A Topological Representation of Branching Neuronal Morphologies

Abstract

Many biological systems consist of branching structures that exhibit a wide variety of shapes. Our understanding of their systematic roles is hampered from the start by the lack of a fundamental means of standardizing the description of complex branching patterns, such as those of neuronal trees. To solve this problem, we have invented the Topological Morphology Descriptor (TMD), a method for encoding the spatial structure of any tree as a "barcode", a unique topological signature. As opposed to traditional morphometrics, the TMD couples the topology of the branches with their spatial extents by tracking their topological evolution in 3-dimensional space. We prove that neuronal trees, as well as stochastically generated trees, can be accurately categorized based on their TMD profiles. The TMD retains sufficient global and local information to create an unbiased benchmark test for their categorization and is able to quantify and characterize the structural differences between distinct morphological groups. The use of this mathematically rigorous method will advance our understanding of the anatomy and diversity of branching morphologies.

Keywords: Branching morphology; Clustering trees; Neuronal morphologies; Topological data analysis.

Fig. 1

Illustration of the separation of similar tree structures into distinct groups, using topological analysis. The colored pie segments show six distinct tree types: three neuronal types (upper half) and three artificial ones (lower half). The thick blue lines show that our topological analysis can reliably separate similar-looking trees into groups. It is accurate both for artificially-generated trees and neuronal morphologies. The dashed green lines show that classification using an improper set of user-selected features (number of branches, total length) cannot distinguish the correct groups

Fig. 2

Application of topological analysis to a neuronal tree (A) showing the largest persistent component (red). The persistence barcode (B) represents each component as a horizontal line whose endpoints mark its birth and death in units that depend on the choice of the function f used for the ordering of the nodes of the tree. In our case, it is radial distance of the nodes from the root (R), so the units are microns. The largest component is again shown in red together with its birth (I) and death (II). This barcode can be equivalently represented as points in a persistence diagram (C) where the birth (I) and death (II) of a component are the X and Y coordinates of a point respectively (in red). The diagonal line is a guide to the eye and marks points with the same birth and death time

Fig. 3

Topological analysis of artificial trees generated using a stochastic process. Three sets of trees are shown (only four individuals out of twenty for clarity). Each group differs from the others only in the tree depth. Each individual of the group is generated using the same tree parameters but a different random number seed. The TMD-distance of the trees allow their accurate separation into groups. The distance matrix indicates the existence of three groups which are identified with high accuracy by a simple dendrogram algorithm

Fig. 4

Topological comparison of neurons from different animal species. Each row corresponds to a species: (I) cat, (II) dragonfly, (III) fruit fly, (IV) mouse and (V) rat. Trees from several exemplar cells for each species are shown in the first column (A). Representative persistence barcodes for the cells in A are shown in the second column (B). The structural differences of the trees are clearly evident in these barcodes. II, III and V have clusters of short components, clearly distinct from the largest component, while I and IV have bars of a quasi-continuous distribution of decreasing lengths. Also, barcodes III, and V show empty regions between dense regions of bars, indicating the existence of two clusters in the morphologies, while barcodes I and IV are dense overall. The unweighted persistence image for each representative barcode in B and its superimposed persistence diagram are shown in the third column (C). By combining the persistence diagrams in (C) for several trees we can define an average unweighted persistence image (D) in order to study and quantify the structural differences between distinct morphological groups. The trees in the first row (cat) are more tightly grouped than those in the second row (dragonfly), and two clusters are visible in the dragonfly trees. Considering rows 1 and 4, the extension of the elliptical peak perpendicular to the diagonal line reflects the variance in the length-scale mentioned earlier for a single cell’s barcode. Note that the trees, barcodes, and unweighted persistence images are not shown to the same scale for clarity: see the scale bar in each case

Fig. 5

Comparison of the TMD of apical dendrite trees extracted from several types of rat pyramidal neuron. Four cell types are shown in (A): UPC, SPC, TPC-A, TPC-B (left to right). The morphological differences between these cell types are subtle, but the unweighted persistence images (B) clearly reveal them, particularly the presence of two clusters in the TPC-A and TPC-B cell types. From these unweighted persistence images we train a decision tree classifier on the expert-assigned groups of cells. The binary classification (C) and the confusion matrix (D) based on the TMD algorithm shows an overlap of TPC-A and TPC-B trees. When those two classes are merged (E, F) the separation between the remaining types is evident. This result shows that the unweighted persistence images objectively support the expert’s classification when the morphological differences between the classes are significant

Fig. 6

Comparison of the TMD of BigNeuron neuronal morphologies. An image stack is used for the manual reconstruction (reference neuron) and for the automatic reconstructions produced by a variety of community supplied algorithms. The results of each algorithm are illustrated in panel C, from best (top left) to worst (bottom right). The reference neuron (in black) is visualized against the density plot of all the automatically reconstructed neurons (A, in blue), and the density plot of the ten best automatic reconstructions (B, in red), ranked according to their TMD-distance from the reference neuron. A comparison between panels A and B shows that the density plot of the ten highest ranked automatic reconstructions closely matches the structure of the reference morphology