A journey in the fields of PDE, probabilities and statistics with point processes

Temporal point processes form a family of probabilistic models commonly used to model the timing of brief events (e.g. stock market orders, infections, action potentials). The presentation will focus on applications in neuroscience.

From a probabilistic point of view, we seek to explain qualitatively certain phenomena observed in large neural networks (synchronisation, wandering of localized brain activity). Furthermore, when the number of neurons is large, these probabilistic models are well approximated by deterministic models (ODE or PDE) in a mean-field framework. This is called the order 0 approximation. A detailed study of the fluctuations associated with this approximation allows us to derive a first-order approximation in the form of an SDE or SPDE. This type of result will be presented in three modeling frameworks.

From a statistical point of view, we seek to quantitatively estimate the underlying network based on observations of brain activity. This type of result will be presented in a mean-field framework with two interacting populations of neurons. The two objectives are: 1) to estimate the network connectivity parameter, 2) to identify the two populations of neurons.

President:

Prof. Clémentine Prieur (Université Grenoble Alpes)

Reporters:

• Prof. Vincent Rivoirard (Université Paris Dauphine - PSL )
• Prof. Delphine Salort (Sorbonne Université )
• Prof. Wilhelm Stannat (TU Berlin )

Examinators:

• Prof. Eric Luçon (Université d'Orléans )
• Prof. Clémentine Prieur (Université Grenoble Alpes )
• DR CNRS Patricia Reynaud-Bouret ( Université Côte d'Azur )