Origin of finite pulse trains: Homoclinic snaking in excitable media

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

Figure 1

Real (dark/blue line) and (negative) imaginary (light/red line) parts of the dispersion relation $\sigma \left(k\right)$ computed from a linear analysis of Eqs. (1) at the instability onset $a_v=a_v^{\text{Hopf}}$.

Figure 2

Bifurcation diagram showing branches of standing (SW, blue/dark line) and traveling (TW, red/light line) waves in terms of the maximum value $u_{\text{max}}$ of the $u$ field, obtained via numerical continuation; the solid red/light line indicates TW that are linearly stable with respect to perturbations with arbitrary wavenumber. The solid black line indicates the stable uniform state $u_0$. The finite wavenumber Hopf instability takes place at $a_v^{\text{Hopf}}$.

Figure 3

(a) Bifurcation diagram for the ${\Xi }_m^1$ [$m=1,2$, thick (dark/blue) line] and ${\Xi }_m^2$ [$m=2,3$, thin (light/red) line] pulse branches and the traveling wave (TW) branch showing the speed $c$ as a function of $a_v$ on a periodic domain with period $L=300$, where $m$ indicates the number of pulses in the wave train. The ${\Xi }_1^1$ and ${\Xi }_2^1$ branches connect at the leftmost saddle node where a trailing pulse is added, and similarly for ${\Xi }_2^2$ and ${\Xi }_3^2$. The insets show the profiles $u\left(\xi \right)$, $\xi \in [0,20]$, at $a_v=0.5$, $c\approx 0.25$, corresponding to the diamond symbol. The Hopf bifurcation occurs at $a_v=a_v^{\text{Hopf}}\approx 0.4497$, $c\approx 0.26$. (b) Left panel: Zoom of the ${\Xi }_{1,2}^1$ and ${\Xi }_{2,3}^2$ snaking region [top rectangle in (a)]. Right panel: Zoom of the leftmost saddle node region [bottom rectangle in (a)]. The insets show profiles $u\left(\xi \right)$ for $\xi \in [0,10]$ at $a_v=0.5$ on the same scale as in (a). Solid (dashed) lines indicate pulse solutions that are stable (unstable) with respect to perturbations of wavelength $L$; they also indicate the stability of the TW with respect to all perturbations with wavelength $\lambda \ge {\lambda }_c\equiv 2\pi /k_c$.

Figure 4

Decay of a slow ${\Xi }_1^1$ pulse with $c\approx 0.09$ into a fast ${\Xi }_1^1$ pulse with $c\approx 0.25$ [black diamond in Fig. 3] when $a_v=0.5$.

Figure 5

2D and 1D solutions to Eqs. (1) with reflecting boundary conditions, showing coexistence of stable fast (top two panels) and stable slow (bottom two panels) ${\Xi }_1^1$ pulses when $a_v=0.5$ and $a_0=-0.11$.

Figure 6

Top panel: Bifurcation diagram for the ${\Xi }_m^1$ ($m=1,2$) pulse branches; the insets correspond to locations indicated in the diagram. The waves travel from left to right. An “extra” trailing pulse is present along the ${\Xi }_2^1$ branch. Bottom panel: Zoom of the bottom leftmost region of the top panel [dashed rectangle] showing the nucleation of the leading pulse. The profiles are shown for $\xi \in [0,120]$ but are computed on a domain of size $L=300$; the amplitude scale for $u$ is the same as in Fig. 3.

Figure 7

A four-pulse train and TW profiles at $a_v\simeq 0.450358$, corresponding to the saddle node (b) in Fig. 6 (top panel). The dotted lines correspond to one wavelength of the coexisting small amplitude (${\lambda }_S$) and large amplitude (${\lambda }_L$) TW. These wavelengths are precisely the wavelengths present in the pulse train in the top panel.

Figure 8

Top panel: nonlinear dispersion relation for TW at $a_v\simeq 0.45064$, corresponding to the location of a saddle node on the pulse train (shown for $\xi \in [0,150]$ in the lower inset) that lies on the ${\Xi }_2^1$ branch [diamond symbol in Fig. 6 (top panel)]. The wavelengths ${\lambda }_S\simeq 4.488$, ${\lambda }_L\simeq 7.192$, ${\lambda }_f\simeq 12.406$, ${\lambda }_r\simeq 10.577$ are computed for $c\simeq 0.28116$ (the speed of the pulse train), with ${\lambda }_{f,r}$ corresponding to the wavelengths associated with the front and rear connections. The top inset shows the dispersion relation as $u_{\text{max}}$ vs. $c$, and depicts the amplitudes of the different TW. The bottom panel compares the profiles over one period of each TW corresponding to the four different wavelengths (solid lines) identified in the top panel with the corresponding profile from within the pulse train (dashed-dotted lines) shown in the lower inset in the top panel.

