Quantifying topological invariants of neuronal morphologies
Résumé
Nervous systems are characterized by neurons displaying a diversity of morphological shapes. Traditionally, different shapes have been qualitatively described based on visual inspection and quantitatively described based on morphometric parameters. Neither process provides a solid foundation for categorizing the various morphologies, a problem that is important in many fields. We propose a stable topological measure as a standardized descriptor for any tree-like morphology, which encodes its skeletal branching anatomy. More specifically it is a barcode of the branching tree as determined by a spherical filtration centered at the root or neuronal soma. This Topological Morphology Descriptor (TMD) allows for the discrimination of groups of random and neuronal trees at linear computational cost. The analysis of complex tree structures, such as neu-rons, branched polymers [1], viscous fingering [2] and fractal trees [3, 4], is important for understanding many physical and biological processes. Yet an efficient method for quantitatively analysing the morphology of such trees has been difficult to find. Biological systems provide many examples of complex trees. The nervous system is one of the most complex biological systems known, whose fundamental units, neurons, are sophisticated information processing cells possessing highly ramified ar-borizations [5]. The structure and size of neuronal trees determines the input sources to a neuron and the range of target outputs and is thought to reflect their involvement in different computational tasks [6–8]. In order to understand brain functionality, it is fundamental to understand how neuronal shape determines neuronal function [9]. As a result, much effort has been devoted to grouping neu-rons into distinguishable morphological classes [10–13], a categorization process that is important in many fields [14]. Neurons come in a variety of shapes with different branching patterns (e.g., frequency of branching, branching angles, branching length, overall extent of the branches, etc). Classifying these different morphologies has traditionally focused on visually distinguishing the shapes as observed under a microscope [15]. This method is inadequate as it is subject to large variations between experts studying the morphologies [10] and is made even more difficult by the presence of an enormous variety of morphological types [11]. An objective method of discriminating between neuronal morphologies could advance progress in generating a parts-list of neurons in the nervous system. For this reason, experts now generate a digital version of the cells structure-a 3D reconstructed model of the neuron [16] that can be studied computa-tionally. The reconstructed morphology is encoded as a set of points in R 3 along each branch and edges connecting pairs of points. The reconstruction forms a mathematical tree representing the neurons morphology. Figure 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 cannot distinguish the correct groups. In general, the properties of geometric trees, described by a set of points, have been rigorously studied in two extreme cases: in the limit of the full complexity of the trees [17], where the entire set of points is used, and in the opposite limit of a feature space [10, 18], where a (typically small) number of selected morphometrics (i.e., statistical features of the branching pattern) [19, 20] are