Program and courses

Courses description

Noufel Frikha

McKean-Vlasov SDEs, Related PDEs on the Wasserstein Space, and Applications

We will explore recent advances concerning nonlinear diffusion processes in the sense of McKean-Vlasov, and their connections to partial differential equations (PDEs) defined on the Wasserstein space, that is, the space of probability measures with finite second order moment. We will discuss recent results on the well-posedness - both in the weak and strong sense - of McKean-Vlasov stochastic differential equations (SDEs) driven by Brownian motion and/or jump processes. These results extend beyond the classical Cauchy-Lipschitz framework.

In the Brownian setting, we will describe the regularization effect of the noise, notably the existence and smoothness of the transition density - particularly in the measure argument - under uniform ellipticity assumptions. These smoothing effects are crucial for establishing the existence and uniqueness of solutions to the Kolmogorov-type PDEs posed on the Wasserstein space, even in the presence of irregular terminal conditions and source terms.

Such infinite-dimensional PDEs play a central role in deriving quantitative propagation of chaos estimates for mean-field approximations via interacting particle systems. Finally, we will discuss the numerical approximation of these equations using the Euler-Maruyama time discretization scheme.

Eva Löcherbach

Modeling spiking neurons by interacting systems of point processes with variable memory

I will introduce a continuous time probabilistic model for systems of interacting and spiking neurons. In this process, neurons spike at a rate depending on their membrane potential value. When spiking, they have a direct influence on their post-synaptic partners, namely, a fixed value, called "synaptic weight", is added to the potential of the postsynaptic neurons. In between successive spikes, due to some leakage effects, the membrane potential process follows a deterministic flow. 

Firstly, I will discuss the construction, well-posedness and the longtime behavior of the process, for a finite number of neurons and for infinite systems of neurons, both in the case with and without reset of the spiking neuron. 

I will then discuss mean field limits for the Hawkes description (without reset) of the model. In particular we will see how in the limit an ODE describing the evolution of the mean firing rate appears and how this approach allows to describe for example oscillatory behavior. If time permits, I will also discuss the influence of delay in the synaptic transmission and quickly speak about short term memory. 

The final part of the course will be devoted to the more difficult case with reset (the membrane potential of the spiking neuron goes back to a resting value, inducing discontinuities in the model). We will see how the limit process and its longtime behavior help us to explain important phenomena in neuroscience such as "metastability".