will be assumed as an activation process. The life time in state $0$ is exponentially distributed and the state will be left with the transition rate $\gamma$.

The backward transition (relaxation from excited to rest) is governed by the waiting time density $w_1(t)$ to remain in state $1$. So far this density is arbitrary except that its mean value shall be $\tau$ and does not depend on any other parameters such as the noise intensity, the size and structure or possible dynamical states of the network. In the simulations two specific choices are made. First, the relaxation is treated as a Markovian rate process with a exponential waiting time distribution $$w_1(t)=\frac{1}{\tau}\exp\left(-\frac{t}{\tau}\right).
\label{eq:w1}$$ Oppositely, when modeling excitable dynamics, a sharply peaked distribution is more suitable, e.g. $$\begin{aligned}
    w_1(t)=\delta(t-\tau)
    \label{eq:w1peak}\end{aligned}$$ would yield a constant waiting time without any variance in state $1$ modeling a fixed delay [@Prager2003; @Prager2007; @Kouvaris2010]. As shown in the appendix \[sec:app\], the bifurcations of the steady states are not affected by the specific choice of $w_1$. It depends on the mean time $\tau$ spent in the excited state. Only, setups with possible transition to a non-stationary behavior reflect on the choice of $w_1$.

The two-state units are located 