 1
6.5.7 Minimum Support Imaging: Minimum ℓ0 Norm Solutions
6.5.8 Non-linear Penalties
6.5.9 Spatio-temporal Approaches
6.5.9.1 Overview
6.5.9.2 Simultaneous Approaches
Overview
Spatio-temporal Component Expansions
Spatio-temporal Regularization
6.5.9.3 Iterative Approach: Dynamic LORETA
6.5.10 Bayesian Approaches to Distributed Source Reconstruction
6.5.10.1 Overview
6.5.10.2 Joint Models of EEG/MEG and fMRI
6.5.10.3 Parametric Empirical Bayes (PEB)
6.5.10.4 Multiple Sparse Priors (MSP)
6.5.10.5 Joint Hierarchical Models
6.5.10.6 Model Selection and Bayesian Model Averaging (BMA)
6.5.11 Recursive Methods
6.5.11.1 Overview
6.5.11.2 The FOCUSS Algorithm
6.5.11.3 Classical LORETA Analysis Recursively Applied (CLARA)
6.5.11.4 Iteratively Reweighted Edge Sparsity Minimization (IRES)
6.5.12 Component-Based Methods
6.5.12.1 Overview
6.5.12.2 Transformation into Lower-Dimensional Space
6.5.12.3 Sparsity in Transformed Spaces
6.5.12.4 Matching Pursuit
6.5.13 Post Hoc Weighting
6.6 Spatial Filters and Scanning Methods
6.6.1 Overview
6.6.2 Multiple Signal Classification (MUSIC) and Related Methods
6.6.2.1 Classical MUSIC
6.6.2.2 Recursive MUSIC
6.6.2.3 First Principal Vectors (FINES) Method
6.6.2.4 The Problems of Correlated Sources and Correlated Noise
6.6.3 Beamformer Methods
6.6.3.1 Overview
6.6.3.2 Linearly Constrained Minimum Variance Beamforming (LCMV)
6.6.3.3 Nulling Beamformer
6.6.3.4 Contrast-Based Beamforming
6.6.3.5 Beamforming in the Frequency Domain
6.7 Identification of Dynamic Source Models
6.7.1 Overview
6.7.2 Dynamic Causal Modeling
6.7.3 Abstract Dynamic Modeling
References
7: Assessment
7.1 Introduction
7.2 Performance Measures
7.2.1 Definitions
7.2.2 Comparison Metrics
7.2.2.1 Overview
7.2.2.2 Extraction of Source Parameters from Distributed Source Representations
7.2.2.3 Parametric Comparison Metrics
7.2.2.4 Vector Comparison Metrics
7.3 Sensitivity and Uncertainty Analysis
7.3.1 Overview
7.3.2 Uncertainty due to Uncertain Model Parameters or Input Variables
7.3.3 Uncertainty due to Uncertain Qualitative Model Choices
7.4 Generation of Reference Data
7.4.1 Computer Simulations
7.4.1.1 Overview
7.4.1.2 Generation of Artificial Noise
7.4.1.3 Source Model Definition
Focal Source Reconstruction Methods
Distributed Source Reconstruction Methods
Multiple Signal Classification and Beamformers
Identification of Dynamic Source Models
Inverse Crime
Examples from Literature
7.4.2 Physical Phantoms
7.4.3 Animal and Human Experiments
7.4.3.1 Overview
7.4.3.2 In Vitro Investigations
7.4.3.3 Artificial Sources in Living Organisms
7.4.3.4 EEG/MEG Measurements with Simultaneous Invasive Recordings
7.4.3.5 EEG/MEG Measurements with Simultaneous or Separate fMRI
7.4.3.6 EEG/MEG Measurements with Known Sources
References
8: Applications
8.1 Introduction
8.2 Focal Solutions Through Dipole Fit Methods
8.3 Smooth Solutions by Minimum ℓ2 Norm Methods
8.4 Sparse Solutions by Minimum ℓ≤1 Norm or Recursive Methods
8.5 Spatio-Temporal Distributed Solutions
8.6 Spatial Filters and Scanning Methods
References
Index
