The research of the mathematical neuroscience lab sits at the frontiers between biology and mathematics. In the team, we work with experimentalists (and some members also perform their own experiments) we build up models and analyze them mathematically.
Currently, we develop several research directions.
The topological organization of the cortex
We also address the functional organization of the cortex using biological experiments and modeling, in order to understand their tropology and how they subtend visual perception. Functional maps are characterized with optical imaging techniques (intrinsic
signals or voltage-sensitive) and electrophysiology.
In order to understand the role of visual experience in shaping functional maps, these are recorded during development, or in pathological conditions of vision (blindness, strabismus, orientation deprivation..)
The functional networks subtending these maps are characterized with dye injection combined with optical imaging or electrophysiology, or are inferred through mathematical models.
Large-scale neuronal networks
In order to understand the emerging properties of large neuronal networks, we analyze the activity of large neural assemblies. We further developed stochastic analysis methods for such equations taking into account the specificity of cortical networks, in particular their topology, spatial extension, and resulting space-dependent delays. In order to understand the role of noise and heterogeneity, we reduced these equations in a particular model (Wilson-Cowan system) in which the dynamics reduces to a simpler deterministic dynamical system. We thus evidenced a surprising phenomenon: noise and heterogeneity govern the qualitative properties of the macroscopic solutions, inducing in particular the emergence of synchronized periodic activity.
Hybrid Dynamical Systems and Single Cells Dynamics
Neurons display a continuous nonlinear dynamics interspersed with discrete events, called spikes, that are meaningful events transmitted to the connected neurons. This structure lead us to analyze hybrid dynamical systems coupling continuous dynamics (the excitable membrane potential) and discrete phenomena (spike emission). We have developed a thorough analysis of the nonlinear bidimensional integrate-and-fire neurons. This lead us to introduce a new, versatile model of neuron, the quartic integrate and fire neuron, supporting subthreshold sustained oscillations. The spike dynamics was rigorously analyzed through the introduction of a specific firing map, linking the spike pattern emitted to the excitable dynamics. We also investigated the well-posedness properties of these models and the precision of numerical simulations for such systems.