Mathematical Analysis of Neuronal Dynamics
Our project, which gathers mathematicians and neuroscientists, aims at developping mathematically rigorous approaches to neuroscience considering single neurons as well as interconnected neuronal populations. Our target is to conduct the mathematical analysis of existing models where there is still much work to be done and to enrich the modelling by proposing new models.

A lot of available studies have been conducted by simulations. Although this approach has certainly been fruitful and must be pursued, we believe that its achievements are necessarily limited and that it needs to be renewed and refueled by new results coming from a profound mathematical analysis. Moreover, there is nowadays a strong demand emanating from neuroscientists for a rigorous mathematical treatment of their models which may lead them to a deep analysis of their simulations or experimental results. Our project gathers internationally renowned competence in mathematics as well as neuroscience (probability theory, partial differential equations, dynamical systems,  neural coding, computational neuroscience, perception and action). Mathematically speaking it is centered on probability and partial differential equations (pdes). Our experience has convinced us that even the classical models in neuroscience (which neuroscientists call the simplest), raise profound mathematical questions at the frontier of present mathematical knowledge and also far beyond this frontier. The richness of the original questions rising from neuroscience modelling makes us expect that our project will lead to the development of new tools and methodology in mathematics. Neuroscience is a young science. What we know about the mecanisms of the brain is still a collection of pieces. While discoveries in biology have exploded, the brain remains poorly understood. For a long time the neurobiological approach to the brain was compartimented according to the methods and techniques employed (neurochemistry, neuropharmacology, etc…). Nowadays the division relies on the considered function of the brain. Roughly speaking we are interested in the following functions : encoding and decoding, transmission of information, perception of the environment, link between perception and action or decision- making. The challenge is to be able to provide to each function a clear mathematical description of the underlying dynamics.

One original aspect of our project is to integrate the various dynamical levels of the nervous system, from the conductance dynamic of channels population in the neuronal membrane, to the network of cerebral areas that include the activity of billions of neurons. The need to enunciate collective principles is standing as much as the need for rigorous precision in the description of the models at each level. Since there is growing evidence that random phenomena are important in these functions we want in particular to investigate the role of noise in the activity of the nervous system. The effect of randomness will be studied both theoretically and numerically.

The development of simple mathematical models can open new biologically important issues previously hidden by the complexity of biological systems. Mathematics can help to decide whether an observed phenomenon is generic or on the contrary, strongly dependent on the experiment or simulation parameters. We believe that mathematicians need to understand properly the biological issues to be able to recognize which mathematical tools are the most relevant and to propose pertinent models. The presence of neuroscientists in our project will provide us access to experimental data and facilitate the validation of the mathematical models. We emphasize that the mathematical analysis should bring an explanation to the observed phenomenon and not only assert the well posedness of the models even though this step is essential beyond the mathematical consistency. Such a flow of skills back and forth is our aim in this project and we plan to develop an active cooperation between the members of our group to increase the value of our global work.