Statistical inference for multivariate partially observed stochastic processes with application to neuroscience

In many signal-processing applications, it is of primary interest to decode/reconstruct the unobserved signal based on some partial observed information. Some examples are automatic speech, face, gesture and handwriting recognition and neuroscience (ion channels modeling). From a mathematical point of view, this corresponds to estimate model parameters of an unknown coordinate based on discrete observations of one or more other coordinates. In this project we consider a partially observed bivariate stochastic process and discuss it in the framework of stochastic modelling of single neuron dynamics. None of the two components is directly observed: The available observations corresponds to hitting times of the first components to the second component and/or measurements at discrete time of the first coordinate. Our aim is to provide statistical inference of the underlying model parameters, as well as developing suitable numerical algorithms. This is particularly difficult since the considered process does not fit into the well-known class of hidden Markov models, requiring the investigation of new ad-hoc mathematical and statistical techniques to handle it. Our background on mathematical modeling, stochastic numerics, statistical inference for (partially observed) stochastic processes, hidden Markov models, and mathematical neuroscience, and our expertise in combining theory and simulations represent a perfect match for the project.