Universal behavior in populations composed of excitable and self-oscillatory elements

We study the robustness of self-sustained oscillatory activity in a globally coupled ensemble of excitable and oscillatory units. The critical balance to achieve collective self-sustained oscillations is analytically established. We also report a universal scaling function for the ensemble’s mean frequency. Our results extend the framework of the “aging transition” [Phys. Rev. Lett. 93, 104101 (2004)] including a broad class of dynamical systems potentially relevant in biology.

Figure 1

Morris-Lecar equations (2) in the excitable (left panel) and in the oscillatory (right panel) regimes. Both dynamical states are linked through a saddle-node bifurcation on invariant circle (SNIC) at ${\varphi }^{*}\simeq 0.076$ [see Eq. (2a)]. The nullclines $\dot{V}=0$ and $\dot{w}=0$ are depicted with gray lines.

Figure 2

(Color online) $(K,p)$ phase diagram for a mixed population of oscillatory and excitable elements: (a) Morris-Lecar (2) with ${\varphi }_A=0.1$ and ${\varphi }_E=0.06$; (b) phase model (3) with $b_A=0$ (oscillatory) and $b_E=\sqrt{2}$ (excitable). Inside the region of global oscillations (GO) the black dashed line separates 1:1 (right) from other $1:r$, $0<r<1$ (left) frequency ratios, within the region GO. The (red, online) dotted curves are the deformation of the solid lines when heterogeneity in each subpopulation is considered (see text): $\gamma =\left(\mathrm{a}\right)$ 0.01 and (b) 0.6.

Figure 3

(a) Normalized ensemble’s frequency $\tilde{\Omega }\left(p\right)$ for different values of the coupling strength $K$ in a population of oscillatory and excitable ML units $(N=500)$. The dashed line for large coupling $(K=5)$ shows the convergence of $p_c$ to the value predicted by Eq. (6). Panels (b)–(e) show the individual frequencies for $K=0.05$ (PO) (top), $K_c\simeq 0.144$, $K=0.2$, and $K=0.3$ (transition GO-$\mathrm{Q}$) (bottom).

Figure 4

Fitting to Eq. (10) for an ensemble of $N=1000$ ML elements, $K_c=0.14386\left(0\right)$. Dashed lines stem from Eq. (10) with fitting parameters $g=5.01$ and $h=0.66$.