Séminaire Grégory Dumont

Jeudi 17 Mars 2016 (14h00, Salle G001)

Grégory Dumont, LNC, Group for Neural Theory, ENS (Ulm).

Titre : An age structural model for neural networks.

Résumé : Neurons are strongly noisy, and stochastic models are almost always required when a system is driven by random events. In accordance with the origin of variability, the sources of noise are classified as intrinsic or extrinsic, and give rise to distinct mathematical frameworks. While the external variability is generally treated by the use of a Wiener process, the internal variability is mostly expressed via random firing events and a non-homogenous Poisson process. Those distinct stochastic processes are completely determined by their probability density function obtained via partial differential equations such as the Fokker-Plank equation and the von-Foerster-McKendrick system. In the first part, we investigate in what way those partial differential equations are related and how their respective solutions can be mapped one to another via integral transforms. In the second part, we investigate the accommodation of finite size fluctuations into the model when the synaptic coupling is taken into account. Thanks to the tau leaping formula, a trick first popularized by Gillespie, we approximate the number of firing events during a time increment as a Poisson random variable. From there, we are able to derive a corrected field equation that encompasses the presence of fluctuations proportional to the mean number of firing events, and therefore fully retains the randomness character of the spike initiation. Our new description is such that it reduces, in the thermodynamic limit, to the classical deterministic mean field equation known as the refractory density equation or von-Foester-McKendrick system. With such a tool in hand, we are capable of computing several statistical information regarding the network’s firing activity in the asynchronous regime.