A Lin's method approach to heteroclinic connections involving periodic orbits - Analysis and numerics

The topic of the thesis is the bifurcation analysis of heteroclinic cycles connecting a hyperbolic equilibrium and a hyperbolic periodic orbit. An extension of Lin's method that is based on the coupling of the global continuous system and the discrete system that describes the dynamics near the periodic orbit is developed in the first part. The method allows to formulate bifurcation equations for the given scenario. The bifurcation equations for homoclinic orbits to the equilibrium and for homoclinic orbits to the periodic orbit are qualitatively solved and discussed, the case of a quadratic tangency is also considered. In the second part a numerical method based on the theoretical results is developed that allows to find and continue in parameters a heteroclinic connection between an equilibrium and a periodic orbit. The method is demonstrated on three selected examples and the theoretical results from the first part are verified. An extension of this numerical method to orbits that connect two periodic orbits is also given.Die Dissertationsschrift beschäftigt sich mit der Bifurkationsanalyse von heteroklinen Zyklen, die eine hyperbolische Gleichgewichtslage und einen hyperbolischen periodischen Orbit miteinander verbinden. Im ersten Teil der Arbeit wird eine Erweiterung von Lins Methode entwickelt, die auf einer Kopplung des globalen kontinuierlichen Systems und des diskreten Systems, das die Dynamik in der Umgebung des periodischen Orbits beschreibt, beruht. Die Methode erlaubt es, Bifurkationsgleichungen für das gegebene Szenario zu formulieren. Die Bifurkationsgleichungen für homokline Orbits an die Gleichgewichtslage und homokline Orbits an den periodischen Orbit werden qualitativ gelöst und diskutiert, ebenso wird der Fall einer quadratischen Berührung behandelt. Im zweiten Teil der Dissertation wird auf Basis der theoretischen Ergebnisse eine numerische Methode entwickelt, die es erlaubt, verbindende Orbits zwischen Gleichgewichtslagen und periodischen Orbits zu finden und im Parameterraum zu verfolgen. Die Methode wird an drei ausgewählten Beispielen demonstriert, dabei werden die theoretischen Ergebnisse aus dem ersten Teil der Arbeit bestätigt. Eine Erweiterung der numerischen Methode auf verbindende Orbits zwischen zwei hyperbolischen periodischen Orbits wird abgeleitet

Lin's method for heteroclinic chains involving periodic orbits

We present an extension of the theory known as Lin's method to heteroclinic chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based on the construction of a so-called Lin orbit, that is, a sequence of continuous partial orbits that only have jumps in a certain prescribed linear subspace, estimates for these jumps are derived. We use the jump estimates to discuss bifurcation equations for homoclinic orbits near heteroclinic cycles between an equilibrium and a periodic orbit (EtoP cycles)

Numerical study of secondary heteroclinic bifurations near non-reversible homoclinic snaking

We discuss the emergence of isolas of secondary heteroclinic bifurcations near a non-reversible homoclinic snaking curve in parameter space that is generated by a codimension-one equilibrium-to-periodic (EtoP) heteroclinic cycle. We use a numerical method based on Lin's method to compute and continue these secondary heteroclinic EtoP orbits for a well-known system

Full System Bifurcation Analysis of Endocrine Bursting Models

Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. For example, in insulin-secreting pancreatic β-cells, plateau bursting is characterized by well-defined spikes during the depolarized phase whereas in pituitary cells, bursting features fast, irregular, small amplitude spikes. The latter has been termed “pseudo-plateau bursting ” because the spikes are transients around a depolarized steady state rather than stable oscillations in the fast subsystem. In this study we systematically investigate the bursting patterns found in endocrine cell models. We show that this class of voltage and calcium gated conductance based models can be reduced to the polynomial model of Hindmarsh and Rose (25). This reduction preserves the main properties of the biophysical class of models that we consider and allows for detailed bifurcation analysis of the full fast-slow system. Our analysis does not require decomposition of the full system into fast and slow subsystems and reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis

Intracellular Tracking of Single-plasmid DNA Particles After Delivery by Electroporation

Electroporation is a physical method of transferring molecules into cells and tissues. It takes advantage of the transient permeabilization of the cell membrane induced by electric field pulses, which gives hydrophilic molecules access to the cytoplasm. This method offers high transfer efficiency for small molecules that freely diffuse through electrically permeabilized membranes. Larger molecules, such as plasmid DNA, face several barriers (plasma membrane, cytoplasmic crowding, and nuclear envelope), which reduce transfection efficiency and engender a complex mechanism of transfer. Our work provides insight into the way electrotransferred DNA crosses the cytoplasm to reach the nucleus. For this purpose, single-particle tracking experiments of fluorescently labeled DNA were performed. Investigations were focused on the involvement of the cytoskeleton using drugs disrupting or stabilizing actin and tubulin filaments as the two relevant cellular networks for particle transport. The analysis of 315 movies (~4,000 trajectories) reveals that DNA is actively transported through the cytoskeleton. The large number of events allows a statistical quantification of the DNA motion kinetics inside the cell. Disruption of both filament types reduces occurrence and velocities of active transport and displacements of DNA particles. Interestingly, stabilization of both networks does not enhance DNA transport

A versatile framework for the analysis of high-throughput screening data

Mitosis is an essential process within the cell life cycle, and research in this field has many applications in medicine. In order to study the proteins involved in mitosis and their function, small molecules (compounds) are required which inhibit the protein under investigation and hence allow to switch off the respective protein. For this purpose, high-throughput screening is performed, where thousands of compounds from commercially available libraries are probed using optical readouts. The vast amounts of data generated in high-throughput screening require dedicated data analysis and visual analytics approaches for evaluation. In this work, a versatile framework is presented which provides data preprocessing and visualization approaches for the analysis of high-throughput screening data

Image Analysis of Nuclear Envelope Breakdown Events using KNIME

Insights into the molecular mechanism underlying nuclear envelope breakdown (NEBD) are crucial to understand dynamic changes of the nuclear envelope (NE) that occur at mitotic entry of eukaryotic cells. In this paper, we present an image processing algorithm and its implementation in KNIME, which allows to automatically quantify image data obtained from the in vitro NEBD assay. The algorithm consists of image alignment via phase correlation, nucleus identification using adaptive thresholding, morphological operations, and intensity measurements. The results of these measurements are verified using images from a known nuclear envelope disassembly assay