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The Cauchy problem for one-dimensional spiking neuron models

Romain Brette 1
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : I consider spiking neuron models defined by a one-dimensional differential equation and a reset—i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on $${\mathbb{R}}$$ for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.
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https://hal.inria.fr/inria-00423347
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Submitted on : Friday, October 9, 2009 - 3:33:32 PM
Last modification on : Thursday, July 1, 2021 - 5:58:02 PM

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Romain Brette. The Cauchy problem for one-dimensional spiking neuron models. Cognitive Neurodynamics, Springer Verlag, 2008, 2, pp.21--27. ⟨10.1007/s11571-007-9032-y⟩. ⟨inria-00423347⟩

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