In this thesis, we introduce several closely related stochastic models which generalize the deterministic Hodgkin-Huxley formalism to an SDE framework. We provide analytical results on the existence and uniqueness of solutions as well as the exact formulas for the moments of a simplified model, with simplifications motivated by the experiments performed by Hodgkin and Huxley in their seminal paper.For more complicated models, we provide an approach for the approximation and simulation of solutions to the corresponding SDEs, and show several realizations of the sample paths and moments of these simulations to verify qualitative behavior in this case. All code for the project is written in the Julia language and can be obtained upon request by the reader.