Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike-adding and chaos

Jonathan Rubin; Justyna Signerska-Rynkowska; Jonathan Touboul; Alexandre Vidal

doi:10.3934/dcdsb.2017204

Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike-adding and chaos

Abstract

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

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Publication version: Accepted or Published Version
DOI:: Digital Object Identifier (open in new tab) 10.3934/dcdsb.2017204
License: Copyright (2017 American Institute of Mathematical Sciences)

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Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B no. 22, edition 10, pages 3967 - 4002,
ISSN: 1531-3492
Language:: English
Publication year:: 2017
Bibliographic description:: Rubin J., Signerska-Rynkowska J., Touboul J., Vidal A.: Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike-adding and chaos// DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B. -Vol. 22, iss. 10 (2017), s.3967-4002
DOI:: Digital Object Identifier (open in new tab) 10.3934/dcdsb.2017204
Verified by:: Gdańsk University of Technology