Statistical shape analysis of simplified neuronal trees

Abstract

Neuron morphology plays a central role in characterizing cognitive health and functionality of brain structures. The problem of quantifying neuron shapes and capturing statistical variability of shapes is difficult because neurons differ both in geometry and in topology. This paper develops a mathematical representation of neuronal trees, restricting to the trees that consist of: (1) a main branch viewed as a parameterized curve in $\mathbb{R}^{3}$, and (2) some number of secondary branches—also parameterized curves in $\mathbb{R}^{3}$—which emanate from the main branch at arbitrary points. It imposes a metric on the representation space, in order to compare neuronal shapes, and to obtain optimal deformations (geodesics) across arbitrary trees. The key idea is to impose certain equivalence relations that allow trees with different geometries and topologies to be compared efficiently. The combinatorial problem of matching side branches across trees is reduced to a linear assignment with well-known efficient solutions. This framework is then applied to comparing, clustering, and classifying neurons using fully automated algorithms. The framework is illustrated on three datasets of neuron reconstructions, specifically showing geodesics paths and cross-validated classification between experimental groups.