On local bifurcations in neural field models with transmission delays

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)

Abstract

Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.
Original languageUndefined
Pages (from-to)837-887
Number of pages51
JournalJournal of mathematical biology
Volume66
Issue number4-5
DOIs
Publication statusPublished - Mar 2013

Keywords

  • EWI-23124
  • Delay equation
  • Neural field
  • Dual semigroup
  • IR-84334
  • Numerical bifurcation analysis
  • Normal form
  • Sun-star calculus
  • METIS-296329
  • Hopf bifurcation

Cite this

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title = "On local bifurcations in neural field models with transmission delays",
abstract = "Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.",
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author = "{van Gils}, {Stephanus A.} and S.G. Janssens and Kouznetsov, {Iouri Aleksandrovitsj} and S. Visser",
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doi = "10.1007/s00285-012-0598-6",
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volume = "66",
pages = "837--887",
journal = "Journal of mathematical biology",
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On local bifurcations in neural field models with transmission delays. / van Gils, Stephanus A.; Janssens, S.G.; Kouznetsov, Iouri Aleksandrovitsj; Visser, S.

In: Journal of mathematical biology, Vol. 66, No. 4-5, 03.2013, p. 837-887.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - On local bifurcations in neural field models with transmission delays

AU - van Gils, Stephanus A.

AU - Janssens, S.G.

AU - Kouznetsov, Iouri Aleksandrovitsj

AU - Visser, S.

N1 - eemcs-eprint-23124

PY - 2013/3

Y1 - 2013/3

N2 - Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.

AB - Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.

KW - EWI-23124

KW - Delay equation

KW - Neural field

KW - Dual semigroup

KW - IR-84334

KW - Numerical bifurcation analysis

KW - Normal form

KW - Sun-star calculus

KW - METIS-296329

KW - Hopf bifurcation

U2 - 10.1007/s00285-012-0598-6

DO - 10.1007/s00285-012-0598-6

M3 - Article

VL - 66

SP - 837

EP - 887

JO - Journal of mathematical biology

JF - Journal of mathematical biology

SN - 0303-6812

IS - 4-5

ER -