Figure 9

Nonlinear dispersion relation for TW at (a) $a_v=0.445<a_v^{\text{Hopf}}$ and (b) $a_v=0.5>a_v^{\text{Hopf}}$. The red diamond symbols in (a) indicate the wave numbers at which $\mathrm{Re}\left[\sigma \right(k\left)\right]=0$ [see Eq. (2)], i.e., the limits of the band of linearly unstable wave numbers around $k=k_c$ when $a_v<a_v^{\text{Hopf}}$ (cf. Fig. 1). Large amplitude TW with wavelength ${\lambda }_c\equiv 2\pi /k_c$ move with speed $c\simeq 0.1924$. The three-pulse state in (b) corresponds to the stable solution on the ${\Xi }_3^3$ branch at $a_v=0.5$ in Fig. 3 and travels with speed $c\simeq 0.25$ corresponding to two intersections with the nonlinear dispersion relation, ${\lambda }_L^s\simeq 6.3$ and ${\lambda }_L^u\simeq 8.2$. The periodic TW with these wavelengths are shown in the inset (for $\xi \in [0,50]$) and are, respectively, stable (superscript $s$) and unstable (superscript $u$) with respect to perturbations with wavelength $\lambda$. The dispersion relation connects to a periodic TW with ${\lambda }_{TW}\simeq 8.225$, $c\simeq 0.276$ at the location indicated by the green diamond symbol.

Figure 10

Nonlinear pulse dispersion relations at $a_v=0.5$ computed by starting from the three-pulse state [black line, bottom inset in Fig. 9], two-pulse state (blue/dark line), and a single-pulse state (green/light line), all in a domain of length $L=100$. The green diamond symbol on the single-pulse branch indicates the termination of the branch of single pulse states ($L\simeq 16.45\simeq 2{\lambda }_{\text{TW}}$) on the corresponding TW solution indicated in Fig. 9 and shown in a black dashed line in the inset, i.e., the green diamond represents a wavelength-doubling bifurcation of the periodic TW.

Figure 11

Top left panel: Zoom of the ${\Xi }_3^3$ branch near its leftmost fold at $a_v\simeq 0.4496\lesssim a_v^{\text{Hopf}}\simeq 0.4497$, indicating the locations (a) $a_v=0.454$, $c\simeq 0.3071$, (b) $a_v=0.453$, $c\simeq 0.2997$, (c) $a_v=0.45$, $c\simeq 0.2937$ corresponding to the pulse profiles shown in the insets in panels (a)–(c). All solutions are computed on a periodic domain with period $L=100$; solid (dashed) lines in the top left panel indicate stable (unstable) ${\Xi }_3^3$ pulses. Panels (a)–(c) show the corresponding nonlinear dispersion relations $c\left(\lambda \right)$ for TW with wavelength $\lambda$. The computed speed $c$ of the pulse state selects specific wavelengths from the dispersion relation: (a) ${\lambda }_L^s\simeq 8.4$, ${\lambda }_L^u\simeq 11$, (b) ${\lambda }_L^s\simeq 8.059$, ${\lambda }_L^u\simeq 10.86$, (c) ${\lambda }_L\simeq 7.9$, ${\lambda }_S\simeq 4.776$. The dashed lines in the inset in (c) correspond to TW with the wavelengths ${\lambda }_{L,S}$ indicated on the dispersion relation. The corresponding $(c,u_{\text{max}})$ plots are shown in the upper insets.

Figure 12

Top panel: Zoom of the ${\Xi }_3^3$ branches for $L=100$ and $L=200$ near their leftmost folds. The black dot marks a 3-pulse state at $a_v=0.45$, $c\simeq 0.309$, shown in the bottom panel. The inset in the bottom panel shows weak spatial growth in the oscillation amplitude that is difficult to observe on shorter domains for which $\mathrm{Re}\left[{\mu }_{3,4}\right]<L^{-1}$, as in the bottom inset in Fig. 11.