Path integration is the ability to navigate to a goal location without using external cues, thus relying entirely on self-motion information. To do so, two components of a path need to be encoded: orientation and distance. While the ability to estimate distance, called linear path integration, is well established in humans, it is unclear whether rodents are equally capable of doing so and the underlying neural circuit mechanisms are only poorly understood. This thesis discusses the development of a virtual reality system and behavioural task to investigate linear path integration in mice, and the results obtained from experiments carried out with this system. The setup provides full control over visual input while de-correlating vestibular and olfactory signals from location. Manipulations of the translation from physical to virtual movement can thus be used to probe relative influences of motor related and visual signals. Chapter 1 reviews the current literature on path integration and provides a background to the technical setup of the system. Chapter 2 describes the design and construction of the virtual reality system, its individual components and the software created to run experiments. It discusses how 3-d modelling and 3-d printing have successfully been combined to allow rapid development and production of custom components in different materials. Chapter 3 discusses the development of behavioural tasks designed to investigate linear path integration. It shows that by using a simple virtual track design and a carefully monitored food-deprivation regime, mice can be trained to successfully associate a visually indicated location with a reward. Chapter 4 describes behavioural experiments carried out using this virtual linear track. I obtained evidence that mice can estimate the distance to the rewarded zone reliably using path integration strategies. To test whether mice rely on motor information or optic flow, I manipulated the gain between physical movement and virtual movement. My results suggest that mice primarily rely on motor information for linear path integration. In the final chapter the results are discussed in the context of other recent work and areas for further development of the system are identified.

As Galileo peered through a lens to see the twinkle of the Jovian moons, and Antonie Van Leeuwenhoek did the same to study the tremulous basis of all life, so the fabric of threads we weave across time and space – the vast net of relations that bind and separate us – is visible only through a lens. Footprints in the snow and the weathered stone steps of buildings hint at the shape of these threads, but the coming of spring and the hardness of stone limit our observations. The Global Positioning System (GPS) now provides us a lens to see the path that individuals, families, and communities take in space-time -- their worldlines. When millions of GPS signatures are collected from hundreds of individuals, heritable patterns emerge that embody particular individual’s ideas and practices, as well as those of the society and the environment in which they operate. Besides providing a tool to test assumptions about how space is used, I argue in this thesis that by allowing us to glimpse a terra incognita, mapping worldlines also provides a unique perspective on our spatial relationship to one another.

As Galileo peered through a lens to see the twinkle of the Jovian moons, and Antonie Van Leeuwenhoek did the same to study the tremulous basis of all life, so the fabric of threads we weave across time and space – the vast net of relations that bind and separate us – is visible only through a lens. Footprints in the snow and the weathered stone steps of buildings hint at the shape of these threads, but the coming of spring and the hardness of stone limit our observations. The Global Positioning System (GPS) now provides us a lens to see the path that individuals, families, and communities take in space-time -- their worldlines. When millions of GPS signatures are collected from hundreds of individuals, heritable patterns emerge that embody particular individual’s ideas and practices, as well as those of the society and the environment in which they operate. Besides providing a tool to test assumptions about how space is used, I argue in this thesis that by allowing us to glimpse a terra incognita, mapping worldlines also provides a unique perspective on our spatial relationship to one another.

Animals are often able to solve complex navigational tasks in very challenging terrain, despite using low resolution sensors and minimal computational power, providing inspiration for robots. In particular, many species of insect are known to solve complex navigation problems, often combining an array of different behaviours (Wehner et al., 1996; Collett, 1996). Their nervous system is also comparatively simple, relative to that of mammals and other vertebrates. In the first part of this thesis, the visual input of a navigating desert ant, Cataglyphis velox, was mimicked by capturing images in ultraviolet (UV) at similar wavelengths to the ant’s compound eye. The natural segmentation of ground and sky lead to the hypothesis that skyline contours could be used by ants as features for navigation. As proof of concept, sky-segmented binary images were used as input for an established localisation algorithm SeqSLAM (Milford and Wyeth, 2012), validating the plausibility of this claim (Stone et al., 2014). A follow-up investigation sought to determine whether using the sky as a feature would help overcome image matching problems that the ant often faced, such as variance in tilt and yaw rotation. A robotic localisation study showed that using spherical harmonics (SH), a representation in the frequency domain, combined with extracted sky can greatly help robots localise on uneven terrain. Results showed improved performance to state of the art point feature localisation methods on fast bumpy tracks (Stone et al., 2016a). In the second part, an approach to understand how insects perform a navigational task called path integration was attempted by modelling part of the brain of the sweat bee Megalopta genalis. A recent discovery that two populations of cells act as a celestial compass and visual odometer, respectively, led to the hypothesis that circuitry at their point of convergence in the central complex (CX) could give rise to path integration. A firing rate-based model was developed with connectivity derived from the overlap of observed neural arborisations of individual cells and successfully used to build up a home vector and steer an agent back to the nest (Stone et al., 2016b). This approach has the appeal that neural circuitry is highly conserved across insects, so findings here could have wide implications for insect navigation in general. The developed model is the first functioning path integrator that is based on individual cellular connections.

