Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

Research output: Contribution to journalJournal article – Annual report year: 2018Researchpeer-review

Standard

Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models. / Granados, Albert; Huguet, Gemma.

In: Communications in Nonlinear Science and Numerical Simulation, Vol. 70, 2019, p. 48-73.

Research output: Contribution to journalJournal article – Annual report year: 2018Researchpeer-review

Harvard

APA

CBE

MLA

Vancouver

Author

Bibtex

@article{881d6d744e1a41c39f63d089309eee4f,
title = "Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models",
abstract = "In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.",
keywords = "Integrate-and-fire, Hybrid systems, Piecewise smooth 2d maps, Quasi-contractions",
author = "Albert Granados and Gemma Huguet",
year = "2019",
doi = "10.1016/j.cnsns.2018.09.006",
language = "English",
volume = "70",
pages = "48--73",
journal = "Communications in Nonlinear Science and Numerical Simulation",
issn = "1007-5704",
publisher = "Elsevier B.V.",

}

RIS

TY - JOUR

T1 - Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

AU - Granados, Albert

AU - Huguet, Gemma

PY - 2019

Y1 - 2019

N2 - In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.

AB - In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.

KW - Integrate-and-fire

KW - Hybrid systems

KW - Piecewise smooth 2d maps

KW - Quasi-contractions

U2 - 10.1016/j.cnsns.2018.09.006

DO - 10.1016/j.cnsns.2018.09.006

M3 - Journal article

VL - 70

SP - 48

EP - 73

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

ER -