## 2 Materials and Methods
### Spiking Network Model  
We considered networks of integrate-and-fire neuron models displaying spike-frequency adaptation, based on two previous papers [6, 16]. We used the Adaptive Exponential (AdEx) integrate-and-fire model [17]. We considered a population of \(N=10^{4}\) neurons, randomly connected with a connection probability of \(p=5\%\). We considered excitatory and inhibitory neurons, with 20% inhibitory neurons. The AdEx model allows us to define two cell types, \"regular-spiking\" (RS) excitatory cells, displaying spike-frequency adaptation, and \"fast spiking\" (FS) inhibitory cells, with no adaptation. The dynamics of these neurons is given by the following equations:  
\[c_{m}\frac{dv_{k}}{dt} = g_{L}(E_{L}-v_{k})+\Delta_{k}e^{\frac{v_{k}-v_{thr}}{\Delta_{k}} }-w_{k}+I_{syn}\] \[\tau_{w}\frac{dw_{k}}{dt} = -w_{k}+b\sum_{ts_{p}(k)}\delta(t-t_{sp}(k))\] \[+a(v_{k}-E_{L}),\]  
where \(c_{m}=200\) pF is the membrane capacitance, \(v_{k}\) is the voltage of neuron \(k\), and whenever \(v_{k}>v_{thr}=-50\) mV (\(v_{thr}\) being the voltage threshold for the action potential) at time \(t_{sp}(k)\), \(v_{k}\) is reset to the resting voltage \(v_{rest}=-65\) mV and fixed to that value for a refractory period \(T_{refr}=5\) ms. The leak term \(g_{L}\) has a fixed conductance of \(g_{L}=10\) nS and the leakage reversal \(E_{L}\) is \(-65\) mV. The exponential term \(\Delta_{k}\) has a different strength for RS and FS cells, i.e., \(\Delta_{e}=2\) mV (\(\Delta_{i}=0.5\) mV) for excitatory (inhibitory) cells. Parameters \(a\) and \(b\) describe the adaptation conductance and current increment, respectively. Inhibitory neurons were modeled as fast-spiking FS neurons without adaptation (\(a_{i}=b_{i}=0\) for all inhibitory neurons), whereas excitatory regular-spiking RS neurons had a lower level of excitability due to the presence of adaptation (for the RS cells, parameter \(b_{e}\) varied in our simulations, while we fixed \(a_{e}=4\) nS and the adaptation time constant \(\tau_{w}=500\) ms, unless otherwise specified). The \(\delta\)-function indicates that whenever a neuron fires at time \(t_{sp}(k)\), the adaptation current \(w_{k}\) is incremented by an amount \(b\). The synaptic current \(I_{syn}\) received by neuron \(i\) is the result of the spiking activity of all neurons \(j\in\mathrm{pre}(i)\) pre-synaptic to neuron \(i\). This current can be decomposed in the synaptic conductances evoked by excitatory E and inhibitory I pre-synaptic spikes, as follows:  
\[I_{syn}=G_{syn}^{e}(E_{e}-v_{k})+G_{syn}^{i}(E_{i}-v_{k}),\]  
where \(E_{e}=0\) mV (\(E_{i}=-80\) mV) is the excitatory (inhibitory) reversal potential. Excitatory synaptic conductances \(G_{syn}^{e}\) were modeled by a decaying exponential function that sharply increases by a fixed amount \(Q_{e}\) at each pre-synaptic spike, that is:  
\[G_{syn}^{e}(t)=Q_{e}\sum_{exc.pre}\Theta(t-t_{sp}^{e}(k))\;e^{-(t-t_{sp}^{e}( k))/\tau_{e}},\]  
where \(\Theta\) is the Heaviside function, \(\tau_{e}=\tau_{i}=5\) ms is the characteristic decay time of excitatory and inhibitory synaptic conductances, and \(Q_{e}=1\) nS (\(Q_{i}=5\) nS) is the excitatory (inhibitory) quantal conductance. Inhibitory synaptic conductances are modeled using the same equation with \(e\to i\). This network displays two different states according to the level of adaptation \(b_{e}\) (hereafter indicated as \(b\)), with \(b=5\) pA for asynchronous irregular states, and \(b=60\) pA for up-down states (Figure 2A; see [6] for details and Table 1 for a synopsis of the model parameters).