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

TL;DR

This paper introduces a novel method for analyzing neuron morphology by fitting neuron traces with branching B-splines to compute curvature and torsion, revealing geometric differences across axonal branch types.

Contribution

The authors developed a spline-based representation of neuron traces using differential geometry, enabling continuous computation of curvature and torsion for detailed morphological analysis.

Findings

Parameters differ between primary, collateral, and terminal branches.

Method validated across two mouse brains with consistent results.

Open-source Python package available for implementation.

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/.