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On local bifurcations in neural field models with transmission delays
Authors:S A van Gils  S G Janssens  Yu A Kuznetsov  S Visser
Institution:1. Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands
2. Mathematical Insitute, Utrecht University, PO Box 80010, 3508, Utrecht, The Netherlands
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.
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