Large Deviation Theory for Parameter Estimation in Simple Neuron Models

To investigate the complex dynamics of a biological neuron that is subject to small random perturbations we can use stochastic neuron models. While many techniques have already been developed to study properties of such models, especially the analysis of the (expected) first-passage time or (E)FPT remains difficult. In this thesis I apply the large deviation theory (LDT), which is already well-established in physics and finance, to the problem of determining the EFPT of the mean-reverting Ornstein-Uhlenbeck (OU) process. The OU process instantiates the Stochastic Leaky Integrate and Fire model and thus serves as an example of a biologically inspired mathematical neuron model. I derive several classical results using much simpler mathematics than the original publications from neuroscience and I provide a few conceivable interpretations and perspectives on these derivations. Using these results I explore some possible applications for parameter estimation and I provide an additional mathematical justification for using a Poisson process as a small-noise approximation of the full model. Finally I perform several simulations to verify these results and to reveal systematic biases of this estimator.