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# Mathematical Aspects of EEG and MEG Based Neuroimaging

2020/2021
Учебный год
ENG
Обучение ведется на английском языке
4
Кредиты
Статус:
Курс по выбору
Когда читается:
1-й курс, 3 модуль

### Course Syllabus

#### Abstract

The course “ Mathematical Aspects of EEG and MEG Based Neuroimaging” aims to introduce masters graduate students to basic theory of inverse modelling used to analyze the distribution of neuronal sources on the basis of EEG or MEG data. This shall prove to be useful for the students who are interested in learning mathematical aspects behind the process of converting non-invasively recor neuronal behavior and show the ways to motivate model choice as well as relations between the features of neuronal activity and dynamical properties of the models.ded data into the dynamic maps of neural activity. During the course we will briefly explore the forward model that describes the way the neuronal sources are mixed into sensor signals. The major portion of the class will be devoted to studying three classes of the approaches used to tackle the underdetermined inverse problem of EEG and MEG that lies in the heart of the transition from the sensor space to source space. The course can be considered as a deep dive into the engineering mathematics behind EEG and MEG based neuroimaging, one of the topics presented during the introductory “Neuroimaging techniques” class. We will start exploration of the inverse modelling from the classification of different types of approaches to reconstruction of neuronal sources from the multichannel EEG and MEG data. Then we will explore several representative solutions for the three main classes of these methods, will see how they behave when applied to modelled and real data, will learn the basic assumptions behind these methods and the effect of their parameters. The course provides students with the basic understanding of the inverse modelling philosophy in application to MEG and EEG, prepares them for comprehending modern methodological literature and attempts to build a landshaft for reasoning to support an educated choice of an inverse solver to apply in a specific study.

#### Learning Objectives

• Gain understanding of the basic terms of forward and inverse models of EEG and MEG: volume conductor, equivalent current dipole, gain matrix, topography, lead field, ill-posed problem, regularization, global vs. local optimization, resolution kernel.
• Gain understanding of principles underlying the classification of EEG and MEG inverse solvers.
• Gain skills in applying an inverse solver to a real MEG\EEG data and tune its parameters to achieve the desired trade-off between the accuracy and the stability\reproducibility of the obtained solution.
• Gain skills in applying the existing software (MNE Python \ Brainstorm) and building pipelines for solving the inverse problem and presenting the results.
• Gain skills in implementing from scratch an inverse solver of their choice.

#### Expected Learning Outcomes

• Know basic terms of forward and inverse models of EEG and MEG: volume conductor, equivalent current dipole, gain matrix, topography, lead field, ill-posed problem, regularization, global vs. local optimization, resolution kernel.
• Know principles underlying the classification of EEG and MEG inverse solvers.
• Possess skills in applying an inverse solver to a real MEG\EEG data and tune its parameters to achieve the desired tradeoff between the accuracy and the stability\reproducibility of the obtained solution
• Know how to apply the existing software (MNE Python \ Brainstorm) and build pipelines for solving the inverse problem and present the results.
• Be capable of implementing from scratch and applying an inverse solver of their choice.

#### Course Contents

• EEG and MEG data origin.
EEG and MEG data origin. Equivalent current dipole. Maxwell Equations. Forward model. Gain matrix. Lead field. EEG and MEG measurement equation. Inverse problem and classification of methods to solve it.
• Inverse problem in non-parametric formulation.
Inverse problem in non-parametric formulation. Ill-posedness. Resolution kernel. Pseudoinverse. Minimum norm solution and its Bayesian interpretation. The role of prior-distribution. Methods to construct priors. Maximally smooth solutions with LORETA. Dynamic statistical parametric maps, dSPM. Neural activity index, NAI.
• Inferring prior distribution from the data.
Inferring prior distribution from the data. Iterative heuristics, FOCUSS. Variational Bayesian principle and related methods. Group inverse with GALA.
• Parametric methods.
Parametric methods. Dipole fitting idea. Residuals minimization. Local minima problem. Nelder-Mead simplex method. Fixed vs. freely oriented dipoles. Moving dipole \ rotating dipole. Data covariance matrix. The notion of subspace correlation. Signal subspace. Subspace methods. MUSIC. RAP-MUSIC.
• Local optimization.
Local optimization. LCMV Beamforming principle. Estimation source distributions with beamformers. Estimation of source timeseries. Signal cancellation in the environment with correlated sources. Dual-core beamformer. Modified covariance beamformer
• EEG and MEG data preprocessing.
EEG and MEG data preprocessing. Sources of artifacts, methods to cope with artifacts. ICA. Extraction of harmonic components from the data. SSD. Full-blown Kalman filter based modeling.
• Brainstorm and MNE-Python software.
Brainstorm and MNE-Python software. Organizing principles. Sample pipelines.

#### Assessment Elements

• Homework assignments
• Final exam

#### Interim Assessment

• Interim assessment (3 module)
0.6 * Classwork + 0.4 * Final exam

#### Recommended Core Bibliography

• Hari, R., & Puce, A. (2017). MEG-EEG Primer. New York, NY: Oxford University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2097017
• Ilmoniemi, R., & Sarvas, J. (2019). Brain Signals : Physics and Mathematics of MEG and EEG. Cambridge: The MIT Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2118154