On local bifurcations in neural field models with transmission delays

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.

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

$$\begin{aligned} \fancyscript{P}(\rho ) = r^T M^T Z \end{aligned}$$

(83)

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

$$\begin{aligned} \begin{bmatrix} 1&\quad k_1^2&\quad k_1^4&\quad \ldots&\quad k_1^{2N-2}&\quad 0\\ 1&\quad k_2^2&\quad k_2^4&\quad \ldots&\quad k_2^{2N-2}&\quad 0\\ \vdots&\quad \vdots&\quad \vdots&\quad&\quad \vdots&\quad \vdots \\ 1&\quad k_N^2&\quad k_N^4&\quad \ldots&\quad k_N^{2N-2}&\quad 0\\ 0&\quad 0&\quad 0&\quad \ldots&\quad 0&\quad 1 \end{bmatrix} \begin{bmatrix} \zeta _0 \\ \zeta _1 \\ \vdots \\ \zeta _{N-1} \\ 1 \end{bmatrix} = \begin{bmatrix} -k_1^{2N} \\ -k_2^{2N} \\ \vdots \\ -k_N^{2N} \\ 1 \end{bmatrix} \end{aligned}$$

(84)

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

$$\begin{aligned} Z =\begin{bmatrix} (-1)^{N-0} \sum _{|P_N|=N-0}{k_1^{2p_1} k_2^{2p_2} \ldots k_N^{2p_N}}\\ (-1)^{N-1} \sum _{|P_N|=N-1}{k_1^{2p_1} k_2^{2p_2} \ldots k_N^{2p_N}}\\ \vdots \\ (-1)^{1} \sum _{|P_N|=1}{k_1^{2p_1} k_2^{2p_2} \ldots k_N^{2p_N}}\\ 1 \end{bmatrix} \end{aligned}$$

From the proof of Lemma 23 we recall the decomposition

$$\begin{aligned} M^T = e^{\lambda \tau _0}(\lambda + \alpha )I + 2\Xi \end{aligned}$$

(85)

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

$$\begin{aligned} \Xi = \sum _{i=1}^{N} c_i k_i \Xi _i, \qquad \Xi _i := \begin{bmatrix} 0&1&k_i^2&\ldots&k_i^{2(N-1)} \\ 0&0&1&\ldots&k_i^{2(N-2)} \\ \vdots&\ddots&\ddots&\vdots \\ 0&&0&1 \\ 0&\ldots&\ldots&\ldots&0 \end{bmatrix} \end{aligned}$$

(86)

Now substitute (85) with (86) into (83) to obtain

$$\begin{aligned} \fancyscript{P}(\rho )&= r^T \left[e^{\lambda \tau _0}(\lambda +\alpha )I+ 2\sum _{i=1}^{N}{c_i k_i \Xi _i}\right] Z\nonumber \\&= e^{\lambda \tau _0}(\lambda +\alpha ) \begin{bmatrix} 1&\rho ^2&\rho ^4&\ldots&\rho ^{2N} \end{bmatrix} Z + 2r^T \sum _{i=1}^{N}{c_i k_i \Xi _i} Z \end{aligned}$$

(87)

We observe that on the one hand,

$$\begin{aligned} e^{\lambda \tau _0}(\lambda +\alpha ) \begin{bmatrix} 1&\rho ^2&\rho ^4&\ldots&\rho ^{2N} \end{bmatrix} Z = e^{\lambda \tau _0}(\lambda +\alpha ) \prod _{i=1}^{N} (\rho ^2-k_i^2) \end{aligned}$$

while on the other hand,

$$\begin{aligned} r^T \sum \limits _{i=1}^{N}{c_i k_i \Xi _i} Z&= \sum \limits _{i=1}^{N}c_i k_i \begin{bmatrix} 1&\rho ^2&\rho ^4&\ldots&\rho ^{2N} \end{bmatrix}\\&\times \begin{bmatrix} (-1)^{N-1} \sum _{|P_{N-1}|=N-1}{k_1^{2p_1} k_2^{2p_2} \ldots k_{i-1}^{2p_{i-1}}k_{i+1}^{2p_i}\ldots k_N^{2p_{N-1}}}\\ (-1)^{N-2} \sum _{|P_{N-1}|=N-2}{k_1^{2p_1} k_2^{2p_2} \ldots k_{i-1}^{2p_{i-1}}k_{i+1}^{2p_i}\ldots k_N^{2p_{N-1}}}\\ \vdots \\ (-1)^{1} \sum _{|P_{N-1}|=1}{k_1^{2p_1} k_2^{2p_2} \ldots k_{i-1}^{2p_{i-1}}k_{i+1}^{2p_i}\ldots k_N^{2p_{N-1}}}\\ 1\\ 0 \end{bmatrix} \end{aligned}$$

Hence by (87) it follows that

$$\begin{aligned} \fancyscript{P}(\rho ) = e^{\lambda \tau _0}(\lambda +\alpha ) \prod _{j=1}^{N} (\rho ^2-k_j^2) + 2\sum _{i=1}^{N} c_i k_i \prod _{\begin{matrix} j=1 \\ j\ne i \end{matrix}}^{N} (\rho ^2-k_j^2) \end{aligned}$$

which is equivalent to (46), in the sense that the two polynomials have identical roots. Hence the proof is complete.