with $r(t)\rightarrow P(1,t|N)$, which has been earlier studied in detail [@Kouvaris2010]. Alternatively, it is possible to consider a discrete number of $K$ degrees $k_i$, $i=1, \dots, K$. The corresponding degree distribution follows as $$\begin{aligned}
   \label{eq:dd}
      p(k) =  \sum_{i=1}^{K} \, \nu_i\, \delta_{k_i,k}\end{aligned}$$ Insertion of the specific rate and degree distribution yields a set of $K$ nonlinear master equations. A qualitative discussion of the possible stationary solutions $P^*(1|k_i)$ which will be approached as $t\to \infty$ will be outlined. Since the equations are nonlinear there might exist a different number of stationary solutions with different stability. If distributed by and with the rate function these stationary states are defined by the set of $K$ coupled nonlinear algebraic equations $$\begin{aligned}
        P^*(1|k_i)\, &= \, \frac{\tau\gamma_i^*}{1+\tau\gamma_i^*},
    \label{eq:SS_gen}\end{aligned}$$ with the stationary spiking rates of the elements with degree $k_i$ $$\begin{aligned}
\label{eq:rate_ss}
\gamma_i^* &= \gamma\left[ \frac{k_i}{N}\,r^* \right]\end{aligned}$$ and the stationary mean-field amplitude with inserted and averaged over the discrete distribution $$\begin{aligned}
  \label{eq:ma_ss}
  r^* &= \frac{1}{\langle k \rangle}\, \left\langle k \,P^*(1|k)
  \right\rangle.\end{aligned}$$

The maximal number of possible stable solutions of this set of equations can be estimated to be of the order $O(2^K)$. The behavior is similar to a spin chain with $K$ elements. Every population is stable in 