Classification of wave regimes in excitable systems with linear cross diffusion

We consider principal properties of various wave regimes in two selected excitable systems with linear cross diffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multienvelope waves, and intermediate regimes appearing as waves propagating at a fixed shape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the “quasisoliton” property of reflection of boundaries and penetration through each other. We also present some examples of the behavior of envelope quasisolitons in two spatial dimensions.

Figure 1

Three typical wave regimes in the cross-diffusion system ((2),(5)) with $k_1=10,\varepsilon =0.01$ for different values of $a$. (a) Simple quasisoliton, $a=0.22$. (b) Envelope quasisoliton, $a=0.12$. (c, d) Multienvelope quasiisoliton, $a=0.04$, at two different time moments.

Figure 2

(a) The parametric regions corresponding to different wave regimes in ((2),(5)) in the $(a,\varepsilon )$ plane at $k_1=10,x_s=4,u_s=2$. The abbreviations in the legend stand for various types of typical wave solutions: SER, single envelope reflecting; SEN, single envelope nonreflecting; MER, multiple envelope reflecting; MEN, multiple envelope nonreflecting; SFR, single fixed-shape reflecting; SFN, single fixed-shape nonreflecting; SIR, single intermediate (between single shape and envelope) reflecting; 2EN, envelope nonreflecting with separate envelopes at the front and at the back with nonoscillating plateau between them; N, no propagation. See Supplemental Material [14] for a movie. (b) Boundaries of the regimes of propagation and decay (of any waves) for different initial conditions.

Figure 3

Density plots of impact episodes for selected regimes designated in Fig. 2. (a) SFR: single fixed-shaped (“simple”) quasisoliton, $a=0.22,\varepsilon =0.01$. (b) SER: single-envelope quasisoliton, $a=0.1,\varepsilon =0.01$. (c) MER: multiple envelope quasisoliton, $a=0.02,\varepsilon =0.01$. In the panel, only the first reflected envelope has almost recovered within the view; other envelopes recover later. (d) SFN: single fixed-shape nonreflecting wave, $a=0.45,\varepsilon =0.004$. (e) SEN: single-envelope nonreflecting wave, $a=0.1,\varepsilon =0.004$. (f) MEN: multiple envelope nonreflecting wave, $a=0.02,\varepsilon =0.004$. In (c) and (f), individual wavelets are not distinguishable at printing resolution so only the envelope is in fact seen; in (f) the fine structure of the wavelets is shown magnified in the inset. White corresponds to $u=-0.3$, black corresponds to $u=1$. Time reference point $t=0$ is chosen arbitrarily at the beginning of the selected episode; point $x=0$ corresponds to the left boundary of the interval. All simulations are done for $\Delta x=0.1,\Delta t=0.001,L=400,k_1=10$.

Figure 4

Dynamics of (a–c) amplitudes and (d–f) velocities of individual wavelets for the three types of quasisolitons in ((2),(5)) for $k_1=10,\varepsilon =0.01$. (a, d) Simple quasisoliton, $a=0.22$; see Fig. 1. (b, e) Envelope quasisoliton, $a=0.12$; see Fig. 1. (c, f) Multienvelope quasisolitons, $a=0.04$; see Fig. 1(c, d).

Figure 5

(a–d) Number of wavelets and (e–h) velocities of the envelope waves in transient and in established regimes. (a, e) The transients for $k_1=10,\varepsilon =0.01$ and three values $a=0.08$ (circles), $0.12$ (squares) and $0.16$ (triangles). (b, f) Established quantities as functions of parameter $a$ for fixed $k_1=10,\varepsilon =0.01$. (c, g) Established quantities as functions $\varepsilon$ for fixed $a=0.12,k_1=10$. (d, h) Established quantities as functions $k_1$ for fixed $\varepsilon =0.01$ and $a=0.12$ (circles) and $a=0.22$ (triangles). In (b–d) and (f–h), filled symbols designate quasisoliton (reflecting) waves, and open symbols designate nonreflecting waves.

