Theory of collective firing induced by noise or diversity in excitable media

Large variety of physical, chemical, and biological systems show excitable behavior, characterized by a nonlinear response under external perturbations: only perturbations exceeding a threshold induce a full system response (firing). It has been reported that in coupled excitable identical systems noise may induce the simultaneous firing of a macroscopic fraction of units. However, a comprehensive understanding of the role of noise and that of natural diversity present in realistic systems is still lacking. Here we develop a theory for the emergence of collective firings in nonidentical excitable systems subject to noise. Three different dynamical regimes arise: subthreshold motion, where all elements remain confined near the fixed point; coherent pulsations, where a macroscopic fraction fire simultaneously; and incoherent pulsations, where units fire in a disordered fashion. We also show that the mechanism for collective firing is generic: it arises from degradation of entrainment originated either by noise or by diversity.

Figure 1

Dynamical trajectories for $10$ typical units in a system of $N=400$ with $\omega =0.95$ and $C=4$. The right-hand column depicts the results without noise $D=0$, and different diversities: $\sigma =0.0$ (top, regime I, no firing), $\sigma =1.6$ (middle, regime II, synchronized firing) and $\sigma =3.0$ (bottom, regime III, desynchronized firing). While the left-hand column shows the dynamical regimes emerging under the presence of noise (and no diversity): $D=0.4$ (top, regime I, no firing), $D=1.0$ (middle, regime II, synchronized firing) and $D=5.0$ (bottom, regime III, desynchronized firing).

Figure 2

Symbols represent $\rho$, $\zeta$, and $J$ as obtained numerically from Eq. (1). Solid lines are the theoretical results. Panel (a) shows the variation with respect to the noise intensity $D$ in absence of diversity, $\sigma =0$, for frequency $\omega =0.95$, coupling $C=4$, and different system sizes: $N=50$ $(\circ )$, $N={10}^2$ $(\times )$, $N={10}^3$ $(▵)$, $N={10}^4$ $(◇)$. Panel (b) displays the same results as a function of $C$ for $D=1.0$. Panel (c) shows the variation with respect to diversity $\sigma$ for $D=0.3$, $C=4$ and $g\left({\omega }_j\right)$ being a uniform distribution. In all cases there are three regimes: (I) no firing, (II) synchronized firing, and (III) desynchronized firing.