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

许多生物过程可以被视为复杂的时空系统。这种观点得到了数据的支持，这些数据通常具有空间分辨率（例如数字照片，医学图像）以及时间演变（例如动态数据集或连续静态数据集）。数学模型帮助我们研究这些过程。可以从模型模拟数据，然后将其与观察到的数据进行比较。然后可以根据比较结果修改模型，从而完善我们对潜在生物过程的理解。提高计算能力意味着模拟和观测数据的质量更高，可用性更强。随着这两种数据变得越来越复杂，需要新的数学技术来分析和总结数据集可能告诉我们的信息。拓扑数据分析（TDA）是一个相对较新且不断扩展的数学领域[1]，它关注数据的空间属性。数据集的空间特征，如连通分量、圆环和三维空隙，可以使用持久同源性来提取。有人认为，这种分析提供了更多的信息，数据集比其他更标准的技术。最近，拓扑数据分析已被用于研究血管生成[2]和动物群体运动[3]。目的和目标我的工作将弥合生物数据和数学技术之间的差距。我将研究生态和生理系统，并找出我们对这些现象的理解差距。我将开发新的拓扑技术来研究这些过程，并展示它们如何导致更好地理解特定领域。除了探索我自己的兴趣之外，我还将在该领域以前和正在进行的工作基础上建立。珊瑚礁生长的新模型正在从文献中的经典模型发展[4]。我将研究TDA如何能够促进对这些模型的理解。我的工作的其他可能的应用领域包括毛细血管的形成[5]，并进一步研究血管生成模型[2]。与EPSRC的策略和研究领域保持一致这个项目与“数学生物学”研究领域保持一致。研究方法的新奇拓扑数据分析中的主要技术之一是持久同源性。除了它的应用之外，在这项技术背后的数学方面已经做了很多工作。Persistent Homology的一个缺点是它不能立即处理动态，振荡或非单调数据。这些数据通常出现在上面提到的生物系统中，因此一个活跃的研究领域是扩展持久同源性来处理这些数据。我的研究将集中在这些新技术以及它们在生物模型中的应用上。公司或合作者参与其中牛津大学数学研究所以外的合作者尚未确定。

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 objectivesMy 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 areasThis project is aligned with the 'Mathematical Biology' research area.Novelty of the research methodologyOne 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 involvedNo collaborators outside of the University of Oxford Mathematical Institute have been identified.