Challenges in Mathematics emerging from Neurosciences – ChaMaNe

The ambition of the ChaMaNe project is to create a research group to achieve significant advances in the field of mathematics for neuroscience. The comprehension of the brain is still far from being achieved and its modeling is extremely complex. In particular, many scales coexist: from proteins and synapses to macroscopic brain areas and functions, from the few milliseconds of an action potential to the hours or years of synaptic plasticity and learning processes.

Hence, the variety of questions, with different ranges of difficulty, emerging from this topic is enormous, and only a very small number of them has been addressed.
The key questions of the ChaMaNe project will be to understand, on the one hand, the intrinsic dynamics of a neuron and their consequences, and on the other hand the qualitative dynamic that emerges from large neural networks with respect to the intrinsic behavior of individual neurons, interactions between neurons, memory effects, spatial structure ...

Answering those very broad questions requires a combination of expertises based on partial differential equation's (PDE) analysis, probability and statistics that will form the consortium. Our strategy will mainly go through a theoretical approach, but we will also rely on data analysis as well as on the development of numerical methods for specific contexts. This approach, based on theory and integrating also applications, will be established in a complementary and fruitful way, allowing us to promote more efficiently interactions with neuroscientists.

In recent years, an important literature has been developed around the mathematical models of neuroscience derived from PDEs or stochastic models. The methods developed recently are very effective in some particular cases, but are far from covering the diversity of issues underlying many models whose complexity is the result of essential biological properties. Our ambition is to better understand the various complex patterns resulting from neural models, by strengthening the existing literature and creating new tools and methods.

The observed patterns that result from the communication between the neurons is very rich: they can notably reveal phenomena of synchronization of neuron discharges which are more or less rhythmic within a population, or phenomena of propagation of waves of neuron discharges in the brain. One of the major questions that will be tackled in the ChaMaNe project is to understand what may be the mechanisms underlying all these interesting observations and how to give them the most rigorous mathematical framework possible. Would this result from the intrinsic dynamics of each neuron? From memory effects or learning? From the spatial complexity of interconnections between neurons? What is the influence of noise ? Understanding fully these issues as a whole is a titanic task, but with an original approach, combining various theoretical expertise's and an applied component related to data analysis, we have the ambition to bring a number of significant answers to these questions.

The models involved are governed by deterministic and stochastic dynamics at different scales, hence, the unification of different theories coming from Partial Differential Equations (PDE), Probability theory and Statistics is essential to elaborate fruitful and promising directions. The strategy of the ChaMaNe project to make significant progress in this area will be through three complementary axes :

1. Single stochastic/deterministic neuron models
2. Scale of large number of neurons : homogeneous models
3. Scale of large number of neurons :spatial interaction or complex connectivity matrix

It is important to stress that these three axes are not at all independent and feed one another. A large component of the project is theoretical, but on each part, comparison with data will be made in a complementary way or to complement the theoretical work.