  One of the essential tasks in connectomics is the morphology analysis of
neurons and organelles like mitochondria to shed light on their biological
properties. However, these biological objects often have tangled parts or
complex branching patterns, which make it hard to abstract, categorize, and
manipulate their morphology. In this paper, we develop a novel topological
nomenclature system to name these objects like the appellation for chemical
compounds to promote neuroscience analysis based on their skeletal structures.
We first convert the volumetric representation into the topology-preserving
reduced graph to untangle the objects. Next, we develop nomenclature rules for
pyramidal neurons and mitochondria from the reduced graph and finally learn the
feature embedding for shape manipulation. In ablation studies, we
quantitatively show that graphs generated by our proposed method align with the
perception of experts. On 3D shape retrieval and decomposition tasks, we
qualitatively demonstrate that the encoded topological nomenclature features
achieve better results than state-of-the-art shape descriptors. To advance
neuroscience, we will release a 3D segmentation dataset of mitochondria and
pyramidal neurons reconstructed from a 100um cube electron microscopy volume
with their reduced graph and topological nomenclature annotations. Code is
publicly available at https://github.com/donglaiw/ibexHelper.
