Wave trains in an excitable FitzHugh-Nagumo model: Bistable dispersion relation and formation of isolas

We investigate the dispersion relations of nonlinear periodic wave trains in excitable systems which describe the dependence of the propagation velocity on the wavelength. Pulse interaction by oscillating pulse tails within a wave train leads to bistable wavelength bands, in which two stable and one unstable wave train coexist for the same wavelength. The essential spectra of the unstable wave trains exhibit a circle of eigenvalues with positive real parts which is detached from the imaginary axis. We describe the destruction of the bistable dispersion curve and the formation of isolas of wave trains in a sequence of transcritical bifurcations unfolding into pairs of saddle-node bifurcations. It turns out that additional dispersion curves of unstable wave trains play an important role in the destruction of the bistable dispersion curve.

Figure 1

Propagation velocity $c$ of a periodic wave train vs wavelength $L$. The upper branch depicts (a) a monotonically increasing and (b) an oscillating dispersion relation, respectively. (a) $a=0.12$, (b) $a=0.02$; $b=0.25$, $ϵ=0.3\times {10}^{-2}$. In both (a) and (b), the lower branch corresponds to unstable waves.

Figure 2

Bistable dispersion relation. Solid (short-dashed) lines indicate stability (instability) of wave trains with $L$-periodic boundary conditions. Profiles of labeled waves are presented in Figs. 3, 4. $a=0$, $b$ and $ϵ$ as in Fig. 1.

Figure 3

Activator $u$ and inhibitor $v$ profiles along one wavelength $L=290$ (a). Propagation from left to right. Labels refer to those in Fig. 2. Appearance of a secondary maximum is enlarged in lower panels; for the activator in (b) and the inhibitor in (c). $a=0$; $b$ and $ϵ$ as in Fig. 1.

Figure 4

Activator $u$ and inhibitor $v$ profile along one wavelength $L=725$ of label 4 in Fig. 2. Six secondary maxima of the profile correspond to six maxima of the dispersion relation counting from label 1 in Fig. 2. Propagation from left to right. $a=0$; $b$ and $ϵ$ as in Fig. 1.

Figure 5

Left panels (a) and (b) display enlargements of the bistable dispersion relation presented in Fig. 2. Solid (short-dashed) line indicates stable (unstable) wave trains with $NL$-periodic boundary conditions for all $N=1,2,3,\dots$. Long-dashed line indicates stability for $N=1$ and instability for $N⩾2$. Right panels (c)–(f) display subsets of essential spectra of pulse trains with reference to labels $c–f$ in (a). Squares indicate eigenvalues for $N=1$. Close to the $P$ point a critical circle crosses the imaginary axis from left to right. We define the $P$ point as the crossing of the eigenvalue belonging to $\gamma =0.5$. The circle detaches from the imaginary axis at the SN point. $a=0$; $b$ and $ϵ$ as in Fig. 1.

Figure 6

(Color online) (a), (b) Additional dispersion curves and sketches of corresponding wave trains (activator profile) which are completely unstable. The dashed curve (blue online) in (a) arises from a period-doubling bifurcation at the primary dispersion curve. A small-amplitude maximum blows up along each plateau whereas a small-amplitude maximum develops within each sawtooth. The dashed curve in (b) (magenta online) is closed, a so-called isola. (c) Dashed curves of (a) and (b) (blue and magenta online) approach the primary dispersion (green online) before partial stable isolas separate. (d) Separated isolas and broken stable branches. A velocity gap opens up for wave trains with secondary maxima. (e) As isolas shrink the velocity gap expands. (f) Between isolas wavelength gaps open up. A solid (dashed) dispersion curve indicates stability (instability) of wave trains with $L$-periodic boundary conditions. $b$ and $ϵ$ as in Fig. 1.