Pattern formation in a two-dimensional array of oscillators with phase-shifted coupling

We investigate the dynamics of a two-dimensional array of oscillators with phase-shifted coupling. Each oscillator is allowed to interact with its neighbors within a finite radius. The system exhibits various patterns including squarelike pinwheels, (anti)spirals with phase-randomized cores, and antiferro patterns embedded in (anti)spirals. We consider the symmetry properties of the system to explain the observed behaviors, and estimate the wavelengths of the patterns by linear analysis. Finally, we point out the implications of our work for biological neural networks.

Figure 1

Phase diagram as a function of $\alpha$ for fixed $R$. (a) The nearest-neighbor coupling $(R=1)$. $\alpha 1\cong 0.29\mathrm{\pi }$. $\mathrm{S}$, spirals; AS, antispirals; $\mathrm{T}$, turbulence; AF-T, antiferro pattern embedded in turbulence; AF-AS (AF-S), antiferro pattern embedded in antispirals (spirals). (b) The finite long-range coupling $(R=6)$. $\alpha 1\cong 0.15\mathrm{\pi }$, $\alpha 2\cong 0.36\mathrm{\pi }$, $\alpha 3\cong 0.57\mathrm{\pi }$, $\alpha 4\cong 0.74\mathrm{\pi }$. $\mathrm{P}$, planar oscillation; S-PRC (AS-PRC), spirals (antispirals) with a phase-randomized cores; IR, irregular pattern induced by expansion of a phase-randomized core; NI, nearly incoherent pattern with a weak correlation of a length scale $\sim R$; CPW, competing plane waves occupying their respective domains of evolvable sizes; PW, plane waves. It is also observed that squarelike pinwheels exist transiently with plane waves.

Figure 2

(Color online) Patterns obtained from simulations with $R=1$: (a) Spirals $(\alpha =0.2\mathrm{\pi })$, (b) turbulence $(\alpha =0.4\mathrm{\pi })$, (c) antispirals $(\alpha =-0.2\mathrm{\pi })$, (d) antiferro pattern embedded in antispirals $(\alpha =0.8\mathrm{\pi })$, (e) antispirals by removing antiferro phases from (d), (f) the color code of the phase used for all figures. Insets of (a), (c), and (d): Magnification of the boxed areas. Arrows in (a) and (c) denote the propagation direction of waves with ${\theta }_{ij}$ in Eq. (2) when $K<\omega$.

Figure 3

(Color online) Patterns with $R=6$: (a) Spirals with a phase-randomized core $(\alpha =0.2\mathrm{\pi })$, (b) antispirals with a phase-randomized core $(\alpha =-0.2\mathrm{\pi })$, (c) irregular pattern induced by the expansion of the phase-randomized core $(\alpha =0.4\mathrm{\pi })$, (d) coexistence of plane waves and squarelike pinwheels $(\alpha =\mathrm{\pi })$.

Figure 4

(Color online) Patterns realized in a two-dimensional array of the Morris-Lecar systems. For the Morris-Lecar equations and the parameters, see Ref. 6. Here we choose $I=0.081$, $k=0.01$. (a) When $R=1$, antiferro patterns embedded in antispirals are observed. The figure shows only antispirals after removing the antiferro phases. (b) If $R$ increases to 6, coexistence of plane waves and squarelike pinwheels is observed.

Figure 5

(a) $\mathrm{Re}\left(\eta \right)$ in Eq. (6) as a function of $\rho$. (b) Wavelengths of plane waves $\left(\mathrm{\lambda }\right)$ as a function of $R$. Predicted values $(2\mathrm{\pi }R∕5.1356=1.2235R)$ fit the observed ones well in our model for $\alpha =\mathrm{\pi }$ and in the Morris-Lecar systems for the same parameters as in Fig. 4, within relative errors of 0.02 and 0.03, respectively. The wavelengths observed in our model are insensitive to $\alpha$ for the plane-wave regime.