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 gemoetry 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/.

中文翻译：

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将样条曲线拟合到轴突轴上可量化分支顺序和几何之间的关系

神经形态学对于识别神经元亚型和理解学习至关重要。它也与神经系统疾病有关。但是，标准的形态学分析着眼于宏观特征，例如分支频率和区域之间的连通性，并且常常忽略了神经元的内部几何形状。在这项工作中，我们将神经元跟踪点视为可微分曲线的采样，并使用一组分支B样条对其进行拟合。我们在设计时就考虑了微分宝石学的Frenet-Serret公式。Frenet-Serret公式完全描述了平滑曲线，并包含两个参数：曲率和扭转。我们的表示法使从封闭形式的神经元迹线计算这些参数成为可能。这些参数沿曲线连续定义，与其他像曲折度一样的参数相反，这些参数取决于起点和终点。我们将我们的方法应用于在两个小鼠大脑中追踪到的皮质投射神经元的数据集，发现参数在初级，侧枝和末端轴突分支之间分布不同，从而量化了轴突乔木的不同组成部分之间的几何差异。结果在两个大脑中都一致，进一步验证了我们的代表性。这项工作中使用的代码可以很容易地应用于SWC格式的神经元痕迹，并且可以在我们的开放源代码Python软件包Brainlit中找到：http：//brainlit.neurodata.io/。侧枝和末端轴突分支，从而量化轴突乔木的不同组成部分之间的几何差异。结果在两个大脑中都一致，进一步验证了我们的代表性。这项工作中使用的代码可以很容易地应用于SWC格式的神经元痕迹，并且可以在我们的开放式Python软件包Brainlit中找到：http：//brainlit.neurodata.io/。侧枝和末端轴突分支，从而量化轴突乔木的不同组成部分之间的几何差异。结果在两个大脑中都一致，进一步验证了我们的代表性。这项工作中使用的代码可以很容易地应用于SWC格式的神经元痕迹，并且可以在我们的开放源代码Python软件包Brainlit中找到：http：//brainlit.neurodata.io/。