activity of the individual nodes has ceased. It is rather the mean activity or flow that is constant (cf. Fig. \[fig:xAS\]). Oscillating behavior of $r^*$ would correspond to synchronization among the units. But without further ingredients like delayed feedback or additional inhibitory coupled nodes such states cannot be reached.

The transition to the disordered state can be studied in more detail. Demanding that the first derivatives of l.h.s. and r.h.s. with respect to $r$ coincide at $r=r^*$ , provides a condition for a saddle-node bifurcation. The homogeneous disordered state becomes unstable and two new stable solutions occur.

Execution of the derivatives in results in $$\begin{aligned}
    \frac{k_1
        \nu}{\langle k\rangle}\frac{\tau\;\frac{\partial{\gamma_{k_1}}^*}{\partial 
            r^*}}{(1+{\gamma_{k_1}}^*\tau)^2} + \frac{k_2
        (1-\nu)}{\langle k\rangle}\frac{\tau\;\frac{\partial{\gamma_{k_2}}^*}{\partial 
            r^*}}{(1+{\gamma_{k_2}}^*\tau)^2} &= 1.
    \label{eq:SNEllipseinR_rbnet} \end{aligned}$$ Using this becomes $$\begin{aligned}
    \nonumber 1=\frac{k_1 \nu}{\langle k\rangle}({P^*(1|k_1)}-({P^*(1|k_1)})^2)\frac{\partial \log({\gamma_{k_1}}^*)}{\partial r^*} &+\\
    \frac{k_2 (1-\nu)}{\langle k\rangle}({P^*(1|k_2)}-({P^*(1|k_2)})^2)\frac{\partial \log({\gamma_{k_2}}^*)}{\partial r^*} &.\\\end{aligned}$$ Changing the variables to $x_{k_1}:={P^*(1|k_1)}-\frac{1}{2}$ and $x_{k_2}:={P^*(1|k_2)}-\frac{1}{2}$ and rearranging the equation gives $$\begin{aligned}
    \frac{x_{k_1}^2}{a_1^2} +\frac{x_{k_2}^2}{a_2^2} &= 1.
    \label{eq:SNEllipse_rbnet}\end{aligned}$$ Equation defines an ellipse with semi-axes $$a_1 = \frac{1}{2}\sqrt{\frac{ \big\langle k\frac{\partial
                    \log(\gamma^*_{k})}{\partial
                    r^*}\big\rangle-4\langle k\rangle}{k_1 \nu \frac{\partial
                    \log(\gamma_{k_1}^*)}{\partial r^*}}}$$ and similarly $a_2$ with the substitutions$k_1
\rightarrow k_2$ and $\nu \rightarrow (1-\nu)$. The ellipse reduces to a point where the two bifurcations merge. It corresponds to ${P^*(1|k_1)}~=~{P^*(1|k_2)}~=~r^*~=~\frac{1}{2}$, at $$\left\langle 