Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model

Jonathan Touboul 1, *
* Auteur correspondant
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 : The quadratic adaptive integrate-and-fire model (Izhikevich 2003, 20060 is recognized as very interesting for its computational efficiency and its ability to reproduce many behaviors observed in cortical neurons. For this reason it is currently widely used, in particular for large scale simulations of neural networks. This model emulates the dynamics of the membrane potential of a neuron together with an adaptation variable. The subthreshold dynamics is governed by a two-parameter differential equation, and a spike is emitted when the membrane potential variable reaches a given cutoff value. Subsequently the membrane potential is reset, and the adaptation variable is added a fixed value called the spike-triggered adaptation parameter. We show in this note that when the system does not converge to its rest value, both variables blow up in finite time if there is no threshold condition. The cutoff is therefore essential for the model to be well defined and simulated. Furthermore, the divergence of the adaptation variable makes the system very sensitive to the cutoff: changing this parameter will dramatically change the spike patterns produced. Furthermore from a computational point of view, the fact that the adaptation variable blows up and the a very sharp slope it has when the spike is emitted implies that the time step of the numerical simulation needs to be very small (or adaptive) in order to catch the value of the adaptation at the time of the spike. It is not the case for the similar quartic (Touboul 2008) and the exponential (Brette and Gerstner 2005) models whose adaptation variable does not blow up in finite time, and therefore which are very robust to changes in the cutoff value.
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Rapport
[Research Report] RR-6634, INRIA. 2008
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Jonathan Touboul. Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model. [Research Report] RR-6634, INRIA. 2008. 〈inria-00319244〉

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