Metrics for comparing neuronal tree shapes based on persistent homology

The geometrical tree structures of axonal and dendritic processes play important roles in determining the architecture and capabilities of neuronal circuitry. Morphological features based on this tree structure have played a central role in classifying neurons for over a century. Yet geometrical trees are not automatically adapted to the basic mathematical tool used widely in data analysis, namely vector spaces and linear algebra, since tree geometries cannot be naturally added and subtracted. Current methods for analysis reduce trees to feature vectors in more or less ad hoc ways. A more natural mathematical object suited to characterizing neuronal tree geometries, is a metric space, where only distances between objects need be defined. In recent years, there have been significant developments in the fields of computational topology and geometry that promise to be useful for the analysis of neuronal geometries. In this paper, we adapt these tools to the problem of characterizing and analyzing neuronal morphology. As more and more neuroanatomical data are made available through efforts such as NeuroMorpho.org and FlyCircuit.org, the need to develop computational tools to facilitate automatic knowledge discovery from such large datasets becomes more urgent. One fundamental question is how best to compare neuron structures, for instance to organize and classify large collection of neurons. We aim to develop a flexible yet powerful framework to support comparison and classification of large collection of neuron structures efficiently. Specifically we propose to use a topological persistence-based feature vectorization framework. Existing methods to vectorize a neuron (i.e, convert a neuron to a feature vector so as to support efficient comparison and/or searching) typically rely on statistics or summaries of morphometric information, such as the average or maximum local torque angle or partition asymmetry. These simple summaries have limited power in encoding global tree structures. Leveraging recent development in topological data analysis, we vectorize each neuron structure into a simple yet informative summary via the use of topological persistence. In particular, each type of information of interest can be represented as a descriptor function defined on the neuron tree, which is then mapped to a simple persistence-signature. Our framework can encode both local and global tree structure, as well as other information of interest (electrophysiological or dynamical measures), by considering multiple descriptor functions on the neuron. The resulting persistence-based signature is potentially more informative than simple statistical summaries (such as average/mean/max) of morphometric quantities – Indeed, we show that using a certain descriptor function will give a persistence-based signature containing strictly more information than the classical Sholl analysis. At the same time, our framework retains the efficiency associated with treating neurons as points in a simple Euclidean feature space, which would be important for constructing efficient searching or indexing structures over them. We present preliminary experimental results to demonstrate the effectiveness of our persistence-based neuronal feature vectorization framework.