title: Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity.
abstract: The Network Noisy Leaky Integrate and Fire equation is among the simplest model allowing for a self-consistent description of neural networks and gives a rule to determine the probability to find a neuron at the potential v. However, its mathematical structure is still poorly understood and, concerning its solutions, very few results are available. In the midst of them, a recent result shows blow-up in finite time for fully excitatory networks. The intuitive explanation is that each firing neuron induces a discharge of the others; thus increases the activity and consequently the discharge rate of the full network. In order to better understand the details of the phenomena and show that the equation is more complex and fruitful than expected, we analyze further the model. We extend the finite time blow-up result to the case when neurons, after firing, enter a refractory state for a given period of time. We also show that spontaneous activity may occur when, additionally, randomness is included on the firing potential VF in regimes where blow-up occurs for a fixed value of VF.
authors.affiliation: Departamento de Matemática Aplicada, Universidad de Granada, E-18071 Granada, Spain. Electronic address: caceresg@ugr.es.
authors.forename: María J
authors.initials: MJ
authors.lastname: Cáceres
authors.affiliation: UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F75005 Paris, France; INRIA Rocquencourt EPI BANG, France. Electronic address: benoit.perthame@upmc.fr.
authors.forename: Benoît
authors.initials: B
authors.lastname: Perthame
mesh_terms: D000200:Action Potentials / Q000502:physiology*; D008959:Models, Neurological; D009415:Nerve Net / Q000502:physiology*; D009716:Numerical Analysis, Computer-Assisted; D012032:Refractory Period, Electrophysiological
keywords: Blow-up; Integrate and fire models; Neural networks; Noise; Spontaneous activity