# Estimation of Synaptic Activity during Neuronal Oscillations
## 3 Results
### Effect of input noise
#### 3.1.4 Estimation without a Reference Base Model  
To fully explore all possibilities of how to extract a relation between the oscillatory period and a constant value of the conductance, apart from considering an approximate model from which we can build up a table or find an explicit expression, we also consider the possibility of having the \(T-g_{sym}\) relationship experimentally. This relationship could be directly obtained by using the dynamic clamp technique or indirectly through the \(I-V\) curve by using other techniques (such as current clamp). Indeed, if the current \(I\) only contains synaptic currents, then the \(T-g_{sym}\) curve can be obtained by computing \(g_{sym}=\frac{I}{V-V_{sym}}\) and extracting the period \(T\) from the membrane potential trace. Note that, if the \(T-g_{sym}\) curve can be obtained with high accuracy (with a sufficiently fine discretization of points), then the estimation will present similar errors than those presented in Section 3.1 (see Table 2). However, if the discretization is poor, errors could be magnified. In this section, we investigate the case where only a small sample of \((g_{sym},T(g_{sym}))\) pairs are obtained. To computationally simulate this specific situation, we assume that the data set is generated from the noisy pyramidal cell model. By using this same model, we generate a small sample of ten \((g_{sym},T(g_{sym}))\) pairs to construct the \(T-g_{sym}\) curve. Moreover, to consider possible errors in the recordings (measurement noise and variability across trials), we have added noise to the resulting set of points. This bias has been included by adding a random normal distributed value with 0 mean and 0.5 standard deviation to the period component. Finally, to estimate the conductance for any given period, and as we are supposing that no extra information is known about the actual data, we have interpolated these data using the PCHIP method (see Figure 6a). Applying now the estimation procedure to the membrane potential shown in Figure 3b (black trace), using the simulated experimental curve instead of an approximate base model to generate the \(T-g_{sym}\) curve, the results show how the estimated conductances are clearly underestimated (see Figure 6b,c) and, even though the frequency of oscillation of conductances is maintained, the amplitude of the oscillation is reduced. This fact is mainly due to misestimations that arise in the \(T-g_{sym}\) experimentally fitted curve (with respect to that obtained with the actual model), specially the underestimation obtained for those values of conductances between 0.02 and 0.04, which is in fact the interval where the actual conductances lie. However, in view of the magnitude of the conductances, these underestimations are not substantially significant, providing similar errors to the cases where the \(T-g_{syn}\) curve is numerically computed from a base model (see fourth row in Table 4). Note that the errors in the \(T-g_{syn}\) curve can alter consistently the overall estimation causing an important increase of the errors in the estimation. One source of these misestimations is the interpolation method. We also tested an exponential fitting, instead of the PCHIP interpolation, which gave us higher misestimations of the \(T-g_{syn}\) curve in the working interval, see Figure 7. Results show, as previously, that the frequency of oscillation is maintained but the amplitude is worst underestimated providing an average relative error of approximately 15%. Therefore, the errors are substantially magnified showing the importance of having a good approximation of the \(T-g_{syn}\) curve to be able to properly estimate the conductances. Figure 6: **Results of the estimation procedure applied to a membrane potential generated through the pyramidal cell model with additive noise, and using a simulated \(T-g_{syn}\) curve in the estimation procedure.