Stochastic compartment models are widely used in modeling processes for biological populations. The residence time has been especially useful in describing the system dynamics in the models. The direct calculation of the distribution for the residence time of stochastic multi-compartment models is very complicated even with a relatively simple model and often impossible to calculate directly. This dissertation presents an analytical method to obtain the moment generating function for stochastic multi-compartment models and describe the distribution of the residence times, especially systems with nonexponential lifetime distributions. A common method for obtaining moments of the residence time is using the coefficient matrix, however it has a limitation in obtaining high order moments and moments for combined compartments in a system. In this dissertation, we first derive the bivariate moment generating function of the residence time distribution for stochastic two-compartment models with general lifetimes. It provides any order of moments and also enables us to approximate the density of the residence time using the saddlepoint approximation. The approximation method is applied to various situations including the approximation of the bivariate distribution of residence times in two-compartment models or approximations based on the truncated moment generating function. Special attention is given to the distribution of the residence time for multi-compartment semi-Markov models. The cofactor rule and the analytic approach to the two-compartment model facilitate the derivation of the moment generating function. The properties from the embedded Markov chain are also used to extend the application of the approach. This approach provides a complete specification of the residence time distribution based on the moment generating function and thus provides an easier calculation of high-order moments than the approach using the coefficient matrix. Applications to drug kinetics demonstrate the simplicity and usefulness of this approach.

Diese Arbeit beschäftigt sich sowohl mit der experimentell motivierten Fragestellung nach der Kontrolle der Einzelzellaktivität kortikaler Neurone sowie mit der theoretischen Beschreibung der neuronalen Dynamik und ihrer Transfereigenschaften anhand einfacher Neuronenmodelle. Hierfür werden in-vivo Daten, die mit Hilfe der juxtazellulären Stimulation mit weißem bandpass limitiertem Gaußschem Rauschen erhoben wurden, verwendet. Mit Parameterfits einfacher Neuronenmodelle werden die experimentell ermittelten Pulszugstatistiken sowie die präzisen Zeitpunkte der einzelnen Aktionspotentiale quantitativ reproduziert. Diese Untersuchungen zeigen, dass mit dynamischen Rauschstimuli in juxtazellulärer Stimulation verlässlich und reproduzierbar Pulszüge in einzelnen kortikalen Neuronen hervorgerufen werden können. Weiterhin offenbart die Analyse der Daten die Eigenschaft der untersuchten Neurone frequenzunabhängig, bishin zu Vielfachen der Feuerrate des Neurons, Information über Signalkomponenten zu transferieren. Diese Eigenschaft steht im Widerspruch zum Verhalten der einfachsten (und populärsten) integrate-and-fire Modelle, die die Zelle ohne Auflösung ihrer räumlichen Struktur näherungsweise beschreiben. Die Erweiterung solcher Ein-Kompartiment Modelle auf ein Zwei-Kompartiment Modell und die damit eingeführte Unterscheidung zwischen Soma und Dendrit ermöglicht es, für einzelne Neuronen sämtliche experimentell erhobenen Statistiken, einschließlich des Hochfrequenz- Transfers, quantitativ zu reproduzieren. Zusätzlich zu den obigen Untersuchungen wird eine Methode vorgestellt, um, anhand von Input-Output Statistiken konkreter Neurone, Gaußsche Stimuli zu berechnen, die in der jeweiligen Zelle einen vorgeschriebenen Pulszug hervorrufen. In Experimenten und Simulationen wird gezeigt, dass diese vorgeschriebenen Pulszüge mit einer Verlässlichkeit erzeugt werden können, die in etwa der intrinsischen Verlässlichkeit des untersuchten Neurons entspricht. / This work elaborates on the question to which extent experimental control about the activity of single cortical neurons can be achieved and deals with the theoretical description of the neuronal dynamics. To this end, in-vivo data that have been recorded from juxtacellular experiments in cortical neurons are used. By means of parameter optimization, simple neuron models are fitted in order to quantitatively reproduce the measured spike train statistics and specific action potential timings. The analysis reveals that dynamic noise-stimuli can be used in juxtacellular stimulation to reliably generate reproducible spike trains in single cortical neurons. The analysis also reveals that the cells show a marked broadband coding of information, up to frequencies that are multiples of the firing rate of the respective neuron. This is in contrast to what is known for the simplest (and most popular) integrate-and-fire models, for which the cellular dynamics are described by a single space-independent variable. The extension of these one-compartment models to two-compartment models introduces a spatially distinction between soma and dendrite and we could show that for particular neurons it is sufficient to quantitatively reproduce all experimentally measured spike-train and input-output statistics, including the highfrequency information-transfer. Therefore, the effect of the spatial structure can be an important (structural) mechanism that can have influence on the neuronal dynamics. Additionally to the above considerations, by means of input-output statistics of particular neurons, we propose a method to compute Gaussian stimuli that are supposed to evoke prescribed spike trains in the respective neuron. Using experiments and simulations, we show that the prescribed spike trains can be evoked with a reliability that is comparable to the intrinsic reliability of the neuron under investigation.