Collective Phase Sensitivity

The collective phase response to a macroscopic external perturbation of a population of interacting nonlinear elements exhibiting collective oscillations is formulated for the case of globally coupled oscillators. The macroscopic phase sensitivity is derived from the microscopic phase sensitivity of the constituent oscillators by a two-step phase reduction. We apply this result to quantify the stability of the macroscopic common-noise-induced synchronization of two uncoupled populations of oscillators undergoing coherent collective oscillations.

Figure 1

Globally coupled SL model. (a) Stationary PDF $f(\varphi )$, (b) zero eigenfunction $u_0^{*}(\varphi )$ of ${\widehat{L}}^{*}$, (c) zero eigenfunction $u_0(\varphi )$ of $\widehat{L}$, and (d) kernel $-u_0^{*}(\varphi {)}^{\prime }f(\varphi )$.

Figure 2

Collective phase sensitivity of the globally coupled SL model. (a) Theoretical curves of ${\zeta }^R(\Phi )$. Single-oscillator phase sensitivity $Z^R(\varphi )$ is also shown with $\varphi =\Phi$. (b) ${\zeta }^R(\Phi )$, found from theory and from numerical simulations of nonlinear FPE and the Langevin equation (LE). For the LE simulation, individual raw responses are shown as dots, and piecewise mean values (25 bins) are shown as a solid line.

Figure 3

Common-noise-induced synchronization of globally coupled SL populations. Time evolution of the real part of $A(t)$ from (a) $t=0$ and (d) $t=600$. Snapshots of $\{W_j\}$ at (b) $t=0$ and (e) $t=600$ ($1/5$ of population shown). PDF of phase at (c) $t=0$ and (f) $t=600$. (g) Comparison between Lyapunov exponent $\Lambda$, found from theory and Langevin simulation.