Summary Progress from description to quantification is essential as a science matures. Yet numerical analysis of the elementary unit of brain circuitry?the individual neuron?continues to pose methodological challenges. Even the definition of a measurement yardstick (a metric) for the tree shape of a neuron remains an open research problem. Without such metrics, researchers cannot accurately classify neurons into cell types, an essential step toward understanding the circuit components and how they work together. Advanced methods from computational topology and geometry, which have only recently made their way from pure mathematics into data analysis, will be used to extract, characterize, and classify neuronal shapes in a way that elegantly incorporates the underlying dynamical electrophysiological properties. The first specific aim will apply new mathematical methods to define and compute metrics on the shapes of a wide variety of neurons. A computational topological analysis called ?persistence summaries? will be used to generate invariant representations of the neurons that can then be compared using different norms. An important strength of this method is that it works flexibly with arbitrary functions defined on the neurons, including purely structural ones (such as distance from the soma) or functions with electrophysiological meaning (such as electrotonic distance or propagation delays) and can therefore incorporate dynamics. A more advanced approach based on the Gromov-Hausdorff distance between metric spaces will be also explored. The metrics so generated will be used for classification and clustering, visualization of the space of neuronal shapes, and shape-based database search for neuronal reconstructions derived from light or electron microscopy. The second aim will use Morse theory to reconstruct individual neurons from light microscopic data, or skeletonize tracer injection data to summarize the structure of projection patterns. This approach retains shape information, which is lost when such data are characterized in a connectivity matrix. Further, these methods will be applied to construct consensus trees, which can be used as a summary of different reconstructions produced by different algorithms. The tools will be freely shared under a suitable open-source software license, and made available via plugins to widely used software platforms as well as web services to a community repository of neuronal morphologies. The team of researchers includes theorists, experimentalists, data scientists, and end users, all with extensive relevant experience. Apart from enabling the understanding of normal brain circuitry in terms of its component neurons, the proposed methods will also allow researchers to characterize changes in the shape of neurons in pathologically altered circuits, with applications to transgenic animal models of disease.