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Diffusion Magnetic Resonance Imaging (dMRI) is a method for measuring molecular diffusion in biological tissue microstructure. This information can be used to predict the location and orientation of white matter fibers in the brain, a process known as tractography. Analysis of the map of neural connections can provide meaningful information about the severity or progression of neurodegenerative diseases such as Alzheimer's, and allow for early intervention to prevent progression. However, tractography has its pitfalls; current fiber-tracking algorithms suffer from generating false-positive connections and affect the reliability of structural connectivity maps. To counter this downside, tractogram filtering methods have been created to remove inaccurately predicted connections. This study aims at evaluating the impact of the novel semi-supervised filtering method FINTA on the brain networks of people with Mild Cognitive Impairment (MCI), which precedes diseases like Alzheimer's. The proposed experiments use the Nipype Neuroimaging Python library for the automation of the entire process. Registration, parcellation, and tracking were performed using MRtrix and FSL. Furthermore, DIPY and NiBabel were used for tractogram processing. Finally, filtering was performed based on code provided by the authors of FINTA, and graph measures were computed using the NetworkX Python library. Experiments were performed on both raw and weighted structural connectivity matrices. Results suggest that filtering has an effect on graph measures such as the clustering coefficient and betweenness centrality for different nodes corresponding to brain regions. / Diffusion magnetisk resonanstomografi (diffusions MRT) är en metod för att mäta den molekylära diffusionen i mikrostrukturen i biologisk vävnad. Denna information kan användas för att förutsäga var fibrerna i den vita substansen i hjärnan befinner sig och hur de är orienterade i den process som kallas traktografi. Analys av kartan över nervförbindelser kan ge meningsfull information om svårighetsgraden eller utvecklingen av neurodegenerativa sjukdomar som Alzheimers och möjliggöra tidiga insatser för att förhindra utvecklingen. Traktografi har dock sina fallgropar och nuvarande algoritmer för fiberspårning lider av att generera falska positiva anslutningar och påverkar de strukturella konnektivitetskartorna som förhindrar tillförlitliga förutsägelser. För att motverka denna nackdel har filtreringsmetoder för traktogram skapats för att ta bort de felaktigt förutsagda anslutningarna.  Denna studie syftar till att utvärdera effekterna av den nya semi-övervakade filtreringsmetoden FINTA på hjärnnätverk hos personer med lindrig kognitiv störning (eng. mild cognitive impairment, MCI) som föregår sjukdomar som Alzheimers. I de föreslagna experimenten används Python-biblioteket Nipype Neuroimaging för automatisering av hela processen. Registrering, parcellering och spårning gjordes med hjälp av MRtrix och FSL, dessutom användes DIPY och NiBabel för traktogrambehandling. Slutligen utfördes filtrering baserat på kod från författarna till FINTA och grafmått beräknades med hjälp av NetworkX Python-bibliotek. Experimenten utfördes på råa och viktade strukturella konnektivitetsmatriser. Resultaten tyder på att filtrering har en effekt på grafmått som klustringskoefficient och betweenness centrality för olika noder som motsvarar hjärnregioner.

Odorous sensations are based on trigeminal and olfactory perceptions. Both trigeminal and olfactory stimuli generate overlapping as well as distinctive activations in the olfactory cortex including the piriform cortex. Orbitofrontal cortex (OFC), an integrative center for all senses, is directly activated in the presence of olfactory stimulations. In contrast, the thalamus, a very important midbrain structure, is not directly activated in the presence of odors, but rather acts as a relay for portions of olfactory information between primary olfactory cortex and higher-order processing centers. The aims of the study were (1) to examine the number of streamlines between the piriform cortex and the OFC and also between the piriform cortex and the thalamus and (2) to explore potential correlations between these streamlines and trigeminal and olfactory chemosensory perceptions. Thirty-eight healthy subjects were recruited for the study and underwent diffusion MRI using a 3T MRI scanner with 67 diffusion directions. ROIs were adapted from two studies looking into olfaction in terms of functional and structural properties of the olfactory system. The “waytotal number” was used which corresponds to number of streamlines between two regions of interests. We found the number of streamlines between the piriform cortex and the thalamus to be higher in the left hemisphere, whereas the number of streamlines between the piriform cortex and the OFC were higher in the right hemisphere. We also found streamlines between the piriform cortex and the thalamus to be positively correlated with the intensity of irritating (trigeminal) odors. On the other hand, streamlines between the piriform cortex and the OFC were correlated with the threshold scores for these trigeminal odors. This is the first studying the correlations between streamlines and olfactory scores using tractography. Results suggest that different chemosensory stimuli are processed through different networks in the chemosensory system involving the thalamus.

