INTERMATHS EMJM FULL PARTNER

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Rémi Catellier

Aims:

The purpose of the course is to teach the basics of the theory of stochastic processes, which has become a standard tool in the modelling of biological neural networks.

The course will focus in particular on the Brownian motion, on stochastic calculus and on diffusion processes, with the integrate and fire model as a benchmark example.

Markov property and martingale theory will be also addressed in this framework, with possible extensions to some jump processes like those used in ion channel models.

Content:

•    Brownian motion
•    Stochastic Calculus
•    Diffusion Processes
•    Markov Property
•    Martingales
•    Integrate and fire model
•    Ion channel models

Pre-requisites:

Probability with measure theory

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Alice Guyon, Ingrid Bethus

Aims:

Neuronal and cognitive systems cannot be modeled without knowledge of the basics of Neurosciences, from the molecular to the integrated level, involved in cognition and behaviors.

The first part of the program focuses on elementary neurophysiology and neuroanatomy. What are the different subparts constituting the nervous system and what are their main roles? How are neurons constituted? How do they generate activity and communicate with other neurons?

The second part of the program explores, with an integrative perspective, the neurobiological basis for higher mental functions through several examples. Sensorimotor functions are at the root of all the other processes. So the study of feeding behaviors is a good way to learn about the bio-logic of elementary behaviors, starting from the physiology of the autonomic nervous system and ending with neuroethological issues. Learning and memory are the basic processes of higher mental functions and also hot topics with applications in many domains.

In addition to all these fundamentals, the course also explains the materials and methods used in cognitive neurosciences to obtain data at the different levels of organization of nervous, cognitive and behavioral systems. This course is taught by a teaching staff member of the Master Programme Mod4NeuCog at UCA.

Content:

• Neuronal and cognitive systems
• Neurophysiology and neuroanatomy
• Neurobiological basis for higher mental functions

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Sylvain Rubenthaler

Aims:

•    This course addresses the basic methods used for simulating random variables and implementing Monte-Carlo and Quasi Monte-Carlo methods.
•    Simulation of stochastic processes used in neuroscience, such as Brownian motion and solutions to stochastic differential equations, will be addressed.
•    The course will introduce sampling methods in finite dimension, discretization of diffusion processes, strong and weak errors.

Content:

• Simulation
• Monte-Carlo methodes
• Discretization schemes
• Error analysis

Pre-requisites:

Probability with measure theory, stochastic calculus, programming 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Patricia Reynaud and Etienne Tanré

Aims:

This course aims at providing a deep understanding of the main stochastic models used in neurocognition (Markov Chains, Integrate and Fire, point processes) and addressed (at least for some of them) in the Stochastic Calculus course and to study their mathematical properties.

The pros and cons of each of them is discussed, especially in terms of the modeling and the statistical inference of real data.

This course is taught by a teaching staff member of the Master Programme Mod4NeuCog at UCA

Content:

•    Stochastic models in neurocognition, statistical inference
•    Statistical inference

Pre-requisites:

Statistics 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: François Delarue

Aims:

The purpose of this course is to address some mathematical and computational aspects of machine learning and of related large networks of artificial and biological neurons. 

The first part of the course will be an introduction to stochastic optimization, control and game theories. This includes stochastic gradient descents, stochastic gradient Langevin dynamics, dynamic programming principle, first order conditions, Hamilton-Jacobi equations and the notion of Nash equilibria. 

The second part will be dedicated to mathematical tools for large networks, including the concept of mean-field models from statistical physics. Illustration will be given in the framework of over-parametrized artificial networks and biological spiking neural networks. 

The third part will address computational aspects of some artificial networks, like generative adversarial networks and deep networks. This will include programming sessions.  

Content:

•           Stochastic optimisation and control 

•           Stochastic gradient descent and Langevin dynamics

•           Dynamic programming principle

•           Mean field models

•           Generative adversarial networks

•           Deep learning

Pre-requisites:

Probability with measure theory, optimization, stochastic calculus