Analysis-ready models of tortuous, tightly packed geometries

Complex networks of cells called neurons in the brain enable human learning and memory. The topology and electrophysiological function of these networks are affected by nano and microscale geometries of neurons. Understanding of these structure-function relationships in neurons is an important component of neuroscience in which simulation plays a fundamental role. This thesis addresses four specific geometric problems raised by modeling and simulation of intricate neuronal structure and behavior at the nanoscale. The first two problems deal with 3D surface reconstruction: neurons are geometrically complex structures that are tightly intertwined in the brain, presenting great challenges in reconstruction. We present the first algorithm that reconstructs surface meshes from polygonal contours that provably guarantees watertight, manifold, and intersection-free forests of densely packed structures. Many algorithms exist that produce surfaces from cross-sectional contours, but all either use heuristics in fitting the surface or they fail when presented with tortuous objects in close proximity. Our algorithm reconstructs surfaces that are not only internally correct, but are also free of intersections with other reconstructed objects in the same region. We also present a novel surface remeshing algorithm suitable for models of neuronal dual space. The last two problems treated by this thesis deal with producing derivative models from surface meshes. A range of neuronal simulation methodologies exist and we offer a framework to derive appropriate models for each from surface meshes. We present two specific algorithms that yield analysis-ready 1D cable models in one case, and proposed "aligned cell" models in the other. In the creation of aligned cells we also present a novel adaptive distance transform. Finally, we present a software package called VolRoverN in which we have implemented many of our algorithms and which we expect will serve as a repository of important tools for the neuronal modeling community. Our algorithms are designed to meet the immediate needs of the neuroscience community, but as we show in this thesis, they are general and suitable for a variety of applications.