[doct2024-cronos] Koopman Analysis of Brain Signals

Le descriptif de l’offre ci-dessous est en Anglais_

**Niveau de diplôme exigé **:Bac + 5 ou équivalent

**Fonction **:Chercheur contractuel

**A propos du centre ou de la direction fonctionnelle**:
The Inria centre at Université Côte d'Azur includes 37 research teams and 8 support services. The centre's staff (about 500 people) is made up of scientists of different nationalities, engineers, technicians and administrative staff. The teams are mainly located on the university campuses of Sophia Antipolis and Nice as well as Montpellier, in close collaboration with research and higher education laboratories and establishments (Université Côte d'Azur, CNRS, INRAE, INSERM...), but also with the regiona economic players.
With a presence in the fields of computational neuroscience and biology, data science and modeling, software engineering and certification, as well as collaborative robotics, the Inria Centre at Université Côte d'Azur is a major player in terms of scientific excellence through its results and collaborations at both European and international levels.

**Contexte et atouts du poste**:
The thesis will take place within the Cronos Inria team.

**Mission confiée**:
**Context**

Electroencephalography (EEG) and magnetoencephalograpny (MEG) are modalities that allow the passive measurement, at the level of the scalp (EEG) or slightly above it (MEG),
of the electric (EEG) or magnetic (MEG) fields generated mainly by the electrical activity of the brain. They are used to characterize different brain pathologies such as,
electrical activities at this same time $t$ of a set of ''sources'' located at the level of the cerebral cortex. Measurement can be acquired at a very high temporal resolution
(typically at a sampling rate from 256Hz to 2kHz) but have a rather poor spatial resolution (a few hundreds of sensors or less), so that they are particularly interesting to characterize
the time organization of brain activity.

**Project description**

The purpose of this project is to estimate neurocognitive states from EEG/MEG data using a Koopman decomposition. A cognitive process is achieved through a succession
of neurocognitive states which are signatures of the neuronal system functioning. The evolution and representation of these states originate from different parallel
dynamical processes. This suggests that these states stem from an elementary dynamical system that we seek to identify and characterize given a cognitive process.

models focus on the study of what is the expected value of $X_t$ given its past while Koopman theory looks at the global evolution of $f(X_t)$ for a whole set of functions
$f$ called observables. The Koopman operator is the linear transformation that associates to an observable $f$ the expected value of $f(X_t)$ given its past.
This linear operator is then amenable to eigen elements decomposition or low rank approximation. The problem is thus to estimate the Koopman operator from a set of
measurements and there exists theoretical guarantees for this procedure [4]. We thus have a framework to study data from a dynamical system without knowing its underlying
equations. It is thus particularly well suited to the study of brain signals and can be complemented with machine learning techniques for example in task segregation.

algorithms to EEG data on selected databases of EEG signals such as the Physionet Sleep-EDF Database Expanded.
The interest of such a dataset is that kooplearn has already been tested on sleep data in the rat (with intracranial
electrodes), and it seems interesting to see whether those results can expand in humans (with non invasive electrodes). In a second step, the formalism will be extended.
So far Koopman decomposition is made for time-homogeneous data and this restricts us to precisely timed tasks. We will thus define a Koopman decomposition for time varying
observations. As a by-product, this will provide methods to assess time-homogeneity (or stationarity) in data. Finally, we will investigate perturbations of the Koopman
operator in order to estimate changes in the underlying dynamical systems. For example, if we were to change external parameters of the task, can we estimate the new Koopman
operator without estimating it from scratch?

**References**

[1] Steven L. Brunton, Marko Budisic, Eurika Kaiser, and J. Nathan Kutz. Modern Koopman Theory for Dynamical Systems. SIAM Review, 64(2):229-340, May 2022. Publisher: Society for Industrial and Applied Mathematics.

[2] Daniel Lehmberg. Operator-informed machine learning: Extracting geometry and dynamics from time series data. page 222, 2022.

**Collaboration**:
The recruited person will be in connection with
- Patricia Reynaud-Bouret, CNRS, Laboratoire JA Dieudonné
- V. Kostic, IIT Genova
- Karim Lounici, Ecole Polytechnique Paris

**Principales activités**:

- Propose methods to study/define dynamical system models of brain activity from data

**Compétences**:
Technical skills and level req