In recent years, the qualitative study of models derived from neuroscience has experienced significant progresses, particularly within the communities of probabilist and PDE specialists. The aim is to use certain models to understand the emergence of complex dynamics observed within interacting neural networks.
In this regard, numerous PDE models have been proposed, presenting structures, and...
Systems of pulse-coupled oscillators model synchronization through singular interactions occurring at discrete times, when particles reach a specific firing phase. They have numerous applications in physics, biology and engineering, for example to cardiac cells, neurons and fireflies. They could also represent a stepping stone for the understanding of networks of voltage-conductance neurons at...
Grid cells, with their striking hexagonal firing patterns, are neurons which play a key role in the internal navigational system of mammals. This talk will concern a nonlocal Fokker--Planck-like PDE which emerged in a pursuit to better understand the effects of noise on grid cell activity. When this model produces hexagonal network activity which persists when translated in accordance with the...
In large-scale excitatory neuronal networks, rapid synchronization manifests as multiple firing events (MFEs), mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset....
Mean-field limits of neural assemblies lead to nonlinear partial differential equations (PDEs) that have attracted considerable interest in recent years; see, for instance, the recent survey \cite{CR2025}. This interest stems from the fact that these equations raise several highly challenging mathematical questions within an unconventional formal framework, including the existence and...