People suffering from congenital anosmia show normal brain architecture although they do not have functional sense of smell. Some studies in this regard point to the changes in secondary olfactory cortex, orbitofrontal cortex (OFC), in terms of gray matter volume increase. However, diffusion tensor imaging has not been explored so far. We included 13 congenital anosmia subjects together with 15 controls and looked into various diffusion parameters like FA. Increased FA in bilateral OFC confirms the earlier studies reporting increased gray matter thickness. However, it is quite difficult to interpret FA in terms of gray matter volume. Increased FA has been seen with recovery after traumatic brain injury. Such changes in OFC point to the plastic nature of the brain.

Ce travail présente l’optimisation d’un système de spectroscopie Raman résolue en temps dédiée aux très hautes températures. Ce dispositif répond au besoin sans cesse croissant d’étudier en temps réels les transformations de phases et des cinétiques de réactions dans des environnements extrêmes. Ce dispositif a été validé dans des conditions d’usages sur des oxydes (Gd₂O3, Y₂O3, ZrO₂ , ZrSiO4 et CeO₂) et des nitrures (h-BN). Le potentiel du système a permis de lever les principaux verrous technologiques et instrumentaux qui limitent l’utilisation de la spectroscopie Raman à haute température. Les trois principaux faits marquants qui illustrent le caractère innovant de ce travail sont les suivants : Le premier correspond au nouveau record mondial d’analyse Raman à haute température à travers l’acquisition du mode E₂g de h-BN jusqu’à 2700°C. La comparaison des performances des deux voies Pockels et ICCD montrent que la voie Pockels est plus performante que l’ICCD, mais plus délicate de mise en oeuvre. Le second fait marquant concerne les autres applications du Raman résolu en temps, comme pour séparer la contribution de la diffusion Raman et de la luminescence. La dernière application quant à elle présente l’étude de la dépendance temporelle comparée des diffusions Raman résonnante et non résonante. Le Raman résonnant se déclenche systématiquement avant le non résonnant. Plus généralement, l’intérêt des méthodes Raman résolues en temps ouvre de nouveaux champs d’application dans la caractérisation de matériaux en condition extrêmes, éventuellement in situ : aéronautique, réfractaire ; sidérurgie, nucléaire, etc… / This work presents the optimization of a time-resolved Raman spectroscopy device dedicated to very high temperatures. This device meets the ever-increasing need to study in real time phase transformations and reaction kinetics in extreme environments. This device has been validated under working conditions on oxides (Gd₂O3, Y₂O3, ZrO₂ , ZrSiO4 et CeO₂) and nitrides (h-BN). The potentialities of the device have enabled the main technological and instrumental locks that limit the use of high temperature Raman spectroscopy to be removed. The three main highlights illustrating the innovative nature of this work are as follows. The first corresponds to the new world record for high temperature Raman analysis through the acquisition of the E₂g mode of h-BN up to 2700°C.A comparison of the performance of the two Pockels and ICCD channels shows that the Pockels channel is more efficient than the ICCD, but more difficult to implement. The second important fact concerns the other applications of time-resolved Raman, as to separate the contribution of Raman scattering and luminescence. The last application presents the study of the comparative time dependence of resonant and non-resonant Raman scattering. The resonant Raman is triggered systematically before the non-resonant. More generally, the interest of time-resolved Raman methods opens new fields of application in the characterization of materials in extreme conditions, possibly in situ: aeronautics, refractories, steel industry, nuclear, etc....

