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The Spike-and-reset dynamics for non-linear integrate-and-fire neuron models

Jonathan Touboul 1, 2 R. 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 : Non-linear integrate and fire neuron models introduced in \cite{touboul08}, such as Izhikevich and Brette-Gerstner neuron models, are hybrid dynamical systems, defined both by a continuous dynamics, the subthreshold behavior, and a discrete dynamics, the spike and reset process. Interestingly enough, the reset induces in bidimensional models behaviors only observed in higher dimensional continuous systems (bursting, chaos,...). The subthreshold behavior (continuous system) has been studied in previous papers. Here we study the discrete dynamics of spikes. To this purpose, we introduce and study a Poincaré map which characterizes the dynamics of the model. We find that the behavior of the model (regular spiking, bursting, spike frequency adaptation, bistability, ...) can be explained by the dynamical properties of that map (fixed point, cycles...). In particular, the sytem can exhibit a transition to chaos via period doubling, which was previously observed in Hodgkin-Huxley models and in Purkinje cells.
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Contributor : Jonathan Touboul <>
Submitted on : Friday, May 2, 2008 - 6:23:40 PM
Last modification on : Thursday, July 1, 2021 - 5:58:02 PM


  • HAL Id : inria-00276927, version 1



Jonathan Touboul, R. Brette. The Spike-and-reset dynamics for non-linear integrate-and-fire neuron models. 2008. ⟨inria-00276927⟩



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