# Estimation of Synaptic Activity during Neuronal Oscillations
## 2 Methods
### Models and Data Treatment
#### 2.1.2 Synaptic Drive  
In our simulations we assume that the target neuron is stimulated through two different currents: (1) an external current, \(I_{app}\), which is a constant value that places the cell in the excitable regime, ready to induce spiking activity within a biologically plausible frequency range, and (2) the synaptic input \(I_{syn}\), which comes from other neurons of the network and eventually leads to the variation of the spike frequency. In the different neural models, the synaptic input is always supposed to be of the form \(I_{syn}=g_{syn}(V-V_{syn})\), where \(g_{syn}=g_{syn}(t)\) is the synaptic conductance trace and \(V_{syn}\) is the synaptic reversal potential; assumed to be excitatory in our simulations. To generate the membrane potential of the neuron, we consider specific prescribed conductances. In our in silico experiments, we inject the corresponding synaptic currents, we extract the voltage response, and, pretending not to know them, we finally apply our methods to estimate the synaptic conductances. For this purpose, we use two different types of conductance traces: a _synthetic_ three-frequency oscillatory drive and a more realistic trace obtained from a computational network. Note that, due to the nature of the problem we tackle here, there are no _actual_ conductance traces available unless dynamic clamp experiments are performed. We define the three-frequency oscillatory drive as  
\[g_{syn}=A_{1}\cos\left(\frac{2\pi}{T_{1}}t\right)+A_{2}\cos\left(\frac{2\pi}{ T_{2}}t\right)+A_{3}\cos\left(\frac{2\pi}{T_{3}}t\right)+A_{4}, \tag{8}\]  
Figure 2: **Approximation of the pyramidal cell model by the EIF model**. Panel (**a**) shows the membrane potential trace obtained from the pyramidal cell model (black curve) given in Equation (1) when no synaptic input is considered, and the approximation of the membrane potential trace obtained by the EIF model (red curve) described in Equation (5). Panel (**b**) shows their respective \(T-g_{syn}\) curve, where \(T\) is the period of oscillation (the black curve corresponds to the pyramidal case and the red curve to the EIF model). being \(A_{1}=0.0022\), \(A_{2}=0.002\), \(A_{3}=0.001\), \(A_{4}=0.025\), \(T_{1}=150\), \(T_{2}=320\), and \(T_{3}=50\). These values have been chosen to ensure a plausible spiking frequency for such kind of neurons. The resulting conductance trace is shown in Figure 3a (black curve); observe that it elicits a membrane potential completely immersed in the spiking regime (without silent periods interspersed), see Figure 3c. On the other hand, to explore the synaptic regime interspersed by short silent periods, we chose a more realistic synaptic conductance input, consisting of a trace with a 1 ms resolution obtained from a computational network that models the layer 4C\(\alpha\) of the primary visual cortex (see [1,38]). A small amount of noise has been added to these synaptic conductances of 0 mean and 0.0001 deviation. The conductances trace is shown in Figure 4a (black curve). Figure 4: **Results of the estimation procedure for a membrane potential from a stellate cell model using the same model as a base model in the estimation procedure. ** Panel (**a**) shows the prescribed conductances (black curve) and the estimated conductances (red curve). Panel (**b**) shows the membrane potential (black curve), obtained from the stellate cell model described in (4) when the prescribed conductances of Panel (**a**) has been used, while the red curve depicts for the reconstructed membrane potential, obtained also from the stellate cell model but using the estimated conductances. Panel (**c**) shows the scatter plot of the prescribed conductances versus the estimated ones. The red line represents the identity line. Figure 3: **Results of the estimation procedure applied to a membrane potential generated through the pyramidal cell model with no additive noise, and using this same model as a base model in the estimation procedure.