Spatial navigation is a cognitive skill fundamental to successful interaction with our environment. Normal aging is associated with weaknesses in this skill, with severe deficits in the context of Alzheimer's disease. Identifying mechanisms underlying how the aged brain navigates is important to understanding these age-related weaknesses and potentially strengthening or preserving spatial navigation ability in the aging population. One understudied aspect of spatial navigation is self-motion perception. Important to self-motion perception is optic flow, which is the pattern of visual motion experienced while moving through our environment. Several brain regions are optic flow-sensitive (OF-sensitive), responding more strongly to optic flow than other types of visual motion. The goal of the experiments in this dissertation was to examine the role of visual motion perception and cortical motion area dynamics in spatial navigation in cognitively intact aged adults. Visual path integration tasks were used because they highlight the use of radial and translational optic flow to keep track of one’s position and orientation, respectively. In the first experiment, a positive relationship between radial optic flow sensitivity and visual path integration accuracy that was stronger in aged adults was found. In the second experiment, brain activity was measured using functional magnetic resonance imaging (fMRI) while participants performed visual path integration (VPI) and turn counting (TC) tasks. Stronger activity in the OF-sensitive regions LMT+ and RpVIP during VPI, not TC, was associated with greater VPI accuracy in aged adults. In the third experiment, the functional connectivity between OF-sensitive regions and the rest of the brain during the VPI and TC tasks was measured using fMRI. Stronger average functional connectivity between the OF-sensitive regions LMT+, RMT+, LpVIP, RpVIP, LpV6 and right supramarginal gyrus and posterior cingulate during VPI, not TC, was associated with greater VPI task accuracy in aged adults. The results demonstrate novel relationships between visual path integration accuracy and radial motion perception, the response of OF-sensitive cortical regions during visual navigation, and the interaction strength between OF-sensitive regions and parietal cortex during visual navigation in aged adults. This work expands our knowledge of mechanisms underlying spatial navigation processes in the aged human brain.

The goal of this thesis is to identify differences and consistencies in the trajectories taken by foraging agents before and after they have learned the location of a target. The challenge is that these agents do not go directly towards the target after learning and keep a certain amount of randomness in their paths. We use different versions of discrete curvature and head angle as tools in this analysis. We also build models of foraging agents using stochastic processes with data supported parameters.

