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Rotation Frequency of Spiral and Scroll Waves in Excitable Media

Daniel Margerit and Dwight Barkley

We have published algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media. Details can be found in [1,2,3]. The predictions for the frequency of spiral and twisted scroll waves can be reduced to particularly simple formulas. These are summarized below. More complete formulas, include spiral and scroll wavelengths, can be found in [2]. We provide a simple C++ computer program to compute frequencies and wavelengths [4].

$\psfig{file=asymptotics_fig1.ps,width=2in}$

The figure above shows in color a spiral solution from Eqs. (1)-(2) for parameters , , $\varepsilon =0.008$ . The spiral rotates counterclockwise with frequency $\omega = 2.72$ . The predicted frequency is $\omega = 2.68$ ; the predicted spiral shape is shown in white.

Model equations

We consider a standard two-species reaction-diffusion model of excitable media:

$\begin{mathletters} \begin{eqnarray} \partial{u}/\partial{t}&=& \nabla^2 u+f(u,v... ...psilon , \\ \partial{v}/\partial{t}&=& g(u,v). \end{eqnarray}\end{mathletters}$

where the following functions model the reaction kinetics

$\begin{mathletters} \begin{eqnarray} f(u,v) & = & u\left(1-u\right)\left(u-\frac{v+b}{a}\right) \\ g(u,v) & = & u-v, \end{eqnarray}\end{mathletters}$

Note that equations for excitable media are sometimes written in other units (scalings). The results below are for the units used above. These can easily be converted to other scalings.

Spiral Frequency

The frequency of spiral solutions to Eqs. (1)-(2) is well approximated by the following very simple formula:

$\begin{mathletters} \begin{eqnarray*} \omega = \frac{0.692 \mu}{\varepsilon ^{1/3}} - \frac{0.926}{a} \end{eqnarray*}\end{mathletters}$

with

$\begin{mathletters} \begin{eqnarray*} \mu & \simeq & 2.70 \left[ \frac{{v^s}(1-v... ...a} \right]^{2/3} v^s = \frac{a}{2} - b. \end{eqnarray*} \end{mathletters}$

This predicts the spiral frequency $\omega$ in terms of the three parameters , , and $\varepsilon$ appearing in Eqs. (1)-(2).

Scroll Frequency

The frequency of scroll wave with twist ${\tau_w}$ is well approximated by the following formula:

$\begin{mathletters} \begin{eqnarray*} \omega = {\omega_0} - a_1 {{\tau_w}}^2 \end{eqnarray*}\end{mathletters}$

where $\omega_0$ is the spiral rotation frequency and $- a_1 {{\tau_w}}^2$ is the correction due to twist (to lowest order in the twist). Specifically:

$\begin{mathletters} \begin{eqnarray*} {\omega_0} & = & \frac{0.692 \mu}{\varepsi... ... 0.373 - \frac{0.748 \varepsilon ^{1/3}}{a\mu}. \end{eqnarray*}\end{mathletters}$

with

$\begin{mathletters} \begin{eqnarray*} \mu & \simeq & 2.70 \left[ \frac{{v^s}(1-v... ...a} \right]^{2/3} v^s = \frac{a}{2} - b. \end{eqnarray*} \end{mathletters}$

This predicts the scroll frequency $\omega$ in terms of the three parameters , , and $\varepsilon$ appearing in Eqs. (1)-(2) and the twist ${\tau_w}$ of the scroll wave.

FitzHugh-Nagumo model

The FitzHugh-Nagumo model is given by Eqs. (1) with the following kinetic functions:

$\begin{mathletters} \begin{eqnarray*} f(u,v) = 3 u -u^3 - v, & & g(u,v) = u-\delta. \end{eqnarray*}\end{mathletters}$

The predicted frequency for spiral and twisted scroll waves in the FitzHugh-Nagumo model is

$\begin{mathletters} \begin{eqnarray*} \omega = {\omega_0} - a_1 {{\tau_w}}^2, \end{eqnarray*}\end{mathletters}$

where

$\begin{mathletters} \begin{eqnarray*} {\omega_0} = \frac{0.692 \mu}{\varepsilon ... ...- 0.373 ( 1 + \frac{\varepsilon ^{1/3}}{\mu}), \end{eqnarray*}\end{mathletters}$

with

$\begin{mathletters} \begin{eqnarray*} \mu & = & 0.744 (3-\delta^2)^{2/3}. \end{eqnarray*} \end{mathletters}$

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Dwight Barkley 2002-07-30