  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.