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Diese Dissertation behandelt Fragen aus der stochastischen Analysis und der Finanzmathematik, die sich unter dem Begriff der Robustheit zusammenfassen lassen. Zunächst betrachten wir finanzmathematische Modelle mit Arbitragemöglichkeiten. Wir identifizieren die Abwesenheit von Arbitragemöglichkeiten der ersten Art (NA1) als minimale Eigenschaft, die in jedem finanzmathematischen Modell gelten muss, und zeigen, dass (NA1) äquivalent zur Existenz eines dominierenden lokalen Martingalmaßes ist. Als Beispiel für Prozesse, die (NA1) erfüllen, studieren wir stetige lokale Martingale, die darauf bedingt werden nie Null zu treffen. Anschließend verwenden wir eine modellfreie Version der (NA1) Eigenschaft, die es erlaubt, qualitative Eigenschaften von “typischen Preistrajektorien” zu beschreiben. Hier konstruieren wir ein pfadweises Itô-Integral. Dies deutet an, dass sich typische Preispfade als rough-path-Integratoren verwenden lassen. Nun entwickeln wir mittels Fourierentwicklungen einen alternativen Zugang zur rough-path-Theorie. Wir zerlegen das Integral in drei Operatoren mit verschiedenen Eigenschaften. So wird offensichtlich, dass Integratoren mit der Regularität der Brownschen Bewegung mit ihrer Lévy-Fläche versehen werden müssen, um ein pfadweise stetiges Integral zu erhalten. Daraufhin bemerken wir, dass die Integration zweier Funktionen gegeneinander äquivalent dazu ist, eine Funktion mit der Ableitung einer anderen (im Allgemeinen eine Distribution) zu multiplizieren. In höheren Dimensionen ist das Multiplikationsproblem jedoch allgemeiner. Wir verwenden Littlewood-Paley-Theorie, um unseren Fourier-Zugang zur rough-path-Theorie auf Funktionen mehrdimensionaler Variablen zu erweitern. Wir konstruieren einen Operator, der für Funktionen mit dem punktweisen Produkt übereinstimmt und in einer geeigneten Topologie stetig ist. Nun lassen sich stochastische partielle Differentialgleichungen lösen, die bisher aufgrund von Nichtlinearitäten nicht zugänglich waren. / This thesis deals with various problems from stochastic analysis and from mathematical finance that can best be summarized under the common theme of robustness. We begin by studying financial market models with arbitrage opportunities. We identify the weak notion of absence of arbitrage opportunities of the first kind (NA1) as the minimal property that every sensible asset price model should satisfy, and we prove that (NA1) is equivalent to the existence of a dominating local martingale. As examples of processes that satisfy (NA1) but do not admit equivalent local martingale measures, we study continuous local martingales conditioned not to hit zero. We continue by working with a model free formulation of the (NA1) property, which permits to describe qualitative properties of “typical asset price trajectories”. We construct a pathwise Itô integral for typical price paths. Our results indicate that typical price paths can be used as integrators in the theory of rough paths. Next, we use a Fourier series expansion to develop an alternative approach to rough path integration. We decompose the integral into three components with different behavior. Then it is easy to see that integrators with the regularity of the Brownian motion must be equipped with their Lévy area to obtain a pathwise continuous integral operator. We now note that integrating two functions against each other is equivalent to multiplying one with the derivative of the other, which will in general only be a distribution. In higher index dimensions however, the multiplication problem is more general. We use Littlewood-Paley theory to extend our Fourier approach from rough path integrals to multiplying functions of a multidimensional index. We construct an operator which agrees with the usual product for smooth functions, and which is continuous in a suitable topology. We apply this to solve stochastic partial differential equations that were previously difficult to access due to nonlinearities.

The hippocampal formation has been implicated in spatial formation for many decades. The hippocampus proper has received the most attention but other regions of the hippocampal formation contribute largely to spatial cognition. This thesis concentrated on one such region, the postsubiculum. The postsubiculum is considered important because it contains head direction cells and because it thought to be a major input to the hippocampus, via the entorhinal cortex. This thesis aims to test the functional role of the rat postsubiculum under two types of situation: one where the rat must rely on idiothetic cues for navigation, and another where the rat has visual cues present and can rely on these for orientation. The thesis also investigates hippocampal place cells and their stability over time after short exposures to novel environments. Chapter 3 of this thesis aimed to test whether the postsubiculum is necessary for path integration during a homing task. Rats were trained on a homing task on a circular platform maze. Once the task was acquired, rats were given lesions of the postsubiculum or sham lesions and then re-tested on the path integration task. The homing performance of rats with lesions of the postsubiculum was as good as that of the sham rats. A series of manipulations suggests that the rats were homing by path integration, confirmed by probe tests. The rats were then tested on a forced-choice delayed alternation T-maze task that revealed a significant impairment in alternation with delays of 5, 30, and 60 seconds. This suggests that the postsubiculum is not necessary for path integration in a homing task but is necessary for avoiding previously visited locations as is necessary in an alternation task. The experiments in Chapters 4 and 5 of this thesis aimed to investigate the effects of postsubiculum pharmacological inactivation on hippocampal CA1 place cells when rats were introduced to a novel environment with visual cues. A necessary first step was to assess place cells without any manipulation of the postsubiculum (Chapter 4) and then use information gained from this in the design of experiments in Chapter 5. Rats chronically implanted with recording electrodes in the CA1 region of the hippocampus were exposed to novel cue-rich environments whilst place fields were recorded. Following delays of 3, 6, or 24 hours, the same cells were recorded again in the same environment but with the cues rotated by 90°. Pixel-by-pixel correlations of the place fields show that stability of the place fields was significantly lower at 24 hours than at 3 hours. Stability after 6 hours was not significantly different from 3 hours. In the third set of experiments, rats were implanted with drug infusion cannulae in the postsubiculum and recording electrodes in CA1. Following infusions of either the AMPA receptor antagonist CXQX, the NMDA receptor antagonist D-AP5 or a control infusion of ACSF, place field stability was assessed as rats were exposed to a cylindrical environment with a single polarising cue card for 3 x 10 minute sessions and then again 6 hours later. There were no differences in place field correlations between the 3 drug conditions, although there was evidence of larger changes in spatial information content between cells in the CNQX and AP5 drug condition, but not the ACSF condition. The results suggest that, under the present testing conditions, place fields stability did not depend upon AMPA receptor-mediated transmission nor did it depend on NMDA receptor-mediated synaptic plasticity.

