Analytical approximations for the speed of pacemaker-generated waves

In oscillatory media, waves can be generated by pacemaker regions which oscillate faster than their surroundings. In many chemical and biological systems, such waves can synchronize the whole medium and as such they are a means of transmitting information at a fixed speed over large distances. In this paper, we apply analytical tools to investigate the factors that determine the speed of these waves. More precisely, we apply singular perturbation and phase reduction methods to two types of negative-feedback oscillators, one built on underlying bistability and one including a time delay in the negative feedback. In both systems, we investigate the influence of time-scale separation on the resulting wave speed, as well as the effect of size and frequency of the pacemaker region. We compare our analytical estimates to numerical simulations which we described previously [J. Rombouts and L. Gelens, Phys. Rev. Research 2, 043038 (2020)].

Figure 1

Setup of the system. (a) A pacemaker of size $S$ is situated in the middle of an oscillatory medium. The oscillation period of the medium is $P_o$; that of the pacemaker is $P_i$. The pacemaker sends out waves of speed $c$, which overtake the medium with an envelope speed $C$. The shade of gray corresponds to the value of $u$. (b) and (c) Limit cycles in the $(u,v)$ plane of bistable (b) and delayed oscillator (c). The cycle is shown for two values of the time-scale separation parameter $\varepsilon$. (d)–(g) Time series of $u$ and $v$ for bistable (left) and delayed (right) oscillator. Small value of $\varepsilon$ in (d) and (e) and a larger value in (f) and (g).

Figure 2

Singular perturbation. (a) The solution consists of slow progress along the branches and fast jumps in between: the outer and inner solutions, respectively. Shaded rectangles indicated the inner solution. The value of $v$ at the jump is $v_0$. (b) The value of $v_0$ determines the level of $v$ at the jump and the limits on the branches of the cubic nullcline that determine the period. (c) The inner solution is a bistable front. (d) Approximation of the roots of $v_0-f\left(u\right)$ when $v_0$ is close to the value at the extremum of the cubic nullcline. The roots are small perturbations of $-2/\sqrt{3}$ and $4/\sqrt{3}$. (d) Dispersion relation obtained by numerically solving for the roots of $v_0-f\left(u\right)$ in green, and with the parabolic approximation in pink. (f) Wave speed in the original system, compared to the result from a numerical simulation (described in Ref. [12]). Here $P_i=9.5$. The horizontal axis has logarithmic scaling.

Figure 3

(a) (Left) The parameters ${\alpha }_u$ and ${\alpha }_v$ as a function of $\varepsilon$ for the bistable and the delayed oscillator. (Right) Values of ${\beta }_u$ and ${\beta }_v$. (b) Sketch of the function $g$ which defines the inhomogeneity, and of the spatial eigenfunction $X^{*}$ associated with the largest eigenvalue ${\omega }^{*}$. (c) Wave speed as a function of $\varepsilon$ for the bistable and the delayed oscillator: numerical simulation (solid line) and approximation through phase reduction method (dotted line). (d) and (e) Wave speed as function of pacemaker size (d) and relative period difference (e) for bistable and delayed oscillator: numerical simulation (solid line) compared to phase reduction approximation (dotted line). (f) and (g) Envelope speed as function of size and period difference. In (d)–(g), $\varepsilon =0.01$. In (d) and (f), $P_i=9.5$. In (e) and (g), $S=20$. The simulations shown in (c)–(g) were described in Ref. [12].

Figure 4

(a) Graphical representation of Eq. (35). The intersection of the curves given by $cosx$ and $\sqrt{\rho }x$ determines the largest eigenvalue ${\omega }^{*}$. (b) Asymptotic approximations of ${\omega }^{*}$ as root of Eq. (31) for small and large pacemakers. The solid black line is the numerically obtained solution. The green lines are approximations for large $d$; the pink lines are the small-$d$ approximation. The solid and dotted colored lines are the approximations obtained by keeping one and two terms in the expansion for ${\omega }^{*}$, respectively. The inset axis shows a zoom for small $d$. Computed with $\alpha =1/2,\beta =2,h=0.05$. (c) and (d) Approximations of the wave speed as a function of pacemaker size and period difference, for the delayed oscillator with $\varepsilon =0.01$. On the left, $P_i=9.5$, on the right $S=20$. Because the size is large, only the large pacemaker approximation is shown in (d). (e) and (f) Envelope speed as a function of size and relative period difference. The simulations whose results are shown in (c)–(f) were described in Ref. [12].

Figure 5

Wave speed as a function of $\varepsilon$ with different approximations.