On local bifurcations in neural field models with transmission delays

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    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

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