Microscopic approach of a time elapsed neural model

Julien Chevallier (1) , Maria J. Caceres (2) , Marie Doumic (3, 4) , Patricia Reynaud-Bouret (5)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 UGR - Universidad de Granada = University of Granada
3 LJLL - Laboratoire Jacques-Louis Lions
4 MAMBA - Modelling and Analysis for Medical and Biological Applications
5 LJAD - Laboratoire Jean Alexandre Dieudonné

The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present article is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.

Mots clés: Structured equations; Hawkes process; Wold process; renewal equation; neural network AMS Subject Classification:35F15; population dynamics

Domaines: Analyse numérique [cs.NA]

Julien Chevallier, Maria J. Caceres, Marie Doumic, Patricia Reynaud-Bouret. Microscopic approach of a time elapsed neural model. Mathematical Models and Methods in Applied Sciences, 2015, http://www.worldscientific.com/doi/10.1142/S021820251550058X. ⟨10.1142/S021820251550058X⟩. ⟨hal-01159215⟩