Ziel dieser Arbeit ist es zwei verschiedene Klassen von Systemen nichtlinearer parabolischer Gleichungen zu homogenisieren, und zwar Reaktions-Diffusions-Systeme mit verschiedenen Diffusionslängenskalen und Gleichungen vom Typ Cahn-Hilliard. Wir betrachten parabolische Gleichungen mit periodischen Koeffizienten, wobei die Periode dem Verhältnis der charakteristischen mikroskopischen zu der makroskopische Längenskala entspricht. Unser Ziel ist es, effektive Gleichungen rigoros herzuleiten, um die betrachteten Systeme besser zu verstehen und den Simulationsaufwand zu minimieren. Wir suchen also einen Konvergenzbegriff, mit dem die Lösung des Ausgangsmodells im Limes der Periode gegen Null gegen die Lösung des effektiven Modells konvergiert. Um die periodische Mikrostruktur und die verschiedenen Diffusivitäten zu erfassen, verwenden wir die Zwei-Skalen Konvergenz mittels periodischer Auffaltung. Der erste Teil der Arbeit handelt von Reaktions-Diffusions-Systemen, in denen einige Spezies mit der charakteristischen Diffusionslänge der makroskopischen Skala und andere mit der mikroskopischen diffundieren. Die verschiedenen Diffusivitäten führen zu einem Verlust der Kompaktheit, sodass wir nicht direkt den Grenzwert der nichtlinearen Terme bestimmen können. Wir beweisen mittels starker Zwei-Skalen Konvergenz, dass das effektive Modell ein zwei-skaliges Modell ist, welches von der makroskopischen und der mikroskopischen Skale abhängt. Unsere Methode erlaubt es uns, explizite Raten für die Konvergenz der Lösungen zu bestimmen. Im zweiten Teil betrachten wir Gleichungen vom Typ Cahn-Hilliard, welche ortsabhängige Mobilitätskoeffizienten und allgemeine Potentiale beinhalten. Wir beweisen evolutionäre Gamma-Konvergenz der zugehörigen Gradientensysteme basierend auf der Gamma-Konvergenz der Energien und der Dissipationspotentiale. / The aim of this thesis is to derive homogenization results for two different types of systems of nonlinear parabolic equations, namely reaction-diffusion systems involving different diffusion length scales and Cahn-Hilliard-type equations. The coefficient functions of the considered parabolic equations are periodically oscillating with a period which is proportional to the ratio between the charactersitic microscopic and macroscopic length scales. In view of greater structural insight and less computational effort, it is our aim to rigorously derive effective equations as the period tends to zero such that solutions of the original model converge to solutions of the effective model. To account for the periodic microstructure as well as for the different diffusion length scales, we employ the method of two-scale convergence via periodic unfolding. In the first part of the thesis, we consider reaction-diffusion systems, where for some species the diffusion length scale is of order of the macroscopic length scale and for other species it is of order of the microscopic one. Based on the notion of strong two-scale convergence, we prove that the effective model is a two-scale reaction-diffusion system depending on the macroscopic and the microscopic scale. Our approach supplies explicit rates for the convergence of the solution. In the second part, we consider Cahn-Hilliard-type equations with position-dependent mobilities and general potentials. It is well-known that the classical Cahn-Hilliard equation admits a gradient structure. Based on the Gamma-convergence of the energies and the dissipation potentials, we prove evolutionary Gamma-convergence, for the associated gradient system such that we obtain in the limit of vanishing periods a Cahn-Hilliard equation with homogenized coefficients.

Die vorliegende Dissertation beschäftigt sich mit der Anwendung der Theorie der großen Abweichungen auf verschiedene Fragestellungen der stochastischen Analysis und stochastischen Dynamik von Sprungprozessen. Die erste Fragestellung behandelt die erste Austrittszeit aus einem beschränkten Gebiet für eine bestimmte Klasse von Sprungdiffusionen mit exponentiell leichten Sprüngen. In Abhängigkeit von der Leichtheit des Sprungmaßes wird das asymptotische Verhalten der Verteilung und insbesondere der Erwartung der ersten Austrittszeit bestimmt wenn das Rauschen verschwindet. Dabei folgt die Verteilung der ersten Austrittszeit einem Prinzip der großen Abweichungen im Falle eines superexponentiellen Sprungmaßes. Wohingegen im subexponentiellen Fall die Verteilung einem Prinzip moderater Abweichungen genügt. In beiden Fällen wird die Asymptotik bestimmt durch eine deterministische Größe, die den minimalen Energieaufwand beschreibt, um die Sprungdiffusion einen optimalen Kontrollpfad, der zum Austritt führt, folgen zu lassen. Die zweite Fragestellung widmet sich dem Grenzverhalten gekoppelter Vorwärts-Rückwärtssysteme stochastischer Differentialgleichungen bei kleinem Rauschen. Dazu assoziiert ist eine spezielle Klasse nicht-lokaler partieller Differentialgleichungen, die auch in nicht-lokalen Modellen der Fluiddynamik eine Rolle spielen. Mithilfe eines probabilistischen Ansatzes und der Markovschen Struktur dieser Systeme wird die Konvergenz auf Ebene von Viskositätslösungen untersucht. Dabei wird ein Prinzip der großen Abweichungen für die involvierten Stochastischen Prozesse hergeleitet. / This thesis deals with applications of Large Deviations Theory to different problems of Stochastic Dynamics and Stochastic Analysis concerning Jump Processes. The first problem we address is the first exit time from a fixed bounded domain for a certain class of exponentially light jump diffusions. According to the lightness of the jump measure of the driving process, we derive, when the source of the noise vanishes, the asymptotic behavior of the law and of the expected value of first exit time. In the super-exponential regime the law of the first exit time follows a large deviations scale and in the sub-exponential regime it follows a moderate deviations one. In both regimes the first exit time is comprehended, in the small noise limit, in terms of a deterministic quantity that encodes the minimal energy the jump diffusion needs to spend in order to follow an optimal controlled path that leads to the exit. The second problem that we analyze is the small noise limit of a certain class of coupled forward-backward systems of Stochastic Differential Equations. Associated to these stochastic objects are some nonlinear nonlocal Partial Differential Equations that arise as nonlocal toy-models of Fluid Dynamics. Using a probabilistic approach and the Markov nature of these systems we study the convergence at the level of viscosity solutions and we derive a large deviations principles for the laws of the stochastic processes that are involved.

