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# Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience

Title | Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience |

Publication Type | Journal Article |

Year of Publication | 2015 |

Authors | Carrillo, JAntonio, Perthame, B, Salort, D, Smets, D |

Journal | Nonlinearity |

Volume | 28 |

Pagination | 3365 |

Abstract | The Noisy Integrate-and-Fire equation is a standard non-linear Fokker–Planck equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential ##IMG## [http://ej.iop.org/images/0951-7715/28/9/3365/non518219ieqn001.gif] {${{V}_{\text{F}}}$} is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L 2 estimates and on the other hand, on the notion of ‘universal super-solution’ and parabolic regularizing effects to obtain ##IMG## [http://ej.iop.org/images/0951-7715/28/9/3365/non518219ieqn002.gif] {${{L}^ınfty}}$} bounds. |

URL | http://stacks.iop.org/0951-7715/28/i=9/a=3365 |