L'analyse de la morphologie axonale est un problème important en neuroscience. Diverses études ont montré que les caractéristiques morphologiques de ces structures donnent des informations sur son fonctionnement et permettent la caractérisation d'états pathologiques. En conséquence, il est important de développer des méthodes pour étudier leurs formes et quantifier leurs différences structurelles.Dans cette thèse on propose une méthode pour la comparaison des arbres axonaux qui inclue des informations topologiques et géométriques. La méthode est fondée sur la théorie des formes élastiques. Avec cette approche, nous pouvons exhiber le chemin géodésique entre deux formes et la forme moyenne d'un ensemble d'échantillons. En outre, nous proposons un schéma de classification à partir de cette métrique que nous comparons à l'état de l'art. Finalement, nous proposons un modèle stochastique pour la simulation de la croissance axonale défini par une chaîne de Markov. Il considère 2 processus principaux qui modélisent l'élongation et forme de l'axone et la génération des branches. Le processus de croissance dépend de différentes variables, dont un champ externe d'attraction généré par certaines molécules dans l'environnement. Les deux techniques proposées ont été validées sur une base d'images de microscopie confocale de neurones chez la Drosophile. Des neurones normaux et modifiés génétiquement ont été considérés. Les résultats montrent que la méthode de comparaison proposée fournit de meilleurs résultats que les méthodes décrites dans la littérature. De plus, les paramètres du modèle donnent des informations sur le processus de croissance de chaque population d'axones. / The morphological analysis of axonal trees is an important problem in neuroscience. It has been shown that the morphological characteristics of thesestructures provide information on their functioning and allows the characterization of pathological states. Therefore, it is of great importance to develop methods to analyze their shape and to quantify differences between structures. In this thesis we propose a method for the comparison of axonal trees that takes into account both topological and geometrical information. Using this method, which is based on the Elastic Shape Analysis Framework, we can compute the geodesic path between two axons and the mean shape of a population of trees. In addition, we derive a classfication scheme based on this metric and compare it with state of the art approaches. Finally, we propose a 2D discrete stochastic model for the simulation of axonal biogenesis. The model is defined by a third order Markov Chain and considers two main processes: the growth process that models the elongation and shape of the neurites and the bifurcation process that models the generation of branches. The growth process depends, among other variables, on an external attraction field. Both techniques were validated on a database of real fluorescent confocal microscopy images of neurons within Drosophila fly brains. Both normal neurons and neurons in which certain genes were inactivated have been considered. Results show that the proposed comparison method obtains better results that other methods found in the literature, and that the model parameter values provide information about the growth properties of the populations.

The Unruh effect is one of the most startling predictions of quantum field theory. Its interpretation has been controversially discussed, since the first publications of Fulling, Davies and Unruh in the 1970ties. In a recent paper Buchholz and Solveen proposed an application of basic thermodynamic definitions to clarify the meaning of temperature and thermal equilibrium in the Unruh effect. As a result the interpretation of the KMS-parameter as an expression of local temperature has been questioned. The main result of my diploma thesis asserts quasi-equivalence of the disputed KMS states on a subregion of Rindlerspace that infinitely extends in the direction of travel of a uniformly accelerated Rindler-observer. Exploring the consequences of this result, I will present new insights on the asymptotic behaviour of such KMS states and how this fits into the picture drawn by Buchholz and Solveen.

This dissertation presents some of the recent developments in the modelling of shape spaces. Forming the basis for a quantitative analysis of shapes, this is relevant for many applications involving image recognition and shape classification. All shape spaces discussed in this work arise from the general situation of a Lie group acting isometrically on some Riemannian manifold. The first chapter summarizes the most important results about this general set-up, which are well known in other branches of mathematics. A particular focus is laid on Hamiltonian methods that explore the relation of symmetry and conserved momenta. As a classical example these results are applied to Kendall’s shape space. More recent approaches of continuous shape models are then summarized and put in the same concise framework. In more detail the square root velocity shape representation, recently developed by Srivastava et al., is being discussed. In particular, the phenomenon of unclosed orbits under the action of reparametrization is addressed. This issue is partially resolved by an extended equivalence relation along with a well defined, non-degenerate, metric on the resulting quotient space.