## Section: New Results

### Modeling the activity of populations of neurons: study of synchronization

Participants : Jacques Henry, Gregory Dumont, Oana Tarniceriu.

Modeling of the interaction of neuron population between basal ganglia is going on within the collaboration with the team “Basal Gang” of UMR 5227. The research of this team is focused at studying the basal ganglia and in particular the mechanisms of selection of action. This joint research is sponsored by the “neuroinformatique” program of CNRS.

The master 2 internship of Gregory Dumont was dedicated to study the numerical approximation to population density equation for the Izhikevich model. It has already been studied by Julien Modolo during his PhD thesis but the numerical scheme he used showed an important numerical diffusion. This is incompatible with the objective of simulating synchronization phenomena which are represented by a concentration of the population density around a point in the state space. G. Dumont tested various numerical schemes on the population density equation in the case of uncoupled neurons. In that case if the initial condition consists in all the neurons being in the same state (Dirac mass) the solution should remain a Dirac mass moving in the state space. This is quite difficult to obtain numerically in particular due to the stiffness of the Izhikevich model where the velocity varies of several porder of magnitude. G. Dumont has experimented a variable discretization in potential and a refinement with respect to the recovery varaiable. The method was of finite volume type with flux limitation. The most satisfactory results were obtained with a WENO scheme of order 5. G. Dumont has now begun a PhD thesis in the team on the modeling on the cortico-basal ganglia-thalamo loop with a financial support of CNRS and Région Aquitaine.

During her one year postdoc O. Tarniceriu investigated the issue of synchronisation of a population of neuron weakly coupled using the population density approach based on the Izhikevich model. Considering neurons having the same rhythmic activity, the model was first transformed to phase density population. Then by analogy to the Malkin's Theorem, a slow phase deviation equation for the weakly coupled population was derived. Sufficient conditions for the stability of the syncrhonized solution were obtained.