Die Wüstenameise, Cataglyphis, orientiert sich vor allem mittels Wegintegration. Über einen Heimvektor, den sie aus Distanz und Richtung einzelner Wegabschnitte berechnet, kann sie auf dem kürzesten Weg zu ihrem Ausgangspunkt zurückkehren. Zur Bestimmung der zurückgelegten Strecken verwendet sie einen Schrittintegrator. Die Laufrichtung wird hauptsächlich über Himmelsinformationen (Polarisationsmuster, Sonnenstand und Spektral- und Intensitätsgradienten) definiert. In dieser Arbeit über die Orientierungsfähigkeit von Cataglyphis fortis wird die Rolle des Polarisationskompasses bei der Bestimmung der Laufrichtung untersucht. In verschiedenen Versuchen wurden der Polarisationskompass der Ameise mit Hilfe eines Polarisationsfilters gezielt manipuliert und künstliche Konfliktsituationen erzeugt. Die Richtungsbestimmung wurde vom Polarisationskompass dominiert, wenn allein die Information des Polarisationskompass und idiothetische Richtungsinformation zur Verfügung standen. Erfuhren die Ameise widersprüchliche Informationen von Sonnen- und Polarisationskompass, berechneten sie eine mittlere Heimlaufrichtung, was eine gemeinsame neuronale Verarbeitung der beiden Signale voraussetzt. Diese These wurde durch Transferexperimente gestützt. In einer weiteren Versuchsreihe wurde die Wahrnehmung des Polarisationsmusters durch direkte Manipulation der entsprechenden Region (DRA) im Ameisenauge untersucht. Standen der Ameise in beiden Augen die frontale oder caudale DRA zur Verfügung führte dies zu einem deutlichen Orientierungsverlust. Die intakte DRA eines Auges erlaubte eine zielgerichtete Fortbewegung, die jedoch von der Erwartungsrichtung abwich. Zusammenfassend zeigen die Ergebnisse der vorliegenden Studie, dass der Polarisationskompass die präziseste Richtungsinformation liefert und den Himmelskompass der Wüstenameise dominiert. / The desert ant, Cataglyphis, navigates predominantly by means of path integration. The information about the distance and direction of individual path segments is integrated into a home vector, which allows the ant to return to the starting point on the shortest way. The distances covered are determined by a stride integrator. The heading direction is inferred mainly via celestial cues: the sky’s polarization pattern, the position of the sun, and the spectral and intensity gradient. This thesis focuses generally on the orientation abilities of Cataglyphis fortis and particularly on the role of the polarization compass to determine the heading direction. In the experiments, the ant’s polarization compass was selectively manipulated using a polarization filter and artificial cue conflict situations were created. The ants relied exclusively on the polarization compass to determine their heading direction if only idiothetic information and information from the polarization compass were available. When the ants experienced contradicting information detected via the sun and the polarization compass systems, an intermediate homing direction was calculated, suggesting a combined neural processing of both signals. This statement was supported by transfer experiments. In a further series of experiments, the input part of the polarization compass was manipulated. Particular regions of the ant’s eye (DRA) that detect polarized light were occluded. Ants with only the frontal or caudal parts of the DRA became disoriented, whereas ants with the entire DRA of one eye were able to perform more precise paths, although deviated from the expected direction. Overall, the results from this thesis suggest that the polarization compass provides the most accurate directional information and dominates the celestial compass system of the desert ant.