PittsburghV1, France, Analysis, bibRecord, 000005

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

Identifieur interne : 000005 ( France/Analysis ); précédent : 000004; suivant : 000006

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

Auteurs : Jonathan Rubin [États-Unis] ; Justyna Signerska-Rynkowska [Pologne] ; Jonathan Touboul [France] ; Alexandre Vidal [France]

Source :

Discrete and Continuous Dynamical Systems - Series B (DCDS-B) [ 1531-3492 ] ; 2017-09-01.

RBID : Hal:hal-01416002

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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https://hal.inria.fr/hal-01416002

Affiliations:

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<front><div type="abstract" xml:lang="en">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.</div>
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Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike adding and chaos

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

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Abstract

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Links to Exploration step

Le document en format XML

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Pour mettre un lien sur cette page dans le réseau Wicri