Neural interfaces that electrically stimulate the peripheral nervous system have been shown to successfully improve symptom management for several conditions, such as epilepsy and depression. A crucial part for closing the loop and improving the efficacy of implantable neuromodulation devices is the efficient extraction of meaningful information from nerve recordings, which can have a low Signal-to-Noise ratio (SNR) and non-stationary noise. In recent years, machine learning (ML) models have shown outstanding performance in regression and classification problems, but it is often unclear how to translate and assess these for novel tasks in biomedical engineering. This paper aims to adapt existing ML algorithms to carry out unsupervised denoising of neural recordings instead. This is achieved by applying bandpass filtering and two novel ML algorithms to in-vivo spontaneous, low-SNR vagus nerve recordings. The performance of each approach is compared using the task of extracting respiratory afferent activity and validated using cross-correlation, MSE, and accuracy in terms of extracting the true respiratory rate. A variational autoencoder (VAE) model in particular produces results that show better correlation with respiratory activity compared to bandpass filtering, highlighting that these models have the potential to preserve relevant features in complex neural recordings.