Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 837–887 | Cite as

On local bifurcations in neural field models with transmission delays

  • S. A. van Gils
  • S. G. Janssens
  • Yu. A. Kuznetsov
  • S. Visser
Article

Abstract

Neural field models 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.

Keywords

Delay equation Neural field Hopf bifurcation Numerical bifurcation analysis Normal form Dual semigroup Sun-star calculus 

Mathematics Subject Classification (2000)

37L10 47H20 37L05 37M20 92C20 

Notes

Acknowledgments

The authors are thankful to Professor Odo Diekmann for informal and formal discussions related and unrelated to the present text. Sebastiaan Janssens and Sid Visser gratefully acknowledge support from The Netherlands Organization of Scientific Research (NWO) through grant 635.100.019: From Spiking Neurons to Brain Waves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • S. A. van Gils
    • 1
    • 2
  • S. G. Janssens
    • 1
    • 2
  • Yu. A. Kuznetsov
    • 1
    • 2
  • S. Visser
    • 1
  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  2. 2.Mathematical InsituteUtrecht UniversityUtrechtThe Netherlands

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