Firing-rate, symbolic dynamics and frequency dependence in periodically driven spiking models: a piecewise-smooth approach

Albert Granados, Maciej Krupa

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing-rate on the frequency of the drive. In an earlier study we have obtained rigorous results on the bifurcation structure in such systems, with emphasis on the relation between the firing-rate and the rotation number of the existing periodic orbits. In this work we study how these bifurcation structures behave upon variation of the frequency of the input. This allows us to show that the dependence of the firing-rate on frequency of the drive follows a devil's staircase with non-monotonic steps and that there is an optimal response in the whole frequency domain. We also characterize certain bounded frequency windows in which the firing-rate exhibits a bell-shaped envelope with a global maximum.
Original languageEnglish
JournalNonlinearity
Volume28
Issue number5
Pages (from-to)1163-1192
ISSN0951-7715
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Firing-rate
  • Integrate-and-fire
  • Piecewise-smooth
  • Period adding
  • Border collision bifurcations

Cite this

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title = "Firing-rate, symbolic dynamics and frequency dependence in periodically driven spiking models: a piecewise-smooth approach",
abstract = "In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing-rate on the frequency of the drive. In an earlier study we have obtained rigorous results on the bifurcation structure in such systems, with emphasis on the relation between the firing-rate and the rotation number of the existing periodic orbits. In this work we study how these bifurcation structures behave upon variation of the frequency of the input. This allows us to show that the dependence of the firing-rate on frequency of the drive follows a devil's staircase with non-monotonic steps and that there is an optimal response in the whole frequency domain. We also characterize certain bounded frequency windows in which the firing-rate exhibits a bell-shaped envelope with a global maximum.",
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Firing-rate, symbolic dynamics and frequency dependence in periodically driven spiking models: a piecewise-smooth approach. / Granados, Albert; Krupa, Maciej.

In: Nonlinearity, Vol. 28, No. 5, 2015, p. 1163-1192.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Firing-rate, symbolic dynamics and frequency dependence in periodically driven spiking models: a piecewise-smooth approach

AU - Granados, Albert

AU - Krupa, Maciej

PY - 2015

Y1 - 2015

N2 - In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing-rate on the frequency of the drive. In an earlier study we have obtained rigorous results on the bifurcation structure in such systems, with emphasis on the relation between the firing-rate and the rotation number of the existing periodic orbits. In this work we study how these bifurcation structures behave upon variation of the frequency of the input. This allows us to show that the dependence of the firing-rate on frequency of the drive follows a devil's staircase with non-monotonic steps and that there is an optimal response in the whole frequency domain. We also characterize certain bounded frequency windows in which the firing-rate exhibits a bell-shaped envelope with a global maximum.

AB - In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing-rate on the frequency of the drive. In an earlier study we have obtained rigorous results on the bifurcation structure in such systems, with emphasis on the relation between the firing-rate and the rotation number of the existing periodic orbits. In this work we study how these bifurcation structures behave upon variation of the frequency of the input. This allows us to show that the dependence of the firing-rate on frequency of the drive follows a devil's staircase with non-monotonic steps and that there is an optimal response in the whole frequency domain. We also characterize certain bounded frequency windows in which the firing-rate exhibits a bell-shaped envelope with a global maximum.

KW - Firing-rate

KW - Integrate-and-fire

KW - Piecewise-smooth

KW - Period adding

KW - Border collision bifurcations

U2 - 10.1088/0951-7715/28/5/1163

DO - 10.1088/0951-7715/28/5/1163

M3 - Journal article

VL - 28

SP - 1163

EP - 1192

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 5

ER -