Boundary in the dynamic phase of globally coupled oscillatory and excitable units

There is a crucial boundary between dynamic phase 1 and dynamic phase 2 of globally coupled oscillatory and excitable units, where the mean field is constant and oscillates in the former and the latter, respectively. This boundary is theoretically derived here for a large population of dynamical units, each having only a phase variable, where it is assumed that both the coupling strength and the distribution width of bifurcation parameters are equally small. This theory, which is applicable only if all or most of the units are intrinsically oscillatory, is confirmed to agree with simulation results for two different distribution densities.

Figure 1

Phase diagrams for the case of a uniform distribution of $a_j$: (a) $\gamma =0.1$ and (b) $\gamma =0.2$. In each panel, DP2 and DP1 are, respectively, situated on the upper and lower sides of the boundary, and the aging transition boundary is not shown (see [17, 18] for details). Moreover, the red pluses show the boundary between DP1 and DP2 based on simulation results $\left(N=10000\right)$, in which the boundary was obtained as the mean of the two nearest values of $a$ with $M>0.05$ and $M<0.05$, as $a$ was decreased for each value of $K$, where the step size of $a$ is 0.02 or 0.01. In addition, the line shows a theoretical boundary obtained from Eq. (29).

Figure 2

Behavior of $M$ against $K$ for $a=4$ (red pluses), 2.5 black crosses), and 1.5 (purple asterisks) with $\gamma =0.1$. The system size is the same as in Fig. 1.

Figure 3

(a) Trajectories of $Z$ on the complex plane and (b) average frequencies of all units, where $N=10000,\gamma =0.1$, and $a=4$. The values of $K$ are 0.1 (red), 0.135 (black), and 0.14 (purple). In (a), the size of a trajectory rapidly grows with $K$, while in (b), the latter two data points appear to perfectly overlap, with all units having a common average frequency.

Figure 4

Phase diagrams for the case of a Gaussian distribution of $a_j$: (a) $\gamma =0.1$ and (b) $\gamma =0.2$. Other details are the same as in Fig. 1.

Figure 5

Behavior of $M$ against $K$ for $\gamma =0.1$. Other details, including the values of $a$ and $K$, are the same as in Fig. 2.

Figure 6

(a) Trajectories of $Z$ on the complex plane and (b) average frequencies of all units, where the values of $K$ are 0.15 (red), 0.17 (black), and 0.2 (purple). In (b), the cluster of mutually frequency entrained units grows when $K$ increases from the middle value (black) to the largest (purple), whereas there is none for the smallest value of $K$ (red). Other details are the same as in Fig. 3.