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

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Abstract

We study a class of planar integrate and fire 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 . 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 .

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Acknowledgements

SR was supported by Ikerbasque (The Basque Foundation for Science), by the Basque Government through the BERC 2018-2021 program and by the Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017- 0718 and through project RTI2018-093860B-C21 funded by (AEI/FEDER, UE) with acronym “MathNEURO”.

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Authors and Affiliations

  1. MathNeuro Team, Inria Sophia Antipolis Méditerranée, Valbonne, France

    Mathieu Desroches

  2. Department of Mathematics, Wrocław University of Science and Technology, Wrocław, Poland

    Piotr Kowalczyk

  3. Ikerbasque - The Basque Foundation for Science, Bilbao, Spain

    Serafim Rodrigues

  4. MCEN Team, BCAM - Basque Center for Applied Mathematics, Bilbao, Spain

    Serafim Rodrigues

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  1. Mathieu Desroches
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  2. Piotr Kowalczyk
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  3. Serafim Rodrigues
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Correspondence to Mathieu Desroches.

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Desroches, M., Kowalczyk, P. & Rodrigues, S. Spike-adding and reset-induced canard cycles in adaptive integrate and fire models. Nonlinear Dyn 104, 2451–2470 (2021). https://doi.org/10.1007/s11071-021-06441-z

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