## Section: New Results

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

Participants : Jacques Henry, Gregory Dumont, Oana Tarniceriu.

During the second year of his PhD thesis G. Dumont continued to develop and refine a simulator of a population of leaky integrate and fire neurons, with a finite jump of potential response to a synaptic stimulation in order to compare with the results from another approach, the Fokker-Planck approach. It would be interesting to show that the Fokker Planck approach can be seen as a limiting case, for high frequency small jump of potential, of the one of B. Knight, L. Sirovich and L. Omurtag. At least it is suggested by the simulations. The second year focused on the study of mathematical aspects of the equation. The methods were inspired by the book of Benoit Perthame: Transport equation in biology. With these tools, the mathematical properties of the model of population density have been established: the existence and uniqueness of a solution. G. Dumont has also been able to establish that in special cases the equation has no solution, or more precisely that the solution blows up, assuming that there is no conduction delay of the spikes within the considered population. This blow up can be related to a synchronization. In case of a conduction delay the condition of existence of a solution are much larger. The situation of inhibiting synapses has also been studied. This has been submitted to the Journal of Mathematical Biosciences.

Using the principles of entropy, G. Dumont has shown that under certain circumstances the solution of the equation converges to a stationary solution and this can be interpreted as the desynchronization of the population of neurons.

Similar sudies have been done on the thetaneuron model which accounts for self spiking neurons.

The question of the synchronization of a population of neurons is also studied in collaboration with O. Tarniceriu. We consider a population of identical self firing neurons that are weakly coupled and we study the long term evolution of the reaprtition of phases. We are now focusing to a population of 1D leaky integrate and fire neurons and we expect more precise results for this simple model.