# Estimation of Synaptic Activity during Neuronal Oscillations
## 3 Results
### Estimation in the Spiking Regime
#### 3.1.1 Estimation When the Neural Model Is Known  
In this case, we apply the estimation procedure given in Algorithm 1 considering the pyramidal neural model as a base model in step 3b of the algorithm. Results show that the estimated synaptic conductance trace follows the actual one before and after interpolating the estimated values (see Figure 3a). After a short transient discrepancy observed in the first 20 ms, due to the the fact that we cannot have an a priori guess of the actual initial conditions, we observe a strong agreement between the estimation and the actual values, as we can see in Figure 3c where the actual conductance trace versus the estimated one (after being interpolated) lies in the vicinity of the identity line, meaning a good approximation. In fact, computing the errors between the two traces, we have that the mean squared error is of order \(O(10^{-9})\) and the relative error is, in mean, of order \(O(10^{-3})\), indicating that the Algorithm 1 accurately estimates the synaptic conductance time course (see Table 2). Note that, in this case, as we are using the same model to generate the data and to estimate the conductances, the only source of errors could arise from the process of inversion, which is also related to the discretization provided by the spiking frequency of the neuron. Nevertheless, the magnitude of the errors obtained corroborates that they do not penalize the estimation (see Figure 3b and the comment about the effects caused by the discretization in the next paragraphs). Moreover, the mean of the conductances after interpolating the data is also well estimated (relative error of \(5.547\times 10^{-4}\)). Therefore, when we reconstruct the membrane potential by using the pyramidal cell model with the estimated synaptic conductances, neither the spike frequency, the spike amplitude nor the spike onset time are modified (see Figure 3b), thus reinforcing the fact that the small errors caused in the estimation are not substantially important. Two different sources of errors can arise during the estimation procedure. First, errors coming from the discretization of the conductances used to build up the \(T-g_{syn}\) table that relates spiking periods with constant synaptic conductance values and, second, the measurement noise, which is embedded in the membrane potential obtained from the recordings. Next, we discuss how these errors are reflected in the estimation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean of** & **MSE** & **MSE** & **Estimation of the** \\
**Base** & **the Relative** & **Estimated** & **Interpolated** & **Interpolated Mean** \\
**Model** & **Error** & **Values** & **Values** & **(Rel. Error)** \\ \hline Pyramidal & \(7.820\times 10^{-2}\) & \(1.491\times 10^{-5}\) & \(3.776\times 10^{-5}\) & \(8.105\times 10^{-2}\) \\ EIF & \(8.768\times 10^{-2}\) & \(2.303\times 10^{-5}\) & \(3.576\times 10^{-5}\) & \(7.331\times 10^{-2}\) \\ Exper. curve & \(3.001\times 10^{-1}\) & \(1.334\times 10^{-4}\) & \(7.171\times 10^{-5}\) & \(1.215\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: **Errors caused by the computation of the \(T-g_{syn}\) curve. ** Data have been generated using the pyramidal cell model with noise (mean 0 and \(\sigma_{V}=0.1\)). The synaptic drive is governed by a synaptic conductance trace extracted from an in silico network (see end of Section 2.1.2) where we have added noise with mean 0 and \(\sigma=0.0001\) (see Figure 4a, black curve). Effects caused by the discretization  
We want to analyze the influence on the estimations of the discretization of synaptic conductances in step 3b of Algorithm 1, that is, the effect of parameter \(\delta_{g}\) used to build up the \(T-g_{syn}\) table. For this reason, and to avoid errors coming from additive noise, the actual membrane potential is also generated without considering noise. Then, we explore the errors (mean square error (MSE)) for different discretization steps (see Table 2).