Developing cures for highly complex diseases, such as neurodegenerative disorders, requires extensive interdisciplinary collaboration and exchange of biomedical information in context. Our ability to exchange such information across sub-specialties today is limited by the current scientific knowledge ecosystem's inability to properly contextualize and integrate data and discourse in machine-interpretable form. This inherently limits the productivity of research and the progress toward cures for devastating diseases such as Alzheimer's and Parkinson's. SWAN (Semantic Web Applications in Neuromedicine) is an interdisciplinary project to develop a practical, common, semantically structured, framework for biomedical discourse initially applied, but not limited, to significant problems in Alzheimer Disease (AD) research. The SWAN ontology has been developed in the context of building a series of applications for biomedical researchers, as well as in extensive discussions and collaborations with the larger bio-ontologies community. In this paper, we present and discuss the SWAN ontology of biomedical discourse. We ground its development theoretically, present its design approach, explain its main classes and their application, and show its relationship to other ongoing activities in biomedicine and bio-ontologies.