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.