# Estimation of Synaptic Activity during Neuronal Oscillations
## 1 Introduction
Once the \(f-g_{syn}\) relationship is obtained, we have an expression of the period \(T=1/f\) depending on the conductances (or vice versa). In cases (I) and (III), where we have a discretization of this relationship, in order to obtain a continuous function \(T(g_{syn})\), one can interpolate the data using, for instance, the piecewise cubic Hermite interpolating polynomial (PCHIP). Therefore, given an interspike interval \(T^{*}\), the corresponding conductance within this interval can be obtained by solving analytically or numerically the equation \(T(g_{syn})=T^{*}\). Notice that for case (I), which involves dynamic clamp, the method can be considered to extract the \(f-g_{syn}\) curve in in vitro experiments but not in in vivo as several injected currents would be required in order to have a precise \(f-g_{syn}\) curve, which could add both variability across trials and noise measurement. Alternatively, we provide different computational strategies (II-III) to obtain the \(f-g_{syn}\) curve. Even though strategies (IIa) and (IIb) provide good approximations to the \(f-g_{syn}\) curve, they should not be applied in general to approximate the \(f-g_{syn}\) curve for a wider range of neurons. For this reason, the computational approach presented in Section 2.2 focuses on the idea of case (III). The paper is organized as follows. In Section 2.1, we present the neuronal model we use to generate data by means of prescribed conductances and its approximation to the EIF model. In Section 2.2, we present the estimation procedure giving a pseudocode of its implementation sketched in Figure 1. Next, in Section 3, we show the results of the estimation procedures. Finally, in Sections 4 and 5 we discuss the results obtained and present our conclusions. 1. **Data collection. ** Let \(\mathcal{V}_{0}:=\{V(t),t\in J\}\) be the membrane potential of a neuron \(\mathcal{N}\) on some time interval \(J\) in absence of synaptic inputs. 2. **Fitting data to a base model. ** Fit \(\mathcal{V}_{0}\) to a _base model_ (for instance, an integrate-and-fire model). Call \(\mathcal{P}\) the set of parameters of the base model that gives the best fitting. 3. **The \(f-g_{syn}\) curve for the base model. ** Build up the function \(G_{syn}(T)\) for the base model with the set \(\mathcal{P}\). Depending on the base model, one possibility is to find either \(G_{syn}\) analytically, or its inverse \(T(g_{syn})\) and then invert it. Otherwise, we will proceed computationally: 1. Consider a discretization of \(N\) equally-spaced synaptic conductances, that is, \(\{g_{syn,i}\}_{i=1}^{N}\) where \(g_{syn,i+1}-g_{syn,i}=\delta_{g}\), for all \(i=1\,\dots\,N-1\), where \(\delta_{g}>0\) is a constant, assuming that the base model is regularly spiking for \(g_{syn}\in[g_{syn,0},g_{syn,N}]\). 2. By numerical integration, for each \(g_{syn,i}\) with \(i=1\,\dots\,N\), obtain the period \(\hat{T}_{i}\) of the corresponding membrane potential. Note that this correspondence could also be obtained using one of the three other options presented in the Introduction (page 3). In all cases, a table containing the relation between \(\hat{T}\) and \(g_{syn}\) is obtained. Then, interpolate it to obtain a continuous function \(G_{syn}(T)\) using, for instance, a piecewise cubic Hermite interpolating polynomial (PCHIP). 4. **Estimation: computing ISIs. ** Let \(\mathcal{V}=\{V(t),t\in J^{\prime}\}\) be the membrane potential trace, for some time interval \(J^{\prime}\), of neuron \(\mathcal{N}\) subject to synaptic input bombardments. Assume that \(\mathcal{N}\) emits \(n_{spk}\) spikes along the time span \(J^{\prime}\) and let \(T_{j}\), for \(j=1,\dots,n_{spk}\), be the \(j\)-_th_ interspike interval of \(\mathcal{V}\). 5. **Estimation: inference. ** For each \(T_{j},j=1\dots n_{spk}\), compute \(G_{syn}(T_{j})\) to obtain the corresponding estimated conductance \(\delta_{syn,j}\). 6. **Estimation: interpolation. ** Interpolate the set \(\{(t_{j},\delta_{syn,j})\}_{j=1}^{n_{spk}}\), where \(t_{j}\) is a point on the \(j-th\) ISI.