#
Math modelling major

Master 2

# 3rd Semester

## Shared courses

**Organisation: **Lectures (12h)

**Coordinator: **Bianca Habermann, Alphée Michelot, Elisabeth Rémy

**Evaluation:** project and continuous monitoring

Following [PRO2] seminars, the students will attend all the Centuri seminars of this semester. For two of them, they will be asked to broaden their knowledges on the subject and present an oral and written synthese.

Choose 2 courses between 3 courses:

- Developmental Biology
- Neurosciences
- Immunology

## COURSE 1: Developmental Biology

**Organisation: **Lectures (6h), TD (6h) TP( 6h)

**Lecturers : ****Thomas Lecuit , Florence Hubert**

**Evaluation:** project

Fundamentals of morphogenesis: molecular, cellular and biophysical basis of tissue forms in animals and plants. The mechanical and biochemical basis of morphogenesis will be addressed to understand the origin of cell and tissue organization. This course is an introduction to the modeling of the emergence of spatial and temporal patterns in morphogenesis and to the description and understanding of Turing instabilities.

## COURSE 2: Neurosciences

**Organisation: **Lectures (6h), TD (6h) TP( 6h)

**Lecturers : **Claudio Riviera

**Evaluation:** final exam

This course covers basic principle in neuroscience. During the lectures special emphasis will be placed on the mechanism of synapse formation and plasticity, functional network maturation, pathophysiology of the brain and role of glia-neuron interaction in network dynamics. Also an introduction to computational methods for analysis and modelling of neurobiological data will be presented. During the TD the students will be challenged in the form of a project where acquired knowledge in mathematical and computational biology will be used to solve specific problems in neuroscience.

## COURSE 3: Immunology

**Organisation: **Lectures (6h), TD (6h) TP( 6h)

**Lecturers : **Guillaume Voisine

**Evaluation:** final exam

The course will focus on the different aspects of immunology as approached by physicists. In particular, the following will be studied: Discrimination of antigens by T cells, communication between cells with cytokines and finally a part on differentiation.

**Organisation: **project during the semester

**Coordinator:** Florence Hubert, Laurence Röder

**Evaluation:** project

At the end of the courses [PRO1], and [BIO1], students will choose a scientific article at the interface of several disciplines on which they will work in groups. They will have to present in a memory and an oral presentation, to explain the biological context and the related basic concepts, to explain the methods used to interpret the biological data, to synthesize the results obtained in the article.

**Organisation: **Lectures (6h), TD (6h), TP (6h)

**Lecturers: ** Pierre Pudlo

**Evaluation:** continuous exam and projects

This course is an introduction to inferential statistics. It will be illustrated with biogical examples. The course will be composed by three parts

- Multiple tests
- Classification
- Time series analysis

**Organisation: **project

**Coordinator: **Florence Hubert, Laurence Röder

**Evaluation:** project

Following the module [PROJ1], the students will do a short internship in laboratory. They will have to propose a modelling or data processing problem at the math-info-bio interface. They will be asked to synthesize their results in a dissertation and an oral presentation.

**Organisation: **Lectures (10h), TD (8h), TP (12h)

**Lecturer: **Anais Baudot

This module will introduce the bases of biological network analysis, from graph theory and algorithms to dynamical modeling. A large part of the module is dedicated to hands-on tutorials, which introduces R and Python packages, as well as softwares widely used in Network Biology, such as Cytoscape and GinSim.

## Math Modelling courses

**Organisation: **Lectures (6h), TD (6h), TP (6h)

**Lecturers**: Guillemette Chapuisat, Assia Benabdallah

**Evaluation:** final exam and projects

We will study optimization tools in finite and infinite dimension (optimal control theory) as well as their numerical implementation using Python. We will apply these different tools to a chemotherapy optimization problem for an *in vitro* model of heterogeneous tumor growth, i.e. we will look for the best way to administer a chemotherapy to optimize the effect on a cancer cell culture mixing sensitive and resistant cells to chemotherapy. We will base our work on experiments conducted at the Faculty of Pharmacy of Timone.

**Scientific Calculus**

**Organisation: **Lectures (8h), TP (15h)

**Lecturer:** Florence Hubert

**Evaluation:** final exam and projects

From a mathematical point of view, this course will focus on the study of equations in 1d space dimension: revisions / complements on the transport equation (Von Neumann stability, decentering, CFL condition) and on unsteady and steady advection-diffusion equation. Finite volume schemes will be used to approximate the solutions. Their properties as consistency, stability, convergence will be illustrated. Much of the time will be devoted to the development of the scientific calculation approach: academic case-tests for validation, error curves, before tests in more general cases. Python will be used for implementation, including advanced use of Numpy, sparse matrix manipulation, and object-oriented design.

**Scientific Calculus advanced**

**Organisation: **Lectures (8h), TP (15h)

**Lecturer: ** Julien Olivier

**Evaluation:** final exam and projects

This course focuses on numerical finite volume schemes for equations in 2d: transport and stationary or unsteady advection-diffusion equations. A important part of the course is devoted to the manipulation of finite volume mesh and to the assembly of matrices through the edges structure. We will study some properties of these numerical schemes (stability L2 or L-infinity, convergence, preservation of the positivity, ...). In this course we will also develop the scientific calculation approach (design of benchmarks, error curves, ...).

**Numerical probabilistic methods**

**Organisation: **Lectures (12h), TP (12h)

**Lecturers: ** Christophe Gomez, Erwan Hillion

**Evaluation:** Continuous exam

This course will present different algorithms based on probabilistic tools, and allowing to solve various problems as simulation of random variables, approximation of integrals, simulation of an invariant measure, stochastic optimization, ...

The course will be divided in four chapters:

- Simulation of Random Variables
- Monte Carlo Method
- MCMC, Metropolis-Hastings Algorithm
- Simulated Annealing.

**Parametric statistics **

**Organisation: **Lectures (12h), TP (9h)

**Lecturers: **Jean-Marc FreYermuth, Oleg Lepski

**Evaluation:** final exam and continuous exam

This teaching will focus on the in-depth study of Bayesian and maximum likelihood methods, exponential inequalities (finite sample) and the optimality of these methods from the minimax approach via local asymptotic normality. Practical work on a computer will illustrate numerically these methods and their limits. Estimators will be calculated in more complicated models requiring the use of numerical methods (EM algorithm, MCMC).