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In der Arbeit wird ein Testverfahren zum Prüfen der Varianzhomogenität der Lebenszeiten eines Erneuerungsprozesses entwickelt. Das Verfahren basiert auf der "Filtered-Derivative"-Methode. Zur Herleitung des Annahmebereichs werden zunächst Bootstrap-Permutationen genutzt, bevor zu einer asymptotischen Methode übergangen wird. Ein entsprechender funktionaler Grenzwertsatz wird skizziert. Aufbauend auf dem Test wird ein Multiple-Filter-Algorithmus zur genauen Detektion der Varianz-Change-Points besprochen. Schließlich folgt die Inklusion von vorher detektierten Ratenänderungen in das Verfahren. Der Test und der Algorithmus werden in Simulationsstudien evaluiert. Abschließend erfolgt eine Anwendung auf EEG-Daten.
Optimierung von Phasen- und Ratenparametern in einem stochastischen Modell neuronaler Feueraktivität
(2014)
In unserem Gehirn wird Information von Neuronen durch die Emission von Spikes repräsentiert. Als wichtige Signalkomponenten werden hierbei die Rate (Anzahl Spikes), die Phase (zeitliche Verschiebung der Spikes) und synchrone Oszillationen (rhythmische Entladungen der Neuronen am selben Zyklus) diskutiert.
In dieser Arbeit wird untersucht, wie Rate und Phase für eine optimale Detektion miteinander kombiniert werden und abhängig vom gewählten Parameterbereich wird der Beitrag der Phase quantifiziert.
Dies wird anhand eines stochastischen Spiketrain-Modell untersucht, das hohe Ähnlichkeiten zu empirischen Spiketrains zeigt und die drei genannten Signalkomponenten beinhaltet. Das ELO-Modell („exponential lockig to a free oscillator“) ist in zwei Prozessstufen unterteilt: Im Hintergrund steht ein globaler Oszillationsprozess, der unabhängige und normal-verteilte Intervallabschnitte hervorbringt (Oszillation). An den Intervallgrenzen starten unabhängig, inhomogene Poisson-Prozesse (Synchronizität) mit exponentiell abnehmender Feuerrate, die durch eine stimulusspezifische Rate und Phase festgelegt ist.
Neben einer analytischen Bestimmung der optimalen Parameter im Falle reiner Raten- bzw. Phasencodierung, wird die gemeinsame Codierung anhand von Simulationsstudien analysiert.
A stochastic model for the joint evaluation of burstiness and regularity in oscillatory spike trains
(2013)
The thesis provides a stochastic model to quantify and classify neuronal firing patterns of oscillatory spike trains. A spike train is a finite sequence of time points at which a neuron has an electric discharge (spike) which is recorded over a finite time interval. In this work, these spike times are analyzed regarding special firing patterns like the presence or absence of oscillatory activity and clusters (so called bursts). These bursts do not have a clear and unique definition in the literature. They are often fired in response to behaviorally relevant stimuli, e.g., an unexpected reward or a novel stimulus, but may also appear spontaneously. Oscillatory activity has been found to be related to complex information processing such as feature binding or figure ground segregation in the visual cortex. Thus, in the context of neurophysiology, it is important to quantify and classify these firing patterns and their change under certain experimental conditions like pharmacological treatment or genetical manipulation. In neuroscientific practice, the classification is often done by visual inspection criteria without giving reproducible results. Furthermore, descriptive methods are used for the quantification of spike trains without relating the extracted measures to properties of the underlying processes.
For that reason, a doubly stochastic point process model is proposed and termed 'Gaussian Locking to a free Oscillator' - GLO. The model has been developed on the basis of empirical observations in dopaminergic neurons and in cooperation with neurophysiologists. The GLO model uses as a first stage an unobservable oscillatory background rhythm which is represented by a stationary random walk whose increments are normally distributed. Two different model types are used to describe single spike firing or clusters of spikes. For both model types, the distribution of the random number of spikes per beat has different probability distributions (Bernoulli in the single spike case or Poisson in the cluster case). In the second stage, the random spike times are placed around their birth beat according to a normal distribution. These spike times represent the observed point process which has five easily interpretable parameters to describe the regularity and the burstiness of the firing patterns.
