# Estimation of Synaptic Activity during Neuronal Oscillations
## 1 Introduction
These traces are approximated by calibrating the membrane potential using dynamic clamp in in vitro experiments of isolated neurons and a quadratic polynomial approximation. More recently, in [26] we introduce a mathematical approach to estimate conductances by characterizing the period of the membrane potential, and in [27] estimations of the conductances of spike trains of the membrane potential are performed. In [26], we present a first proof-of-concept to estimate synaptic conductances in the spiking regime based on information about the \(f-I\) curve of the neuron. The idea was exemplified with the McKean model, a piecewise linear system that allows for a quasi-analytic expression of the \(f-I\) curve. More specifically, we approximated the firing period, \(T\), as a function of the parameters of the system and, in particular, of the synaptic conductance parameter, \(g_{syn}\). Therefore, for a given interspike interval (ISI) of length \(T^{*}\), one can numerically solve the implicit equation \(T(g_{syn})=T^{*}\) to obtain an estimation, \(g_{syn,est}\), of the synaptic conductances during that ISI. In order to obtain the time course of the synaptic conductances, the procedure is repeated for each ISI. By interpolation, a continuous estimation of \(g_{syn,est}(t)\) can be obtained, which is proved to be an accurate estimation of the prescribed conductances provided that they have slower variation (in time) than the membrane potential. Although the results obtained in [26] show that using the \(f-I\) to estimate synaptic conductancesin spiking regimes can be a suitable strategy, there were only applied to the McKean model, a very specific and non-realistic model, which is nothing but a caricature of the Fitzhugh-Nagumo system. An improvement of this estimation procedure is to consider a different underlying mathematical model that could better fit experimental data. On the one hand, desirable candidates would be _conductance-based_ models as per the Hodgkin-Huxley formalism but, even though some of them (pyramidal cells [28], stellate cells [29], etc.) have been investigated in the context of estimation of conductances in subthreshold regimes, it is extremely difficult to devise a procedure for a generic conductance-based model. On the other hand, _integrate-and-fire_ (IF) models have also been considered as good approximations to the neuronal activity of different type of neurons (see [30; 31; 32; 33; 34]) and are simple enough. We found that the _exponential integrate-and-fire_ (EIF) model represents an optimal balance between adaptability to experimental data and computational \(f-l\) curves, and model simplicity. In this paper, following the ideas introduced in [26], we propose a procedure to estimate conductances under a regular spiking regime. This procedure is based on a modification of the standard frequency-input curve, which consists of replacing the input constant current by a constant synaptic conductance (\(f-g_{syn}\) curve). We aim to estimate the global synaptic conductance time course from a membrane potential time series of a spiking neuron, for which we can have an accurate \(f-g_{syn}\) curve. For simplicity, here we assume that all synaptic input is excitatory. The \(f-g_{syn}\) curve can be provided in several ways: (I) from dynamic clamp experiments, using in vitro recording; (II) approximating the real data by a base model (IIa) for which we can obtain an analytic expression for the \(f-g_{syn}\) curve, such as the leaky or the quadratic integrate-and-fire models (see [35; 36]), or (IIb) allowing an approximate expression for the \(f-g_{syn}\) curve, such as the McKean model (see [26]) or the exponential integrate-and-fire (see Section 2.1); finally, (III) using a more detailed model that can be numerically integrated in order to construct a \(f-g_{syn}\) curve table, that is, providing a set of period-conductance pairs extracted from numerical experiments (see Section 2.2).