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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Notes
In this paper we adopt the ‘postfix notation’ for the action of a functional on a vector. That is, if \(W\) is a Banach space with dual space \(W^{*}\), \(w \in W\) and \(w^{*} \in W^{*}\), then \(\langle w,w^{*} \rangle _{} := w^{*}(w)\).
It is customary to suppress the identity operator and write \(\lambda - S\) instead of \(\lambda I - S\).
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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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Dedicated to Odo Diekmann, on the occasion of his 65th birthday.
Appendix A: Proof of Proposition 26
Appendix A: Proof of Proposition 26
We use the same notation as in Lemma 23 and its proof. We recall that the vector \(Z = [\zeta _0, \zeta _1,\ldots ,\zeta _{N-1},1]\) is chosen such that the vector \(\beta = [\beta _0,\beta _1,\ldots ,\beta _N]\), whose elements are the coefficients of the characteristic polynomial \(\fancyscript{P}\), is given by \(\beta = M^TZ\). Introducing \(r := [1,\rho ^2, \rho ^4,\ldots ,\rho ^{2N}]\) we see that
First we determine the vector \(Z\), thereafter we split \(M\), and we conclude the proof by determining how \(Z\) acts on each factor in this decomposition.
Although \(Z\) can be obtained by applying the inverse of the Vandermonde matrix \(W\), we will proceed in a different manner. We start by rewriting (42) as
For \(m\in \mathbb N \) we define \(P_m := [p_1,p_2,\ldots ,p_m]\) with \(p_i \in \{0,1\}\) for \(i=1,\ldots ,m\). We set \(|P_m| = \sum _{i=1}^m p_i\) equal to the number of 1’s in \(P_m\). Using Gaussian elimination the solution of (84) is found to be
From the proof of Lemma 23 we recall the decomposition
where \(I\) is the \((N+1) \times (N+1)\) identity matrix and \(\Xi \) was defined in the proof of Lemma 23. Expanding the bilinear forms in the matrix \(\Xi \) and moving the summation in front of the matrix yields
Now substitute (85) with (86) into (83) to obtain
We observe that on the one hand,
while on the other hand,
Hence by (87) it follows that
which is equivalent to (46), in the sense that the two polynomials have identical roots. Hence the proof is complete.
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van Gils, S.A., Janssens, S.G., Kuznetsov, Y.A. et al. On local bifurcations in neural field models with transmission delays. J. Math. Biol. 66, 837–887 (2013). https://doi.org/10.1007/s00285-012-0598-6
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DOI: https://doi.org/10.1007/s00285-012-0598-6
Keywords
- Delay equation
- Neural field
- Hopf bifurcation
- Numerical bifurcation analysis
- Normal form
- Dual semigroup
- Sun-star calculus