Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system. We completely describe the dynamics of wavefronts and discuss the properties of the speed equation. The nonequilibrium Ising-Bloch bifurcation of traveling fronts is found to occur in the hyperbolic case as well as in the parabolic system. Waves in the hyperbolic case typically propagate with lower speeds, in absolute value, than waves in the parabolic one. We find the interesting feature that the hyperbolic and parabolic front trajectories coincide in the phase plane for the FitzHugh-Nagumo model with a diagonal diffusion matrix, which is the case of self-diffusion, and differ for the system with cross diffusion.

Figure 1

Self-diffusion case. Front speed $c$ versus excitation threshold $a$ relation described by Eq. (14) for (a) $\varepsilon =0.1$ and (b) $\varepsilon =0.5$, $D=1$, and $\alpha =1$. The curves for the hyperbolic case ($\tau =1$) and the parabolic case ($\tau =0$) are marked by thick and thin lines, respectively. Circles and diamonds represent results of numerical simulations for the hyperbolic and parabolic systems, respectively.

Figure 2

Self-diffusion case. Front profile with monotonic tails for the (a) $u=u\left(\xi \right)$ and (b) $v=v\left(\xi \right)$ components and (c) the $u\text{−}v$ phase plane. The parameters are fixed at $D=1$, $\alpha =1$, $\varepsilon =0.1$, and $a=0.3$. The curves for the hyperbolic ($\tau =1$ and $c\approx 0.812$) and parabolic case ($\tau =0$ and $c\approx 1.392$) are marked by thick and thin lines, respectively.

Figure 3

Self-diffusion case. Wavy front profile for the (a) $u=u\left(\xi \right)$ and (b) $v=v\left(\xi \right)$ components and (c) the $u\text{−}v$ phase plane. The parameters are fixed at $D=1$, $\alpha =0.1$, $\varepsilon =0.05$, and $a=0.8$. The curves for the hyperbolic ($\tau =1$ and $c\approx 0.948$) and parabolic case ($\tau =0$ and $c\approx 2.969$) are marked by thick and thin lines, respectively.

Figure 4

Cross-diffusion case. Wavy front profile for the (a) $u=u\left(\xi \right)$ and (b) $v=v\left(\xi \right)$ components and (c) the $u\text{−}v$ phase plane. The parameters are fixed at $D=1$, $h=4$, $\alpha =1$, $\varepsilon =1$, and $a=0.4$. The curves for the hyperbolic ($\tau =0.03$ and $c\approx 5.526$) and parabolic case ($\tau =0$ and $c\approx 5.772$) are marked by thick and thin lines, respectively.