Early diagnosis of neurodegenerative diseases, such as Alzheimer's disease, improves treatment and care outcomes for patients. Early signs of cognitive decline can be detected using functional scales, which are written records completed by a clinician or carer, detailing a patient's capability to perform routine activities of daily living. For example, tasks requiring planning, such as meal preparation, are some of the earliest affected by early mild cognitive impairment. In this article, we describe work towards the development of a system to automatically discriminate and objectively quantify activities of daily living. We train a selection of neural networks to discriminate a novel list of 14 activities, specially selected to overlap with those measured by existing functional scales. Our dataset consists of eight hours of development data captured from four individuals wearing the Continuous Ambulatory Vestibular Assessment (CAVA) device, which was originally developed to aid the diagnosis of vertigo. Using frequency domain recognition features derived from eye-movement and accelerometer data, we compare several classification approaches, including three bespoke neural networks, and two established network architectures commonly applied to time-series classification problems. In 10-fold cross-validation experiments, a peak mean accuracy of 64.1% is obtained. The highest accuracy across all folds is 75.3%, produced by networks comprising Gated Recurrent Units. The addition of eye-movement data is shown to improve discrimination compared to using accelerometer data alone, by close to 9%. Classification accuracy is shown to degrade if the system is trained such that test subjects are excluded from the training data, with the small size of the dataset given as a likely explanation. Our findings demonstrate that the addition of eye-movement data can significantly improve the discrimination of daily activities, and that neural networks are well suited to this task.