# Spike-adding and reset-induced canard cycles in adaptive integrate and fire models

1 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
4 Ikerbasque and BCAM
BCAM - Basque Center for Applied Mathematics
Abstract : We study a class of planar integrate and fire (IF) models called adaptive integrate and fire (AIF) models, which possesses an adaptation variable on top of membrane potential, and whose subthreshold dynamics is piecewise linear (PWL). These AIF models therefore have two reset conditions, which enable bursting dynamics to emerge for suitable parameter values. Such models can be thought of as hybrid dynamical systems. We consider a particular slow dynamics within AIF models and prove the existence of bursting cycles with $N$ resets, for any integer $N$. Furthermore, we study the transition between $N$- and $(N+1)$-reset cycles upon vanishingly small parameter variations and prove (for $N=2$) that such transitions are organised by canard cycles. Finally, using numerical continuation we compute branches of bursting cycles, including canard-explosive branches, in these AIF models, by suitably recasting the periodic problem as a two-point boundary-value problem.
Document type :
Journal articles
Domain :

https://hal.inria.fr/hal-03129713
Contributor : Mathieu Desroches Connect in order to contact the contributor
Submitted on : Wednesday, February 3, 2021 - 9:00:17 AM
Last modification on : Thursday, January 20, 2022 - 4:13:16 PM

### Citation

Mathieu Desroches, Piotr Kowalczyk, Serafim Rodrigues. Spike-adding and reset-induced canard cycles in adaptive integrate and fire models. Nonlinear Dynamics, Springer Verlag, In press, ⟨10.1007/s11071-021-06441-z⟩. ⟨hal-03129713⟩

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