Courses

Emeric Bouin (Université Paris Dauphine) 

Transport phenomena in Mathematical Biology

Many phenomena in mathematical biology can be described by using the mathematical properties of the transport equation and more generally a kinetic description of a large population of individuals. One main example is the pattern formation that occurs naturally while observing a flock of birds, a school of fish or a swarm of bees. A possible mathematical description of this behaviour uses the kinetic gas description of statistical mechanics to describe the interaction of a large number of individuals. With this perspective, swarming would be a consequence of an equilibrium between large-range attraction between moving individuals of the same species and a long-rangerepulsion intended to avoid collisions. The mathematical framework of these models involves the use of the transport equation and the associated collision and alignment alignment models, based on the Boltzmann equation. The goal of this course is to give an introduction to these tools.

Diane Peurichard (INRIA Paris) 

Simulation and numerical treatment of PDEs in Mathematical Biology

The partial differential equations describing biological phenomena may or may not possess exact solutions, and even if they do, they are frequently difficult to compute. For practical purposes it is sometimes enough to produce accurate simulations that describe a fair approximation of the original model and possibly give some useful qualitative or quantitative information about the biological model itself. In order to produce these computational approximations, a number of numerical methods can be used, all with some particular advantages and flaws depending on the underlying partial differential equation dynamics. The goal of this course is to introduce some basic numerical methods, such as finite differences, finite volumes and Monte-Carlo methods, that can be very useful to get a complementary insight into the equations introduced in the courses and simulation.

Maria José Caceres (Universidad de Granada) 

Mathematical Models in Neurobiology

In recent decades, mathematical models employed by the neuroscience community—whose properties were not yet well understood—have been studied in greater depth. Modeling the behavior of large populations of neurons using microscopic approaches often entails a high computational cost. To overcome this, models based on partial differential equations (PDEs) have been developed, which describe the dynamics of neuron populations at the mesoscopic or macroscopic scale.These advancements are providing deeper insights into brain functioning and its disorders, such as neurodegenerativediseases and neuropsychiatric conditions.In this course, we will focus on the mathematical study of some of these PDE-based models. The course is structured as follows:

1. Introduction to Neuronal Physiology. We begin with an introduction to the fundamental aspects of neuronal physiology, discussing how neurons transmit signals through action potentials and how these signals can be mathematically modeled. This physiological context is essential for understanding the models we will study later.

2. Microscopic Models for Neuronal Dynamics. We examine how neurons are modeled at the microscopic scale, beginning with the detailed Hodgkin-Huxley model and moving towards simpler approaches, such as the integrate-and-fire model.

3. Some PDE-based Models (Meso/Macroscopic Scale). In this part of the course, we focus on two models formulated as partial differential equations (PDEs) that describe the dynamics of neuronal networks at the mesoscopic macroscopic level: the Nonlinear Leaky Integrate-and-Fire (NNLIF) model and the Age-Structured Equation model. We will study their stability and asymptotic behavior, focusing on the system’s equilibria and how they depend on various parameters such as synaptic delay and network connectivity. To analyze these models, we will use advanced mathematical tools, including the entropy dissipation method, commonly used in kinetic theory, as well as a new technique that involves rewriting the problem as a Volterra-type equation. This allows us to explore the long-term behavior and stability of the system in a more powerful way. Throughout the course, students will explore how microscopic models at the neuronal level can be translated into macroscopic descriptions of neural networks, and how mathematical techniques provide a deep understanding of the asymptotic behavior in these systems. 

Noemi David (CNRS & LMRS - Université de Rouen) 

Singular limits arising in mechanical models of tissue growth

Based on the mechanical viewpoint that living tissues present a fluid-like behaviour, PDE models inspired by fluid dynamics are nowadays well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (visco-elastic models) or Darcy’s law (porous-medium equations (PME)). Moreover, the stiffness of the pressure law plays a crucial role in distinguishing density-based (compressible) models from free boundary (incompressible) problems where saturation of the density holds.
This course aims to analyse how to relate different mechanical models of living tissues through singular limits. In particular, we will address the inviscid limit towards the PME, the incompressible limit from the PME towards Hele-Shaw free boundary problems and their joint limit for equations including convective effects as well as for systems of coupled nonlinear equations. The techniques that will be employed during the course include classical tools in the theory of the PME (Aronson-Bénilan estimate, uniform BV-bounds, energy bounds) as well as techniques that are reminiscent of the theory of gradient flows, such as, for instance, the energy dissipation inequality (EDI) formulation.

Frank Ernesto Alvarez Borges (LJLL, Sorbonne Université) and Jorge Estrada Hernández (Universidad de la Habana)

Mathematical models in population dynamics: applications in agroecology

Due to the high number of diseases that they transmit, mosquito is considered as the most dangerous animal species for human. In particular, Aedes mosquitoes are one of the main vector for diseases like dengue, chikungunya, zika. Due to climate change, such mosquitoes are present and invasive on every continent in the world. Since there is no vaccine available for such diseases, one of the most promising technique to limit the transmission consists in acting on the mosquitoes population. Several techniques of control of the mosquitoes dynamics are under study, among them we may cite the sterile insect technique (SIT) and the replacement technique. These two techniques consist in releasing specific mosquitoes which will interact with the host population to reduce the size of the population (SIT) or to replace it by a population carrying a bacteria blocking the transmission of arboviruses. Obviously, a safe, proper and optimal use of such techniques requires a careful mathematical analysis. In this course we will review the main aspect of the life cycle of mosquitoes and present the main techniques that may be used to control the mosquitoes population. Then, we will present some mathematical models and mathematical tools that will be useful to answer to several questions : How to optimize such strategies of control ? How to guarantee the success of such strategies ? What could be the influence of the spatial heterogeneities on the success of these control strategies?

Teaching assistents for the excercise sessions:  

- Claudia Fonte Sanchez (INRIA Grenoble)

- Frank Ernesto Alvarez Borges (LJLL, Sorbonne Université)

- Jorge Estrada Hernández (Universidad de la Habana)