No abstract

Electroencephalography (EEG) is a widely used cerebral activity measuring device for both clinical and everyday life applications. In addition to denoising and potential classification, a crucial step in EEG processing is to extract relevant features. Topological data analysis (TDA) as an emerging tool enables to analyse and understand data from a different angle than traditionally used methods. As a higher dimensional analogy of graph analysis, TDA can model rich interactions beyond pairwise relations. It also distinguishes different dynamics of EEG time series. TDA remains largely unknown to the EEG processing community while it fits well the heterogeneous nature of EEG signals. This short review aims to give a quick introduction to TDA and how it can be applied to EEG analysis in various applications including brain-computer interfaces (BCIs). After introducing the objective of the article, the main concepts and ideas of TDA are explained. Next, how to implement it for EEG processing is detailed, and lastly the article discusses the benefits and limitations of the method.

Brain-computer interfaces (BCIs) allow direct communication between one’s central nervous system and a computer without any muscle movement hence by-passing the peripheral nervous system. They can restore disabled people’s ability to interact with their environment, e.g. communication and wheelchair control. However, to this day their performance is still hindered by the non-stationarity of electroencephalography (EEG) signals, as well as their susceptibility to noise from the users’ environment and from their own physiological activity. Moreover, a non-negligible amount of users struggle to use BCI systems based on motor imagery. In this paper, a new method based on the path signature is introduced to tackle this problem by using features which are different from the usual power-based ones. The path signature is a series of iterated integrals computed from a multidimensional path. It is invariant under translation and time reparametrization, which makes it a robust feature for multichannel EEG time series. The performance can be further boosted by combining the path signature with the gold standard Riemannian classifier in the BCI field exploiting the geometric structure of symmetric positive definite (SPD) matrices. The results obtained on publicly available datasets show that the signature method is more robust to inter-user variability than classical ones, especially on noisy and low-quality data. Hence, this study paves the way towards the use of mathematical tools that until now have been neglected, in order to tackle the EEG-based BCI variability issue. It also sheds light on the lead-lag relationship captured by path signature which seems relevant to assess the underlying neural mechanisms.

Neural oscillations captured by electroencephalography (EEG) can be used by Brain-Computer Interfaces (BCIs) to reveal the underlying mental processes and enable explicitly or implicitly interacting with one's environment. Most feature extraction techniques are based on spatial filters and power analyses in multiple frequency bands. The global geometry feature is seldom investigated. In this paper, the spatial aspects of EEG signals are studied using the Laplace-Beltrami operator. The eigenvectors of the Laplace-Beltrami operator form an orthonormal basis for square-integrable functions over the scalp and capture the geometry of electrodes' position in a hierarchical way. The signals are decomposed into different spatial frequency components by the projection into the eigenspaces of the Laplace-Beltrami operator. Dimensionality reduction could be done by using only the low frequency components. This method is compared with Principal Component Analysis (PCA) filtering on publicly available motor imagery BCI data and achieved comparable results while being unsupervised, dataindependent and requiring 33.7% less computation time.