(fast variable). Indeed, since we measure the membrane potential $y_i(t) = u_i(t)$, the slow variable is given as $
z_i(t) = \mu \sum_{k=1}^t (y_i(k) - \sigma)
$ and for a good choice of parameters $\mu$ and $\sigma$ this can be identified with the slow variable of the model $w$. In the methods section, we obtain an equation for the slow variable and analyze the effect of the network of the dynamics of the slow variable.

We can use the data on the network and the dynamics recovered from the time-series recorded at $\Delta \alpha = 0.3$ to predict that at the value $\Delta \alpha \approx 0.42$ the rich-club will develop a burst synchronization (details in the Methods). To capture such a transition in the bursts of the rich club, we introduce a phase $\theta_j(t)$ for the slow variable, the definition of $\theta_j(t)$ can be found in the Methods section. Once we have the phase we compute the order parameter $$r(t) e^{i \psi(t)} = \frac{1}{N_c} \sum_{j=1}^{N_c} e^{i \theta_j (t)}$$ Loosely speaking, a small value of the order parameter, $r \approx 0$, means that no collective state is present in the system, whereas $r(t)\approx 1$ means that the bursts are synchronized. Figure 