# Estimation of Synaptic Activity during Neuronal Oscillations
## 2 Methods
### Models and Data Treatment  
#### 2.1.1 Neuronal Models  
Along the paper, we consider different neuronal models both for the purpose of generating membrane potential traces and to use them as base models for the estimations of synaptic conductances. We keep the choice of models minimal; as we mention in other parts of the paper, we can adapt our procedure to many other models. Pyramidal cell model  
We first consider a simplified version of a model for a pyramidal neuron (see [28]), where only the soma compartment has been considered. It is a spiking model driven by the interaction of sodium (\(I_{Na}\)) and potassium (\(I_{K}\)) currents. The membrane potential \(V\) follows the differential equation  
\[C_{m}\frac{dV}{dt}=-I_{L}-I_{Na}-I_{K}+I_{app}-I_{syn}, \tag{1}\]  
where \(C_{m}\) is the capacitance, \(I_{syn}\) is the synaptic current, \(I_{app}\) is the applied current, \(I_{L}\) is the leak current, and \(I_{Na}\) and \(I_{K}\) are the ionic currents, which are described by equations  
\[ \begin{array}{l}I_{L}=g_{L}(V-V_{L}),\\ I_{Na}=g_{Na}m_{\infty}^{3}(V)h(V-V_{Na}),\\ I_{K}=g_{K}n^{4}(V-V_{K}),\end{array} \]  
where \(V_{x}\) and \(g_{x}\), with \(x\in\{L,Na,K\}\), represent the specific reversal potentials and maximal conductances, respectively. The variables \(h\) and \(n\) are gating variables governed by first-order kinetics of type  
\[\dot{w}=\frac{dw}{dt}=\phi[\alpha_{w}(V)(1-w)-\beta_{w}(V)w]=\phi\frac{w_{ \infty}(V)-w}{\tau_{w}(V)}, \tag{2}\]  
where \(w\) represents either \(h\) or \(n\); \(w_{\infty}(V):=\alpha_{w}(V)/(\alpha_{w}(V)+\beta_{w}(V))\) is the \(w\) steady state and \(\tau_{w}(V)=(\alpha_{w}(V)+\beta_{w}(V))^{-1}\) is the \(w\) time decay. In our version of this model, the \(m\)-type (fast) variable is considered to be at the steady state \(m=m_{\infty}(V)\). The oscillatory regime for this model is approximately \(I_{app}\in(0.225653,108.285)\)\(\mu\)A/cm\({}^{2}\). By default, we take \(I_{app}=0\)\(\mu\)A/cm\({}^{2}\) (see Appendix B in [20] for a justification of this choice). For more details about the biophysical parameters of the model, see Appendix A. We will also explore how the random synaptic input or measurement noise affects the estimation of conductances, and so we will add a noise term to the membrane potential Equation (1). For the sake of reference, we present the resulting system as a separate model. Noisy pyramidal cell model  
Considering a white noise process \(\eta(t)\) having zero mean and standard deviation \(\sigma_{V}\), system (1) writes as  
\[C_{m}\frac{dV}{dt}=-I_{L}(t)-I_{Na}(t)-I_{K}(t)+I_{app}-I_{syn}(t)+\eta(t). \tag{3}\]Stellate cell model  
The stellate cell model is taken from [29]. The membrane potential dynamics is given by  
\[C\frac{dV}{dt}=-I_{L}-I_{Na}-I_{K}-I_{NaP}-I_{h}+I_{app}-I_{syn}, \tag{4}\]  
where \(C\) is the membrane capacitance, \(I_{syn}\) the synaptic current, and \(I_{app}\) the applied current. The leakage and ion currents are given by  
\[ \begin{array}{l}I_{L}=g_{L}(V-V_{L}),\\ I_{Na}=g_{Na}m^{3}(V)h(V-V_{Na}),\\ I_{K}=g_{K}n^{4}(V-V_{K}),\\ I_{NaP}=g_{P}p(V-V_{Na}),\\ I_{h}=g_{h}(0.65r_{f}+0.35r_{s})(V-V_{h}),\end{array} \]  
where \(V_{x}\) and \(g_{x}\), for \(x\in\{L,Na,K,NaP,h\}\), denote the corresponding ion reversal potentials and maximal conductances, respectively. The gating variables \(w\), which can be either \(m\), \(h\), \(n\), \(p\), \(r_{f}\), or \(r_{s}\), follow the same first-order dynamics (2) as for the pyramidal cell model. In absence of synaptic inputs, this model presents spikes for an applied current greater that \(I_{T}=-1.46\,\upmu\)A/cm\({}^{2}\). In our case, to have a high frequency of spikes, we inject a constant current equal to \(I_{app}=8\,\upmu\)A/cm\({}^{2}\). For more details about the biophysical parameters of the model, see Appendix A.  
Exponential Integrate-and-Fire (EIF) model with constant adaptation  
We chose the EIF model as representative of _base models_ due to its well-known properties of being able to shape its parameters in order to fit to diverse \(f-I\), see [30].