Die Forschung an der Hefe Saccharomyces cerevisiae – auch als Bäckerhefe bekannt – hat sich für die biologische Grundlagenforschung als unentbehrlich erwiesen und führte zu wichtigen Erkenntnissen in der Erforschung von Krankheiten wie Krebs. Am Beispiel der Paarung von Hefezellen werden in dieser Arbeit wesentliche Aspekte der eukaryotischen Zellbiologie untersucht. In der Haplophase des Lebenszyklus der Hefe, treten haploide Zellen als Paarungstyp MATa oder MATα auf. Diese Paarungstypen kommunizieren über Pheromone, die in ein extrazelluläres Medium abgesondert werden und von Zelloberflächenrezeptoren des komplementären Paarungstyps erkannt werden. Hefezellen wachsen in die Richtung eines möglichen Paarungspartners, da sie sich nicht aktiv bewegen können. Die Auswertung von empirischen Daten aus der Fluoreszenzmikroskopie und Rasterkraftmikroskopie (AFM) mit mathematischen Modellen ermöglichte die Rekonstruktion wesentlicher Prozesse der Hefepaarung: (i) Interzelluläre Kommunikation über die Sezernierung und Rezeption von Pheromonen, (ii) Aufbau der Zellpolarität als Reaktion auf die Pheromonantwort, (iii) Induktion und Mechanik der Zellformänderung. Folgende Modelle wurden dazu entwickelt: (i) Die interzelluläre Kommunikation wurde unter Verwendung von zellulären Automaten mit Hilfe von Reaktions-Diffusions (RD) Gleichungen modelliert. Das Modell zeigte, dass die gegenseitige Stimulierung und erhöhte Pheromonabsonderung zu einer verbesserten Abstimmung in der Paarung in der Zellpopulation führt. (ii) Ein Turing- und ein Phasenseparations- Mechanismus wurden als Modelle zum Aufbau der Zellpolarität verwendet. Volumen-Oberflächen gekoppelte RD Gleichungen wurden analytisch und numerisch mit der Finite-Elemente-Methode (FEM) untersucht. (iii) Die Zellwandveränderung wurde mit klassischer Kontinuumsmechanik und der FEM Methode modelliert. Dies ermöglichte eine Beschreibung der reversiblen elastischen und der irreversiblen plastischen Verformungen der Zellwand. / Research on the yeast Saccharomyces cerevisiae – also known as baker’s yeast – has been essential not only for fostering basic biological knowledge but even more so for contributing towards understanding diseases such as cancer. In this thesis, general biological phenomena occurring in eukaryotic cells are investigated, exemplified by the mating process of yeast. In the haploid phase of their life cycle, yeast cells occur as mating type MATa or MATα, both of which communicate via pheromones that are secreted in an extracellular medium and can be sensed by cell-surface receptors of the complementary mating type. In order to mate, yeast cells grow towards a potential mating partner, since they are not able to actively move. Mathematical models on the basis of fluorescence and atomic force microscopy (AFM) data were developed. The key aspects of the yeast mating process that I examined were (i) intercellular communication of cells via pheromones, (ii) the initial symmetry break and implementation of cell polarity, and (iii) subsequent morphogenetic changes. The methods used and findings were as follows: (i) Pheromone secretion and sensing motifs were modelled using cellular automata models based on reaction-diffusion (RD) equations. My models show that mutual stimulation and increased pheromone secretion between cells improves mating efficiency in cell populations. (ii) To explain yeast mating decisions, two possible model types for cell polarity were tested: a Turing-type and a phase-separation mechanism. Bulk-surface RD equations were investigated analytically and numerically using the finite element method (FEM). Typical cell shapes were reconstructed in 2D and 3D. (iii) The cell wall was modelled using classical continuum mechanics that allows for reversible elastic and irreversible plastic cell wall deformation. Mathematical modelling demonstrated that all three processes investigated are precisely orchestrated and interlocked during yeast mating.

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles. In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem. The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’). The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion. In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.