It turns out that the point process is stationary, simple and ergodic. It can be characterized as a cluster process and for the bursty firing mode as a Cox process. Furthermore, the distribution of the waiting times between spikes can be derived for some parameter combination. The conditional intensity function of the point process is derived which is also called autocorrelation function (ACF) in the neuroscience literature. This function arises by conditioning on a spike at time zero and measures the intensity of spikes x time units later. The autocorrelation histogram (ACH) is an estimate for the ACF. The parameters of the GLO are estimated by fitting the ACF to the ACH with a nonlinear least squares algorithm. This is a common procedure in neuroscientific practice and has the advantage that the GLO ACF can be computed for all parameter combinations and that its properties are closely related to the burstiness and regularity of the process. The precision of estimation is investigated for different scenarios using Monte-Carlo simulations and bootstrap methods.
The GLO provides the neuroscientist with objective and reproducible classification rules for the firing patterns on the basis of the model ACF. These rules are inspired by visual inspection criteria often used in neuroscientific practice and thus support and complement usual analysis of empirical spike trains. When applied to a sample data set, the model is able to detect significant changes in the regularity and burst behavior of the cells and provides confidence intervals for the parameter estimates.
A multiple filter test for the detection of rate changes in renewal processes with varying variance
(2014)
The thesis provides novel procedures in the statistical field of change point detection in time series.
Motivated by a variety of neuronal spike train patterns, a broad stochastic point process model is introduced. This model features points in time (change points), where the associated event rate changes. For purposes of change point detection, filtered derivative processes (MOSUM) are studied. Functional limit theorems for the filtered derivative processes are derived. These results are used to support novel procedures for change point detection; in particular, multiple filters (bandwidths) are applied simultaneously in oder to detect change points in different time scales.
Neuronal activity in the brain is often investigated in the presence of stimuli, termed externally driven activity. This stimulus-response-perspective has long been focussed on in order to find out how the nervous system responds to different stimuli. The neuronal response consists of baseline activity, so called spontaneous activity1, and activity which is caused by the stimulus. The baseline activity is often considered as constant over time which allows the identification of the stimulus-evoked part of the neuronal response by averaging over a set of trials.
However, during the last years it has been recognized that own dynamics of the nervous system plays an important role in information processing. As a consequence, spontaneous activity is no longer regarded only as background ’noise’ and its role in cortical processing is reconsidered. Therefore, the study of spontaneous firing pattern gains more importance as these patterns may shape neuronal responses to a larger extent as previously thought. For example, recent findings suggest that prestimulus activity can predict a person’s visual perception performance on a single trial basis (Hanslmayr et al., 2007). In this context, Ringach (2009) remarks that one can learn much about even the quiescent state of the brain which “underlies the importance of understanding cortical responses as the fusion of ongoing activity and sensory input”.
Taking into account that spontaneous activity reflects anything else but noise, new challenges arise when analysing neuronal data. In this thesis one of these problems related to the analysis of neuronal activity will be adressed, namely the nonstationarity of firing rates.
The present work consists of four chapters. First of all the introduction gives neurophysiological background information to get an idea of neuronal information processing. Afterwords the theory of point processes is provided which forms the basis for modeling neuronal spiking data. In the last section of the introduction a statement of the problem is given. Chapter 2 proposes an easily applicable statistical method for the detection of nonstationarity. It is applied to simulations and to real data in order to show its capabilities. Thereafter, four other approaches are presented which provide useful illustrations concerning the nonstationarity of the firing rate but share the problem that one cannot make objective statements on the basis of their results. They were developed in the course of establishing a suitable method. In chapter 4 the results are discussed and suggestions for further study are given.
This work proposes to employ the (bursty) GLO model from Bingmer et. al (2011) to model the occurrence of tropical cyclones. We develop a Bayesian framework to estimate the parameters of the model and, particularly, employ a Markov chain Monte Carlo algorithm. This also allows us to develop a forecasting framework for future events.
Moreover, we assess the default probability of an insurance company that is exposed to claims that occur according to a GLO process and show that the model is able to substantially improve actuarial risk management if events occur in oscillatory bursts.