Irregular excitation patterns in reaction-diffusion systems due to perturbation by secondary pacemakers

Spatiotemporal excitation patterns in the FitzHugh-Nagumo model are studied, which result from the disturbance of a primary pacemaker by a secondary pacemaker. The primary and secondary pacemakers generate regular waves with frequencies $f_{\mathrm{pace}}$ and $f_{\mathrm{pert}}$, respectively. The pacemakers are spatially separated, but waves emanating from them encounter each other via a small bridge. This leads to three different types I–III of irregular excitation patterns in disjunct domains of the $f_{\mathrm{pace}}-f_{\mathrm{pert}}$ plane. Types I and II are caused by detachments of waves coming from the two pacemakers at corners of the bridge. Type III irregularities are confined to a boundary region of the system and originate from a partial penetration of the primary waves into a space, where circular wave fronts from the secondary pacemaker prevail. For this type, local frequencies can significantly exceed $f_{\mathrm{pace}}$ and $f_{\mathrm{pert}}$. The degree of irregularity found for the three different types is quantified by the entropy of the local frequency distribution and an order parameter for phase coherence.

Figure 1

Sketch of the simulation area. In region R the primary pacemaker excites planar waves (red dash-dotted line) propagating in the $y$ direction with frequency $f_{\mathrm{pace}}$. In region L the secondary pacemaker generates planar waves (blue dashed line) propagating in the $x$ direction with frequency $f_{\mathrm{pert}}$. Here $P_1$, $P_2$, and $P_3$ mark the points for which time series of $u$ are shown in Fig. 2. The red solid rectangle in the bridge ($10<x<11$ and $4<y<6$) indicates the region of detachment referred to in Sec. 3.

Figure 2

Time evolution of the activation variable $u$ at three points marked in Fig. 1, $P_1=(11.49,5.34)$ (black solid line), $P_2=(11.33,6.34)$ (red dashed line), and $P_3=(11.94,6.15)$ (blue dash-dotted line), for a pacing frequency $f_{\mathrm{pace}}=0.091$ and two different perturbation frequencies (a) $f_{\mathrm{pert}}=0.1$ and (b) $f_{\mathrm{pert}}=0.105$. It was checked that the behavior remains qualitatively the same after a three times longer simulation time, which gives confidence that the obtained features are not a transient phenomenon.

Figure 3

The three domains in the $f_{\mathrm{pace}}-f_{\mathrm{pert}}$ plane, where different types of irregularities occur. The symbols mark the points where solutions of the FHN equations (1) have been generated. Circles refer to solutions where regular excitation patterns have been obtained and squares, diamonds, and triangles refer to solutions where irregularities of types I, II, and III, respectively have been found. The scale of the frequencies is limited to $f=0.1176$. Above this frequency planar waves cannot be excited if the frequency of generated wave fronts corresponds to the pacing frequency in a one-to-one manner.

Figure 4

Time evolution of $u$ for irregularity type I for $f_{\mathrm{pace}}=0.117$ and $f_{\mathrm{pert}}=0.091$. Time steps between the pictures are (a)–(e) $\Delta t=4$ and (f)–(h) $\Delta t=3$. The color coding is given by the bar on the right. Yellow refers to the excited state and red to the excitable or refractory state.

Figure 5

Time evolution of $u$ for irregularity type II for $f_{\mathrm{pace}}=0.091$ and $f_{\mathrm{pert}}=0.105$. Time steps between the pictures are (a)–(d) $\Delta t=2$ and (e)–(h) $\Delta t=4$. The color coding is given by the bar on the right. Yellow refers to the excited state and red to the excitable or refractory state.

Figure 6

Time evolution of $u$ for irregularity type III for $f_{\mathrm{pace}}=0.067$ and $f_{\mathrm{pert}}=0.100$. Time steps between the pictures are (a)–(d) $\Delta t=2$ and (e)–(h) $\Delta t=4$. The color coding is given by the bar on the right. Yellow refers to the excited state and red to the excitable or refractory state.

Figure 7

Maximal local frequency in region R as a function of $f_{\mathrm{pert}}$ for different pacing frequencies $f_{\mathrm{pace}}=0.050$ (black circles), $0.067$ (red squares), $0.091$ (blue diamonds), $0.10$ (orange stars), and $0.111$ (green triangles). The dotted line marks the critical frequency $f_c$ and the case $f_{\max }=f_{\mathrm{pert}}$ is indicated by the dashed line. The solid lines connecting the data points are drawn as a guide for the eye.

Figure 8

Order parameter $\Phi$ for phase coherence and entropy $S$ of the local frequency distribution dependent on the perturbation frequency $f_{\mathrm{pert}}$, calculated for areas (a) and (c) $y>4$ and (b) and (d) $y<4$ in region R. Results are shown for different pacing frequencies: $f_{\mathrm{pace}}=0.050$ (black circles), $0.067$ (red squares), $0.091$ (blue diamonds), $0.10$ (orange stars), and $0.111$ (green triangles). The vertical dashed line marks the critical frequency $f_c$.

Figure 9

Transient time $t_{\mathrm{tr}}$ before onset of irregularities in regime III as a function of $f_{\mathrm{pert}}$ for $f_{\mathrm{pace}}=0.050$ (black circles), $0.059$ (red diamonds), $0.067$ (blue squares), and $0.077$ (green triangles). Dashed lines are calculated according to Eq. (5) with $t_a=0$ for $f_{\mathrm{pace}}=0.05$ (black line), $t_a=9$ for $f_{\mathrm{pace}}=0.0588$ (red line), $t_a=18$ for $f_{\mathrm{pace}}=0.0667$ (blue line), and $t_a=36$ for $f_{\mathrm{pace}}=0.077$ (green line).