Effects of Repulsive Coupling in Ensembles of Excitable Elements

Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berücksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und Stabilität periodischer Zwei-Cluster-Lösungen für generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen für eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie für solche Systeme. Mit dieser können wir Rückschlüsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen für das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "Stabilitätstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild für solch kompliziertere und realistischere Neuronenmodelle liefern.We study the collective dynamics of class I excitable elements, which can be described within the theory of nonlinear dynamics as systems close to a saddle-node bifurcation on an invariant circle. The focus of the thesis lies on the study of active rotators as a prototype for such elements. In part one of the thesis, we motivate the classic model of repulsively coupled active rotators by Shinomoto and Kuramoto and generalize it by considering higher-order Fourier modes in the on-site dynamics of the rotators. We also discuss the mathematical methods which our work relies on, in particular the concept of Watanabe-Strogatz (WS) integrability which allows to describe systems of identical angular variables in terms of Möbius transformations. In part two, we investigate the existence and stability of periodic two-cluster states for generalized active rotators and prove the existence of a continuum of periodic orbits for a class of WS-integrable systems which includes, in particular, the classic active rotator model. We show that (i) this continuum constitutes a normally attracting invariant manifold and that (ii) one of the solutions yields splay state dynamics. We then develop a perturbation theory for such systems, based on the averaging method. By this approach, we can deduce the asymptotic dynamics of the generalized active rotator model. As a main result, we find that periodic two-cluster states and splay states are robust periodic solutions for systems of identical active rotators. We also investigate a 'transfer of stability' between these solutions by means of so-called broken-symmetry states. In part three, we study ensembles of higher-dimensional class I excitable elements in the form of Morris-Lecar neurons and find the asymptotic dynamics of such systems to be similar to those of active rotators, which suggests that our results from part two yield a suitable qualitative description for more complicated and realistic neural models

“When Will the Time Finally Come When There Will Only Be Humans”: An Unknown Letter from Beethoven to Heinrich von Struve

Transcription, translation, and discussion of a recently discovered letter from Ludwig van Beethoven to Heinrich von Struve from 1795. In the letter, Beethoven expresses a yearning for a time when there are only humans, without distinctions of rank and class. This sentiment accords with Schiller’s ode An die Freude, which Beethoven set to music more than once

Extracting Hadamard Coefficients from Green's Operators

We develop a formula for the diagonal values of the Hadamard coefficients associated to a normally hyperbolic operator on a globally hyperbolic spacetime in terms of the advanced and retarded Green's operators. We develop a local formula as well as formulae for integrals over (parts of) the diagonal. Furthermore, we develop analogues of the Hadamard expansion for powers of the advanced/retarded Green's operators and an analogue of a resolvent.Comment: This is the Author's Ph. D. thesis (with minor typo corrections), advised by Prof. Matthias Lesch and unofficially also advised by Dr. Koen van den Dungen. A shorter and more easily readable version of the part on additional expansions is in the makin

Characterization of the mechanical properties and microstructural evolution of martensitic steel in repeated tempering cycles

The purpose of this study was to understand the behavior of martensitic H13 steel in accordance with the microstructural evolution, mechanical properties and wear in repeated tempering cycles. The microstructures were characterized by axio image observer microscope, scanning electron microscope (SEM), x-ray diffraction (XRD). Uniaxial tensile test, charpy v-notch impact test, rockwell hardness test and wear test were conducted to analyze the changes in mechanical properties, impact properties, hardness and wear in repeated tempering cycles. The specimen prepared were subjected to the hardening at 1030 °C for 20 minutes, oil quenched and subjected to repeated tempering cycles at 570 °C for 2hrs holding time each. The mechanical properties recorded indicates that the maximum ultimate tensile strength obtained was at double tempering due to secondary hardening effect i.e., alloy carbides precipitation offering strength to the matrix and corresponding wear was found to be minimum. The annealed specimen revealed bainitic microstructure and with hardening and repeated tempering cycles, fine needle like structure and carbides was observed in microstructure and retained austenite was converted into martensite and martensite was converted into tempered martensite. Carbide size and martensite lath distribution controls the strength and fracture rate

Plio-Pleistocene variability of the East Pacific Thermocline and the Intertropical Convergence Zone

The transition from the Pliocene to the Pleistocene was accompanied by major tectonic reorganizations of key oceanic gateways. In particular, the gradual closure of the Panama Gateway and the constriction of the Indonesian Gateway significantly affected the structure of the Pacific thermocline. In the East Pacific, the thermocline shoaled from an early Pliocene El Niño‐like depth to its modern state, which had significant implications for global climate. Here we use Mg/Ca temperature estimates from subsurface and thermocline dwelling foraminifera to reconstruct the meridional Plio‐Pleistocene evolution of the Southeast Pacific thermocline, in relation to atmospheric circulation changes. In combination with similar reconstructions from the north‐equatorial Pacific, our data indicate a change in the thermocline, responding to the northward displacement of the Intertropical Convergence Zone/South Pacific High system between ~3.8 and 3.5 Ma. After 3.5 Ma, we record a second major phase of thermocline shoaling, which points to the Intertropical Convergence Zone/South Pacific High‐system movement toward its modern position along with the gradual cooling of the Northern Hemisphere and its associated glaciation. These findings highlight that a warming globe may affect equatorial regions more intensively due to the potential temperature‐driven movement of the Intertropical Convergence Zone/South Pacific High and their associated oceanic systems

Continuous phase stabilization and active interferometer control using two modes

We present a computer-based active interferometer stabilization method that can be set to an arbitrary phase difference and does not rely on modulation of the interfering beams. The scheme utilizes two orthogonal modes propagating through the interferometer with a constant phase difference between them to extract a common phase and generate a linear feedback signal. Switching times of 50ms over a range of 0 to 6 pi radians at 632.8nm are experimentally demonstrated. The phase can be stabilized up to several days to within 3 degrees.Comment: 3 pages, 2 figure