Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Seminar Assistant

Duc Nguyen

Date and Place

Talks (90 min each) will be made in one or two day blocks in the middle or at the end of the summer term.

Prerequisites

The seminar covers a range of topics with varying levels of mathematical complexity. Participants are expected to have a background in:

• Basic probability and statistics
• Linear algebra (matrix decompositions, eigenvalues, projections)
• Fourier analysis and basic signal processing is desirable, but not mandatory.

Most advanced concepts will be introduced and explained during the seminar.

Intended Audience

Bachelor and Master students in mathematics, statistics, data science engineering, biostatistics, or related fields who are interested in multivariate time series analysis and its application to EEG signals.

Content

This seminar provides an in-depth exploration of multivariate time series analysis for EEG signals, combining both theoretical and applied perspectives. The goal is to equip students with mathematical tools and statistical techniques to analyze EEG data effectively. The course will focus on time-frequency analysis, filtering techniques, connectivity measures, and statistical testing for EEG signals, all framed within the multivariate time series framework.

The seminar is structured into eight talks, each covering a fundamental aspect of EEG time-series analysis. The topics are arranged in a logical progression, starting with foundational concepts and moving towards advanced statistical methods.

The first two talks introduce essential mathematical tools for EEG time-series analysis. These include the Fourier Transform, which allows decomposition of signals into frequency components, and the Wavelet Transform, which provides a time-frequency representation more suitable for nonstationary EEG data.

Talk 1: Introduction & Fourier Transform for EEG
This talk provides an introduction to EEG as a multivariate time series and explores fundamental spectral analysis techniques. It covers the discrete Fourier transform (DFT), fast Fourier transform (FFT), power spectral density (PSD), and the role of frequency-domain analysis in EEG. The talk will also discuss the limitations of Fourier analysis in dealing with nonstationary signals.

Talk 2: Wavelet Transform & Time-Frequency Analysis
The second talk introduces the wavelet transform, an alternative to Fourier analysis for EEG signals. We will discuss the Morlet wavelet, continuous wavelet transform (CWT), and how wavelets improve time-frequency resolution compared to the short-time Fourier transform (STFT). This talk will also cover the trade-off between time and frequency resolution.

Talk 3: Filtering, STFT & Multitaper
This talk discusses various filtering techniques used in EEG preprocessing, including bandpass filtering and the Hilbert transform for extracting instantaneous phase and amplitude. We will then introduce the short-time Fourier transform (STFT) for analyzing nonstationary EEG signals and compare it with the multitaper method for spectral estimation, which reduces variance in power spectral density estimation.

Talk 4: Time-Frequency Power Transformations & ITPC
In this talk, we focus on power transformations in EEG analysis. We will distinguish between total power, phase-locked power, and non-phase-locked power and introduce the concept of intertrial phase clustering (ITPC), a measure of phase consistency across trials. Applications of ITPC in event-related studies and its role in detecting neural synchronization will be discussed.

Following that, we will explore spatial filtering techniques and EEG source imaging, followed by an in-depth discussion of EEG connectivity measures.

Talk 5: Spatial Filters & Source Imaging
This talk covers spatial filtering techniques to improve EEG signal localization. Topics include the surface Laplacian, which enhances spatial resolution, and principal component analysis (PCA) for dimensionality reduction. We will also introduce EEG source imaging techniques, such as dipole fitting and distributed source imaging, which attempt to reconstruct the neural sources of EEG signals.

Talk 6: Phase-Based & Power-Based Connectivity
Connectivity analysis is crucial in EEG research. This talk explores two main approaches: phase-based connectivity, which measures synchronization between EEG channels using metrics such as phase-locking value (PLV) and inter-site phase clustering (ISPC), and power-based connectivity, which examines correlations in power fluctuations across channels. We will discuss their applications in cognitive neuroscience.

The final two talks will focus on advanced statistical methods for EEG data analysis. These methods are crucial for extracting reliable conclusions from large-scale EEG datasets.

Talk 8: Statistical Analyses in EEG
The final talk focuses on statistical challenges in EEG analysis. We will introduce nonparametric permutation testing, multiple comparisons correction techniques (false discovery rate, cluster-based correction), and methods for dealing with large-scale hypothesis testing. These methods are essential for ensuring the validity of EEG research findings.

The seminar will consist of eight talks, but we are open to adding more topics if there is sufficient interest from students. Participants are encouraged to expand upon key mathematical concepts by referring to additional sources beyond the main textbook.

Contact and Registration

To register for the seminar, please send an email to tran-1.nguyen@uni-ulm.de by April 30th, 2025. In your email, include: Your name; Matriculation number; Your program of studies; Relevant coursework in probability, statistics, or signal processing

After the registration deadline, we will schedule an initial meeting to discuss topic preferences and finalize the seminar structure.

Criteria to pass the seminar

Each participant (or group) is required to prepare and present one talk. A written summary of the presentation is also expected. Talks will be held in English, and a preliminary version of the slides must be submitted two weeks before the talk for feedback. Successful completion of these requirements will qualify the participant for passing the seminar.

Literature

[1] M. X. Cohen, Analyzing Neural Time Series Data: Theory and Practice, The MIT Press, 2014, ISBN 978-0262319553, doi.org/10.7551/mitpress/9609.001.0001.

Since the main book is more explanatory and application-focused, students are encouraged to explore additional mathematical resources for deeper theoretical insights. Some suggested references include:

[2] D. R. Brillinger, Time Series, Society for Industrial and Applied Mathematics, 2001, doi.org/10.1137/1.9780898719246.

[3] A. Oppenheim, R. Schafer, Discrete-Time Signal Processing, 3rd ed., Pearson, 2013, ISBN 978-1-292-02572-8 (Print), ISBN 978-1-292-03815-5 (Electronic).

[4] R. H. Shumway, D. S. Stoffer, Time Series Analysis and Its Applications: With R Examples, 5th ed., Springer Texts in Statistics, Springer Cham, 2025

Additional references will be cited in specific talks based on the mathematical background required.