Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Figure 1

Traveling-pulse solution of the FHN model. Snapshots of (a) ${\mathit{X}}_0(x,\theta =0)=(X_u,X_v)$, (b) $\mathit{U}(x,\theta =0)=(U_u,U_v)$, and (c) $\mathit{Z}(x,\theta =0)=(Z_u,Z_v)$. Both $u$ and $v$ components are plotted in each figure. Some of the curves are enlarged for visibility.

Figure 2

Optimized interaction functions for the traveling pulse. (a) Optimal nonlocal interaction $\mathit{G}(x,\theta =0)=(G_u,G_v)$ and (b) optimal nonlinear interaction $\mathit{H}(x,\theta =0)=(H_u,H_v)$. Both $u$ and $v$ components are plotted. Results for the $u$ components are enlarged for clarity.

Figure 3

Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling functions for the traveling pulse. Results for direct interaction, optimal nonlocal interaction, and optimal nonlinear interaction are shown.

Figure 4

Synchronization dynamics between traveling pulses. Results for direct interaction and for optimal nonlocal interaction are compared. (a) Evolution of synchronization error. (b) Evolution of phase difference.

Figure 5

Oscillating-spot solution of the FHN model. Snapshots of (a) ${\mathit{X}}_0(x,\theta =0)=(X_u,X_v)$, (b) $\mathit{U}(x,\theta =0)=(U_u,U_v)$, and (c) $\mathit{Z}(x,\theta =0)=(Z_u,Z_v)$. Both $u$ and $v$ components are shown.

Figure 6

Evolution of the limit-cycle solution ${\mathit{X}}_0(x,\theta )=(X_u,X_v)$ and the phase sensitivity function $\mathit{Z}(x,\theta )=(Z_u,Z_v)$ for one period of oscillation, $0\le \theta \le 2\pi$. (a) $X_u$, (b) $X_v$, (c) $Z_u$, and (d) $Z_v$.

Figure 7

Correlation matrix $\widehat{W}(x,x^{\prime })=\left(\begin{array}{cc}{W}_{uu}& W_{uv}\\ W_{vu}& W_{vv}\end{array}\right)$. (a) $W_{uu}$, (b) $W_{uv}$, (c) $W_{vu}$, and (d) $W_{vv}$.

Figure 8

Optimized interaction functions of the oscillating spot. (a,b) Optimal nonlocal interaction $\mathit{G}(x,\theta )=(G_u,G_v)$. (a) $G_u$ and (b) $G_v$. (c,d) Optimal nonlinear interaction $\mathit{H}(x,\theta )=(H_u,H_v)$. (c) $H_u$ and (d) $H_v$. Evolution for one period of oscillation are shown ($0\le \theta \le 2\pi$). In each figure, the lines indicate the locations where the interaction function vanishes and changes the sign.

Figure 9

Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling functions for the oscillating spot. Results for direct interaction, optimal nonlocal interaction, and optimal nonlinear interaction are shown.

Figure 10

Synchronization dynamics between oscillating spots. Results for direct interaction and for optimal nonlocal interaction are compared. (a) Evolution of synchronization error. (b) Evolution of phase difference.

Figure 11

Spiral solution of the FHN model. (a,b) Snapshots of the limit-cycle solution ${\mathit{X}}_0(x,y,\theta =0)=(X_u,X_v)$. (a) $X_u$ and (b) $X_v$. (c,d) Snapshots of the phase sensitivity functions $\mathit{Z}(x,y,\theta =0)=(Z_u,Z_v)$. (c) $Z_u$ and (d) $Z_v$. All patterns constantly rotate around the center of the system in the direction shown by the arrow.

Figure 12

Phase sensitivity function and optimized interaction functions near the spiral tip. (a,b) Snapshots of the phase sensitivity function $\mathit{Z}(x,y,\theta =0)=(Z_u,Z_v)$. (a) $Z_u$ and (b) $Z_v$. (c,d) Snapshots of the optimal nonlocal interaction $\mathit{G}(x,y,\theta =0)=(G_u,G_v)$. (c) $G_u$ and (d) $G_v$. (e,f) Snapshots of the optimal nonlinear interaction $\mathit{H}(x,y,\theta =0)=(H_u,H_v)$. (e) $H_u$ and (f) $H_v$.

Figure 13

Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling functions for the spiral. Results for direct interaction, optimal nonlocal interaction, and optimal nonlinear interaction are shown.

Figure 14

Synchronization dynamics between spirals. Results for direct interaction and for optimal nonlocal interaction are compared. (a) Evolution of synchronization error. (b) Evolution of phase difference.