# Estimation of Synaptic Activity during Neuronal Oscillations
## 1 Introduction  
Estimating non-measurable connectivity parameters from experimental measurements is an important challenge to understand brain function. Brain activity is mainly driven by the intrinsic dynamics of neurons and their mutual connections, which are essentially regulated by synapses. A thorough knowledge of individual membrane potentials and synaptic terms is, therefore, crucial to understand how the brain operates. However, we are far from this utopia: current recording methods can only retrieve partial information, such as the time evolution of membrane potential of one or a few neurons. Unfortunately, other quantities that shape the connectivity, like synaptic conductances, are impossible to be directly recorded in experiments. The time course of synaptic conductances arriving at a single cell not only provides local information about its functional connectivity, but it often constitutes an indicator to validate more general theories such as obtaining information about the architecture of the cortex, discerning between phase insensitive or spatial phase selective coupling [1], unveiling whether the integrative properties of cortical neurons are altered under high-conductance states [2], as well as studying many neuronal problems related to the balance between excitation and inhibition [3]. Unfortunately, there are no direct methods to estimate synaptic conductances, and thus they have to be deduced from experimentally observable quantities. As no direct methods are known to estimate synaptic conductances, they have to be deduced from experimentally observable quantities. This leads to the need of using inverse mathematical methods. As for many inverse problems, it has been proved to be an arduous analytical and computational task for which no general procedures are yet available. Different strategies have been developed to estimate the time course of the synaptic conductances from time series of the membrane potential. Most strategies are based on deterministic linear estimation procedures (see [3; 4; 5; 6; 7]). However, researchers are aware of possible misestimations coming from the measurement noise produced during recordings. Therefore, new methods have been also devised to solve this problem from different approaches: stochastic linear procedures (see [8; 9; 10; 11; 12; 13]), sophisticated filtering techniques (see [14; 15; 16; 17; 18]), and techniques using the \(I-V\) relation extracted experimentally (see [19]). Even though all these strategies provide circumstantial solutions, it is also well known that they can only be applied in the subthreshold regime, where ionic currents are negligible (see [20; 21]), a situation for which simplified models (close to linear) are representative enough to account for the voltage dynamics. However, it has also been demonstrated that these linear procedures can also provide wrong estimations when ionic currents are active in the subthreshold regime causing non-spikes in the membrane potential of the postsynaptic neuron. This fact motivates the introduction of new strategies to estimate conductances under these nonlinear paradigms (see [22; 23]). More recently, in [24], the authors identify that another artifact that can provide wrong estimations is to assume that the synaptic conductances are voltage-independent; to avoid this problem, they provide a new estimation procedure, also in the linear framework. Despite the many efforts in the literature to get rid of artifacts that can contaminate the estimation of conductances (influence of noise, activity of subthreshold ionic currents, variability across trials, etc. ), one of the most resilient troubles are the nonlinearities that the \(f-I\) curve exhibits in the presence of spiking mechanisms. On the other hand, a line of research is devoted to look for possible strategies to estimate conductances in the spiking regime. For instance, in [25], the authors use a firing clamp technique where a description of the spike is given in terms of the membrane potential threshold and the amplitude of the spike.