PhD_course_Roux_2.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

PhD_course_Roux_1.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

PhD_course_Roux_0.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

PhD_course_Roux.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

Permanent geological carbon storage will reduce greenhouse gas emissions and help mitigate climate change. Storage security is increased by CO₂ capillary trapping in cm-to-m scale, layered rock heterogeneities; features that are ubiquitous across storage sites worldwide. This talk will outline the challenges associated with modelling the impact of small-scale heterogeneity on large scale saturation distributions and trapping during geological CO₂ storage, including the difficulties in incorporating petrophysical and geological uncertainty into field-scale numerical models. Experimental results demonstrate the impact of cm-scale heterogeneity on pore-scale processes, which in turn influence large scale behaviour. Heterogeneity is shown to have a leading order impact on saturation distribution and storage capacity during geological CO₂ storage.

We study the positional information conferred by the morphogens Sonic Hedgehog and BMP in neural tube patterning. We use the mathematics of information theory to quantify the information that cells use to decide their fate. We study the encoding, recoding and decoding that take place as the morphogen gradient is formed, triggers a nuclear response and determines cell fates using a gene regulatory network.

When a biological system is modelled using a mathematical process, the next step is often to estimate the system parameters. Although computational and statistical techniques have been developed to estimate parameters for complex systems, this can be a difficult task. As a result, researchers have focused on revealing parameter-independent dynamical properties of a system. In this talk, we will discuss the study of qualitative behaviors of stochastic biochemical systems using reaction networks, which are graphical configurations of biochemical systems. The goal of this talk is to (1) introduce the basic modelling aspects of stochastically-modelled reaction networks and (2) discuss important results in this literature, including the random time representation, relationships between stochastic and deterministic models, and derivation of stability via network structures.