Fitting Splines to Axonal Arbors Quantifies Relationship Between Branch Order and Geometry

Abstract

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit: http://brainlit.neurodata.io/.

nAdder: A scale-space approach for the 3D analysis of neuronal traces

“…23]. In practice, the local 3D scale of neurons (as measured with nAdder and observed in this study) ranges from 0 to 100-200 μm.…”

“…An array of geometric algorithmic methods and associated software has already been developed to process neuronal reconstructions [20][21][22][23]. Morphological features enabling the construction of neuron ontologies, described for instance by the Petilla convention [24], have been used for machine learning-based automated neuronal classification [25].…”

Preserving Derivative Information while Transforming Neuronal Curves

“…Neurons have a tree-like structure, and we split them into non-branching curves in order to apply our mapping methods. We follow a method introduced previously [1] where the root to leaf path with the longest arc length is recursively removed until the tree is reduced to non-bifurcating "branches". The whole-brain image was registered to the Allen Reference atlas [21] using CloudReg [6] (Figure 3).…”