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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.

Poster presentation from Twentieth Annual Computational Neuroscience Meeting: CNS*2011 Stockholm, Sweden. 23-28 July 2011. In statistical spike train analysis, stochastic point process models usually assume stationarity, in particular that the underlying spike train shows a constant firing rate (e.g. [1]). However, such models can lead to misinterpretation of the associated tests if the assumption of rate stationarity is not met (e.g. [2]). Therefore, the analysis of nonstationary data requires that rate changes can be located as precisely as possible. However, present statistical methods focus on rejecting the null hypothesis of stationarity without explicitly locating the change point(s) (e.g. [3]). We propose a test for stationarity of a given spike train that can also be used to estimate the change points in the firing rate. Assuming a Poisson process with piecewise constant firing rate, we propose a Step-Filter-Test (SFT) which can work simultaneously in different time scales, accounting for the high variety of firing patterns in experimental spike trains. Formally, we compare the numbers N1=N1(t,h) and N2=N2(t,h) of spikes in the time intervals (t-h,t] and (h,t+h]. By varying t within a fine time lattice and simultaneously varying the interval length h, we obtain a multivariate statistic D(h,t):=(N1-N2)/V(N1+N2), for which we prove asymptotic multivariate normality under homogeneity. From this a practical, graphical device to spot changes of the firing rate is constructed. Our graphical representation of D(h,t) (Figure 1A) visualizes the changes in the firing rate. For the statistical test, a threshold K is chosen such that under homogeneity, |D(h,t)|<K holds for all investigated h and t with probability 0.95. This threshold can indicate potential change points in order to estimate the inhomogeneous rate profile (Figure 1B). The SFT is applied to a sample data set of spontaneous single unit activity recorded from the substantia nigra of anesthetized mice. In this data set, multiple rate changes are identified which agree closely with visual inspection. In contrast to approaches choosing one fixed kernel width [4], our method has advantages in the flexibility of h.