linear factor in the signal function, cf. . In the master equation for nodes with the degree $k$ it reads $$\label{eq:coup}
   \gamma[f]=\gamma\left[\frac{k}{N} r(t)\right].$$

Stationary behavior of excitable units {#sec:qual}
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Depending on the specific structure of $\gamma$ this equation can be highly non-linear. Here in this manuscript, the activation rate $\gamma$ is assumed to follow Arrhenius’ law [@Rice1954; @Hanggi1990]. The two-state system shall mimic the behavior of stochastic excitable dynamics [@Lindner2004] with state $0$ being the rest state and state $1$ the excited one, respectively. Transitions to the excited state $1$ is achieved by overcoming a threshold with barrier $\Delta U$ under the influence of noise with intensity $D$. The corresponding Arrhenius’ law of the rate reads $$\gamma  =  \gamma_0 \,\exp\left(-\frac{\Delta U}{D}\right),$$ with a constant $\gamma_0$ defining the time scale.

The coupling between units is assumed to be purely excitatory and thus each coupled unit that is already in the excited state will lower the potential barrier by an amount proportional to the coupling strength $\sigma$, which is the same for every unit throughout this paper. The following ansatz for the potential $\Delta U$ combines the above information $$\Delta U = 1-\sigma\,\frac{k}{N}r(t).
    \label{eq:g_net}$$ Setting $p(k)=\delta_{k,N}$ restores the globally connected network 