Synchronization of two interacting populations of oscillators

We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators. It includes three types of bistabilities, higher order entrainment, and the existence of states with unusual stability properties. All possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically. The impact of these findings for neuroscience is discussed.

Figure 1

$(K,\gamma )$ phase diagram of system (1) assuming Lorentzian frequency distributions, $\Delta \omega =0.5,K_p=1$, and for different values of $\alpha$ (in rad): (a) $\alpha =0$, (b) $\alpha =0.1$, (c) $\alpha =\pi ∕4$, (d) $\alpha =1.15$. Numerical stability boundaries $(N=1000)$ are indicated as solid lines. Dotted lines represent analytical stability boundaries ${\gamma }_{c\pm }$ obtained from Eq. (6). Note that ${\gamma }_{c+}$ fully overlaps with numerical results. Region I: synchronization. Region II: coexistence. Region III: incoherence. Insets: bistability in the dashed regions around $A$ (see text).

Figure 2

Coupling-modified frequencies ${\tilde{\omega }}_i$ of populations 1 (black) and 2 (gray) as a function of the oscillator’s index $i$: oscillator $i$ has natural frequency ${\omega }_i^{(1,2)}={\overline{\omega }}^{(1,2)}+\gamma \tan [(\pi ∕2)(2i-N-1)∕(N+1)]$ (Lorentzian) ($\Delta \omega =0.5,\overline{\omega }=0,K_p=1$, and $N=1000$). First row: $\alpha =0,\gamma =0.4$ and (a) $K=0$, (b) $K=0.4$, (c) $K=0.41$. Second and third rows: $\alpha =\pi ∕4,K=0.53$ and (d) $\gamma =0.5$, (e) $\gamma =0.47$, (f) $\gamma =0.187$, (g) $\gamma =0.15$, (h) $\gamma =0.12$, (i) $\gamma =0.118$.

Figure 3

Order parameters $R^{\left(1\right)}$ (black) and $R^{\left(2\right)}$ (gray) and phase difference $\Delta \psi$ as a function of time $(N=1000,\Delta \omega =0.5,\overline{\omega }=0,K_p=1)$. At $t=0$ the phases were equally spaced in (0, $2\pi$]. (a) $\alpha =\pi ∕4,\gamma =0.12,K=0.53$ (region II) [Fig. 2h]; (b) and (c) represent regions ${\mathrm{II}}^{\prime }(\alpha =1.2,K=0.8)$ and ${\mathrm{I}}^{\prime }(\alpha =1.2,K=1.05)$, respectively ($\gamma =0$, see Fig. 4).

Figure 4

Phase diagram $(K,\alpha )$ for $\gamma =0,\Delta \omega =0.5$. Boundaries $K_c$ and $K_c^{\mathrm{I}}$ are obtained analytically from Eqs. (3, 9), respectively, whereas the symbols ◻ and ▵ correspond to numerical results. All other boundaries are determined numerically. Regions I (synchronization) and II (drift) are characterized by $R^{(1,2)}=1$. Within regions ${\mathrm{I}}^{\prime }$ ($\Delta \psi$ bounded) and ${\mathrm{II}}^{\prime }$ ($\Delta \psi$ not bounded) $R^{\left(1\right)}<1$ whereas $R^{\left(2\right)}=1$ (see text). Dashed regions present bistability between states I and ${\mathrm{II}}^{\prime }$ (horizontal dashes) and between I and ${\mathrm{I}}^{\prime }$ (vertical dashes).