# Estimation of Synaptic Activity during Neuronal Oscillations
## 4 Discussion  
Estimations of the time course of synaptic conductances arriving at a single cell (or a population) reveal relevant information to fully understand the functional connectivity of the brain. Several strategies have been developed for the estimation during subthreshold activity regimes [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 21; 22; 23; 25]. However, the estimation during oscillatory (spiking) regimes has received less attention [26; 27; 25; 40]. The main drawback is the predominance of ionic currents over synaptic ones in this regime. Nevertheless, as we already showed in a caricature model (see [26]), if one  
Figure 8: **Results of the estimation procedure applied to a non-tonic membrane potential from a noisy pyramidal cell model. ** Panel (**a**) shows the prescribed conductances (black curve) and the estimated conductances using the same model (blue trace), the EIF model (green trace), and the experimentally fitted curve (red trace). Panels (**b–d**) show the reconstructed membrane potential when each of the mentioned estimated conductances in panel (**a**) has been, respectively, used as input to the base model (the noisy pyramidal cell model). gets the main information about the contribution of ionic currents and is able to track their activity during the spiking regime, then it is possible to infer the synaptic activity. In this paper, we extend this idea to more realistic spiking models and provide a systematic procedure that can be applied to a variety of neural data, coming both from computational simulations and from experiments. The idea is to exploit the input-response properties of the cell; more precisely, the relation between constant synaptic conductances and the spiking frequency (the \(f-g_{syn}\)_curve_). In the spiking regime, the most natural measurable quantity is the oscillation frequency, which constitutes a valuable piece of information for our purpose. The frequency-input relationship encompasses the average response of the neuron, which includes both the direct effect of the input stimulus and the resulting activation of ionic currents. The fact that \(f-I\) curves are usually monotone implies the one-to-one correspondence between frequency and input, and favors the resolution of the inverse problem. However, while in the case of a steady input (\(I\) constant) that induces a constant-frequency repetitive firing, it is straightforward to infer the input current from the output frequency; the presence of the neuron's voltage in the synaptic current (\(I=g_{syn}(t)\left(v(t)-V_{syn}\right)\) in their minimal expression) makes the estimation of the total input current highly sensitive to the fluctuations in \(g_{syn}\). Here, we propose to consider an alternative frequency-input relationship, the \(f-g_{syn}\) curve, which is more appropriate for the estimation problem since it implicitly incorporates the voltage dependency of the synaptic current and mitigates the effects of the fluctuations. In a mathematical model, we can obtain the \(f-g_{syn}\) or \(T-g_{syn}\) curve, \(T=1/f\), numerically (in some cases, even analytically or quasi-analytically, see [26] and Section 2.1.1). For experimental studies, retrieving the \(T-g_{syn}\) curve requires the use of dynamic clamp techniques on the isolated cell for fixed \(g_{syn}\) values within a physiologically plausible interval. In Algorithm 1, we summarize the steps to perform the estimation of synaptic conductances from the interspike intervals of the voltage traces and the knowledge of the \(T-g_{syn}\) curve. The second ingredient of the method presented in this paper is the choice of a reference/base model, which must be both flexible to fit to a variety of neural data (choosing a suitable parameter set in each case) and simple enough to be able to compute the \(f-g_{syn}\) curve in a precise way.