Bridging the gap between biological data and mathematical models, with Topological Data Analysis

Many biological processes may be viewed as complex, spatio-temporal systems. This viewpoint is supported by data, which often has spatial resolution (e.g. digital photographs, medical images), as well as temporal evolution (e.g. dynamical datasets, or successive static datasets). Mathematical models help us to study such processes. Data may be simulated from a model, and then compared to observed data. The model can then be modified as a result of this comparison, thus refining our understanding of the underlying biological process.
Advancing computational power means that both simulated and observed data is of higher quality and greater availability. As both kinds of data become more complex, new mathematical techniques are required to analyse and summarise what a dataset may be telling us.
Topological Data Analysis (TDA) is a relatively new and expanding field of mathematics [1], which is concerned with the spatial properties of data. Spatial features of a dataset, such as connected components, circular loops and three-dimensional voids, may be extracted using Persistent Homology. It is argued that such analysis gives more information about the dataset than other, more standard techniques.
Recently, Topological Data Analysis has been used to study angiogenesis [2] and animal group locomotion [3].

Aims and objectives
My work will bridge the current gap between biological data and mathematical techniques. I will study ecological and physiological systems, and identify gaps in our understanding of such phenomena. I will develop new topological techniques to study these processes and demonstrate how they lead to a better understanding of specific areas.
I will build on previous and ongoing work in the field, in addition to exploring my own interests. New models of Coral Reef growth are being developed from classical models in the literature [4]. I will investigate how TDA may be able to advance understanding of such models. Other possible areas for applications of my work include the formation of capillaries [5], and further work into angiogenesis models [2].


Alignment to EPSRC's strategies and research areas
This project is aligned with the 'Mathematical Biology' research area.

Novelty of the research methodology
One of the primary techniques within Topological Data Analysis is Persistent Homology. Much work has already been done on the mathematics behind this technique, in addition to its applications. One drawback to Persistent Homology is that it cannot immediately deal with dynamic, oscillating, or non-monotone data. Such data appears commonly in the biological systems mentioned above, and so an active area of research is extending Persistent Homology to work on such data.
My research it will focus on such new techniques as well as their applications to biological models.

Companies or collaborators involved
No collaborators outside of the University of Oxford Mathematical Institute have been identified.