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.

Neuronal networks produce reliable functional output throughout the lifespan of an animal despite ceaseless molecular turnover and a constantly changing environment. The cellular and molecular mechanisms underlying the ability of these networks to maintain functional stability remain poorly understood. Central pattern generating circuits produce a stable, predictable rhythm, making them ideal candidates for studying mechanisms of activity maintenance. By identifying and characterizing the regulators of activity in small neuronal circuits, we not only obtain a clearer understanding of how neural activity is generated, but also arm ourselves with knowledge that may eventually be used to improve medical care for patients whose normal nervous system activity has been disrupted through trauma or disease. We utilize the pattern-generating pyloric circuit in the crustacean stomatogastric nervous system to investigate the general scientific question: How are specific aspects of rhythmic activity regulated in a small neuronal network? The first aim of this thesis poses this question in the context of a single neuron. We used a single-compartment model neuron database to investigate whether co-regulation of ionic conductances supports the maintenance of spike phase in rhythmically bursting “pacemaker” neurons. The second aim of the project extends the question to a network context. Through a combination of computational and electrophysiology studies, we investigated how the intrinsic membrane conductances of the pacemaker neuron influence its response to synaptic input within the framework of the Phase Resetting Curve (PRC). The third aim of the project further extends the question to a systems-level context. We examined how ambient temperatures affect the stability of the pyloric rhythm in the intact, behaving animal. The results of this work have furthered our understanding of the principles underlying the long-term stability of neuronal network function.

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