Diversity of coupled oscillators can enhance their synchronization

The heterogeneity of coupled oscillators is important for the degree of their synchronization. According to the classical Kuramoto model, larger heterogeneity reduces synchronization. Here, we show that in a model for coupled pancreatic $\beta$-cells, higher diversity of the cells induces higher synchrony. We find that any system of coupled oscillators that oscillates on two time scales and in which heterogeneity causes a transition from chaotic to damped oscillations on the fast time scale exhibits this property. Thus, synchronization of a subset of oscillating systems can be enhanced by increasing the heterogeneity of the system constituents.

Figure 1

(a) Synchrony index $\lambda$ for different cell diversities. (b) Time courses of a typical $\beta$-cell membrane potential $V$ in a homogeneous network, i.e., with identical cells.

Figure 2

Time courses of membrane potential $V$ with each gray shade (color) representing one $\beta$-cell. (a) Low diversity $\left(\sigma =0.002\right)$, (c) high diversity $\left(\sigma =0.04\right)$, and (b) and (d) zoomed view of the region depicted by the dashed line in (a) and (c), respectively.

Figure 3

Time courses of intracellular calcium C corresponding to Fig. 2 with each gray shade (color) representing one $\beta$-cell for (a) low $\left(\sigma =0.002\right)$ and (b) high diversity $\left(\sigma =0.04\right)$. Intracellular calcium is plotted as a ratio to the resting concentration of $C_0=0.1 \mu \mathrm{M}$.

Figure 4

Steady-state solutions of $r_1$ (a) and $r_2$ (b) as $\mathrm{\Delta }$ changes for different values of $\mathrm{\Gamma }$. Each gray shade (color) refers to one value of $\mathrm{\Gamma }=1,\cdots ,5$. Solid and dotted lines correspond to stable and unstable steady-state solutions, respectively. For each $\mathrm{\Gamma }$, depending on the value of $\mathrm{\Delta }$, the reduced model (20) may have a unique stable steady state or show bistability (i.e., the existence of two stable steady states). Note that, for each $\mathrm{\Gamma }$, flipping the ${\overline{r}}_1$ curve at the line $\mathrm{\Delta }=1$ gives the corresponding curve of ${\overline{r}}_2$ and vice versa.

Figure 5

Steady-state solutions of $r_1$ as $\mathrm{\Gamma }$ changes for $\mathrm{\Delta }=0.98$ (a) and $\mathrm{\Delta }=1.02$ (b). Solid and dotted lines correspond to stable and unstable steady-state solutions, respectively. The vertical lines delineate a region of $\mathrm{\Gamma }$ values with NSSs. The results for ${\overline{r}}_2$ are the same.