Mathematical Modelling in Neuroscience

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My research work consists in developing mathematical models that help better understand the normal and pathological behiavor of neurons. I  work in collaboration with several experimental researchers so that my models have tangible applications in neuroscience. My research program is divided into the following axes. 1) Develop models of ordinary differential equations (ODEs) to describe ionic concentration fluctuations and volume changes in neural cells. With my team and collaborators, we have developed mathematical models that describe the fluctuations of ionic concentrations, especially chloride,  in the context of the study of neuropathic pain. We are currently working to improve our models so that they also describe water fluxes across cell membranes and volume changes. Volume fluctuations play a crucial but often overlooked role in cellular signaling.  They are little investigated partly due to difficulties in performing accurate experimental measurements.  We will use digital holography microscopy measurements from our collaborators to validate and calibrate our mathematical models.   2) Modeling the tetrapartite synapse. To fully understand synaptic signaling, it is essential to understand the four parts of the synapse that is 1) the presynaptic button, 2) the postsynaptic receptors, 3) the extracellular space and 4) the glial cells.  With the PhD candidate Xiaoting Li and my collaborator Antoine Godin, we have developed a mathematical model to study the impact of nanocolumns which are nanoscopic structures that promote the alignment of postsynaptic receptors with the opening sites of presynaptic vesicles. From a mathematical point of view, this model uses a Monte Carlo approach, discrete Markov chains and a description of the synaptic electrical potential using a finite volume approach. In the continuation of this project, we will develop models that take into account the extracellular space surrounding the synapse, the distribution of fixed electrical charges in this space and the impact of glial cells on neurotransmitters diffusion. 3) Application of the finite element method to study the node of Ranvier. We have developed models that use the Poisson Nernst-Planck system of partial differential equations to calculate the electric field distribution and the electrodiffusion of ionic species in neural structure. These models allow, among other things, to study the impact of a myelin loss on the electrical signaling as characterized by the shape of an action potential. We will take advantage of a collaboration with Antoine Godin to obtain experimental measurements on the fluctuations of ionic concentrations during action potentials. In these three projects, the determination of parameters in non-linear dynamical models is an important issue. The PhD student Pierre-Louis Gagno is developing new mathematical methods to better solve this problem.