Computation of the response functions of spiral waves in active media

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spiral’s rotation center and frequency, i.e., drift. The response functions (RFs) of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues $\lambda =0,\pm i\omega$. The RFs describe the spiral’s sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spiral’s drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e., discretization steps and size of the medium. The obtained RFs are localized at the spiral’s core.

Figure 1

Solutions of the nonlinear problem (3) and the linearized problem [Eqs. (4, 5)], i.e., the Goldstone modes, at the best parameters, ${\rho }_{\text{max}}=25$, $N_{\rho }=1280$, and $N_{\theta }=64$ as density plots. Numbers under the density plots are their amplitudes $A$: white of the plot corresponds to the value $A$ and black corresponds to the value $-A$ of the designated field. Upper row: first components; lower row: second components.

Figure 2

Same visualization as in Fig. 1, for the adjoint linearized problem [Eqs. (13, 15)], i.e., the response functions.

Figure 3

(Color online) Radial dependence of the angle-averaged solutions for the (a) spiral wave, (b) Goldstone modes, and (c) response functions. In (c), the dependence of $\Delta \omega \left({\rho }_{\text{max}}\right)=\widehat{\omega }\left({\rho }_{\text{max}}\right)-\widehat{\omega }\left(25\right)$ is shown for comparison, where $\widehat{\omega }\left({\rho }_{\text{max}}\right)$ is the numerically found spiral angular velocity in the disk of given radius ${\rho }_{\text{max}}$.

Figure 4

(Color online) Convergence in numerical parameters of deviation of the numerical eigenvalues from theoretical (upper row), of $L_2$ distance between numerical and theoretical eigenfunctions (second row), and of orthogonality, i.e., Frobenius norm of the difference of the matrix of scalar products of eigenfunctions and adjoint eigenfunctions from the unity matrix (third row), all in logarithmic scales, as dependencies of disk radius (first and second columns, linear scale), radius discretization step (third column, logarithmic scale), and polar angle discretization step (fourth column, logarithmic scale). In the first column, ${\rho }_{\text{max}}$ is changed while the values of $\Delta \rho$ and ${\rho }_{\text{max}}\Delta \theta$ are kept constant. In the second column, ${\rho }_{\text{max}}$ is changed while $\Delta \rho$ and $\Delta \theta$ are kept constant.

Figure 5

(Color online) [(a) and (b)] Effect of the accuracy of the unperturbed spiral wave solution on the convergence: (a) Newton-iteration tolerance ${10}^{-8}$ and (b) Newton iterations until the norm of the residual stopped decreasing. (c) Convergence of the response functions in $\Delta \rho$.