Figure 6

Profiles of established propagating waves at selected moments of time for $k_1=10,\varepsilon =0.01,L=\infty$. The origins of the $t$ and $x$ axes are chosen arbitrarily. (a) Simple quasisoliton, $a=0.22$. (b) Envelope quasisoliton, $a=0.12$.

Figure 7

Formation and evolution of a multienvelope quasisoliton regime on a circle (one-dimensional cable with periodic boundary conditions), for $a=0.03,k_1=10,\varepsilon =0.01$. (a) Snapshots of the profiles at selected moments of time. The waves and wavelets propagate counterclockwise. (b) Density plots of the $u$ component of the solution for selected time intervals; white corresponds to $u_{min}=-0.2$, black corresponds to $u_{max}=1$ corresponds to black. See also the movie in the Supplemental Material [14].

Figure 8

The dynamics of (a–c) numbers of wavelets and (e–h) velocities of the envelope waves in multienvelope quasisoliton regimes. (a, d) The quantities measured in a finite $L$ with periodic boundary conditions; the same simulation as shown in Fig. 7. (b, e) The quantities are measured for the headmost $L=800$ interval of a wave train in an “infinite length” computations, for $\varepsilon =0.01,k_1=10$ and selected values of $a$ as indicated in the legend. (c, f) The quantities are measured for $\varepsilon =0.01,k_1=10$, and $a=0.02$, with different measurement length $L$ as indicated in the legend. In panel (f), the velocities measured for different $L$ are indistinguishable.

Figure 9

Quasisoliton interaction of an envelope soliton with an impermeable boundary in LE model ((2),(6)), $A=6.35,B=0.045$.

Figure 10

Formation of a multienvelope quasisoliton and its interaction with an impermeable boundary in LE model ((2),(6)), with $A=6.45,B=0.045$.

Figure 11

Parametric regions in the $(A,B)$ plane for different wave regimes in LE model ((2),(6)),. The nomenclature of the wave regimes is the same as in Fig. 2.

Figure 12

Dynamics of quasisolitons in the transitional area “SIR” in Figs. 2 and 11. (a, b) amplitudes and (d, e) velocities of individual wavelets of quasisolitons. (a, d) FHN model ((2),(5)), $a=0.18,\varepsilon =0.006$. (b, e) LE model ((2),(6)), $A=6.25,B=0.025$. (c, f) Dependence of the quasisoliton shape change period in the FHN model as a function of parameter $a$ at fixed $\varepsilon =0.005$, and its best fits by the theoretical dependencies (9) and (10), respectively. See the last episode of the Fig02 movie in the Supplemental Material [14].

Figure 13

(a) Profiles of established wave regimes for parameters from the “Z” zones in Figs. 2 and 11. Left column: FHN model ((2),(5)) for $\varepsilon =0.001$ and $a=0.25,a=0.15$ and $a=0.1$ as shown by the legends. Right column: LE model ((2),(6)) for $B=0.001$ and $A=6.1,A=6.25$ and $A=6.4$, as shown by the legends. (b, c) Fragment of a density plot of, respectively, the $u$ and $v$ variables for the FHN model with $a=0.1$.

Figure 14

Dynamics of the (a–c) amplitudes and (d–f) velocities of individual wavelets, (a, d) of the $u$ variable in the front zone, (b, e) of the $v$ variable in the front zone, (c, f) of the $v$ variable in the back zone, for the FHN model ((2),(5)) at $k_1=10,a=0.1,\varepsilon =0.001$; see top bottom left panel of Fig. 13.

Figure 15

Selected snapshots of “wave flocks” of envelope quasisolitons in 2D FHN model with $a=0.02,\varepsilon =0.01,k_1=5,h_1=h_2=0.3$, box size $140\times 140$. See Supplemental Material [14] for a movie.

Figure 16

Selected snapshots of “wave grid” of quasisolitons in 2D FHN model with $a=0.02,k_1=30,\varepsilon =0.01,h_1=h_2=0.1$, box size $140\times 140$. See Supplemental Material [14] for a movie.

Figure 17

Selected snapshots of “wave flocks” of envelope quasisolitons in 2D LE model with $A=6.4,B=0.04,h_1=0.3,h_2=0.3$, box size $140\times 140$. See Supplemental Material [14] for a movie.