## 2 Materials and Methods
### Mean-Field Models  
We considered a population model of a network of AdEx neurons, using a master equation formalism originally developed for balanced networks of integrate-and-fire neurons [18]. This model was adapted to AdEx networks of RS and FS neurons [6], and later modified to include adaptation [8]. The latter version is used here, which corresponds to the following equations:  
\[T\frac{\partial v_{\mu}}{\partial t} = (F_{\mu}-v_{\mu})+\frac{1}{2}c_{\lambda\eta}\frac{\partial^{2}F_{ \mu}}{\partial v_{\lambda}\partial v_{\eta}}\] \[T\frac{\partial c_{\lambda\eta}}{\partial t} = \delta_{\lambda\eta}\frac{F_{\lambda}(1/T-F_{\eta})}{N_{\lambda}} +(F_{\lambda}-v_{\lambda})(F_{\eta}-v_{\eta}) \tag{2}\] \[+\frac{\partial F_{\lambda}}{\partial v_{\mu}}c_{\eta\mu}+\frac{ \partial F_{\eta}}{\partial v_{\mu}}c_{\lambda\mu}-2c_{\lambda\eta}\] \[\tau_{w}\frac{\partial W}{\partial t} = -W+b\tau_{w}v_{e}+a(\mu_{V}(v_{e},v_{i},W)-E_{L}),\]  
where \(T\) is the time constant, \(\mu=\{e,i\}\) is the population index (excitatory or inhibitory), \(v_{\mu}\) is the population firing rate, and \(c_{\lambda\eta}\) is the covariance between populations \(\lambda\) and \(\eta\). \(W\) is a population adaptation variable [8]. \(\delta_{\lambda\eta}\) = 1 if \(\lambda=\eta\) and zero otherwise. The function \(F_{\mu=\{e,i\}}=F_{\mu=\{e,i\}}(v_{e},v_{i},W)\) is the transfer function, which describes the firing rate of population \(\mu\) as a function of excitatory and inhibitory inputs (with rates \(v_{e}\) and \(v_{i}\)) and adaptation level \(W\). These functions were estimated previously for RS and FS cells and in the presence of adaptation [8]. At the first order, i.e., neglecting the dynamics of the covariance terms \(c_{\lambda\eta}\), this model can be written simply as follows:  
\[T\frac{dv_{\mu}}{dt} = (F_{\mu}-v_{\mu}), \tag{3}\]  
together with Equation (3). This system is equivalent to the well-known Wilson-Cowan model [19], with the specificity that the functions \(F\) need to be obtained according to  
Figure 2: Asynchronous irregular and slow oscillatory states in AdEx networks and mean-field models (**A**). Raster plots of excitatory RS (blue) and inhibitory FS (red) AdEx neurons in a network during asynchronous irregular (AI) state (**top**), and slow oscillations with up–down states (**bottom**). The two states differed from the values of adaptation parameter \(b\) (\(b\) = 1 pA for AI state, and 20 pA for slow oscillations shown here) (**B**). Corresponding mean firing rates of the two populations (**C**). Mean-field model of AdEx networks with adaptation (scheme in inset). The mean-field model reproduces the two states and the corresponding firing rate variations (adaptation current \(W\) shown in yellow). Adapted from [8,13]. the specific single-neuron model under consideration. These functions were obtained previously for AdEx models of RS and FS cells [6; 8] and the same are used here. For a cortical volume modeled as two populations of excitatory and inhibitory neurons, the equations (at first order) can be written as follows:  
\[T\frac{dv_{e}}{dt} = \mathcal{F}_{e}(v_{e}+v_{aff}+v_{drive},v_{i})-v_{e}\] \[T\frac{dv_{i}}{dt} = \mathcal{F}_{i}(v_{e}+v_{aff},v_{i})-v_{i} \tag{4}\] \[\tau_{w}\frac{dW}{dt} = -W+b\tau_{w}v_{e}+a(\mu_{V}(v_{e},v_{i},W)-E_{L}),\]  
where \(v_{aff}\) is the afferent thalamic input to the population of excitatory and inhibitory neurons and \(v_{drive}\) is an external noisy drive. The function \(\mu_{V}\) is the average membrane potential of the population and is given by the following equation:  
\[\mu_{V}=\frac{\mu_{Ge}E_{e}+\mu_{Gi}E_{i}+g_{L}E_{L}-W}{\mu_{Ge}+\mu_{Gi}+g_{L }},\]  
where the mean excitatory conductance is \(\mu_{Ge}=v_{e}K_{e}\tau_{e}Q_{e}\). Similar to inhibition \(K_{e}=(1-g_{ei})Np\), \(g_{ei}\) is the fraction of inhibitory neurons in the network, \(N\) is the total size of the network, and \(p\) is the probability of connection.