Unveiling Mechanisms for Electrical Activity Patterns in Neurons and Pituitary Cells Using Mathematical Modeling and Analysis

Computational neuroscience is a relatively new area that utilizes the computational analyses of neural systems as well as development of mathematical models. Analyses of neural systems help us to gain a deeper understanding of how different dynamical variables contribute to generate a given electrical behavior and modelling helps to explain experimental results or make predictions that can be tested experimentally. Due to the complexity of nervous system behavior, mathematical models often have many variables, however simpler lower-dimensional models are also important for understanding complex behavior. The work described herein utilizes both approaches in two separate, but related, studies in computational neuroscience. In the first study, we determined the contributions of two negative feedback mechanisms in the Hodgkin-Huxley model. Hodgkin and Huxley pioneered the use of mathematics in the description of an electrical impulse in a squid axon, developing a differential equation model that has provided a template for the behavior of many other neurons and other excitable cells. The Hodgkin-Huxley model has two negative feedback variables. The activation of a current (n), subtracts from the positive feedback responsible for the upstroke of an impulse. We call this subtractive negative feedback. Divisive feedback is provided by the inactivation of the positive feedback current (h), which divides the current. Why are there two negative feedback variables when only one type of negative feedback can produce rhythmic spiking? We detect if there is any advantage to having both subtractive and divisive negative feedback in the system and the respective contributions of each to rhythmic spiking by using three different metrics. The first measures the width of a parameter regime within which tonic spiking is a unique and stable limit cycle oscillation. The second metric, contribution analysis, measures how changes in the time scale parameters of the feedback variables affect the durations of the "active phase" during the action potential and the inter-spike interval "silent phase" of a tonically spiking model. The third metric, dominant scale analysis, measures a sensitivity of the voltage dynamics to each of the ionic currents and ranks their influence. xi In the second study, we used electrophysiology data provided from the collaborating lab of Mike Shipston combined with mathematical modelling to show how two different neurohormones regulate patterns of electrical activity in corticotrophs. Corticotroph cells of the anterior pituitary are electrically excitable cells and are an integral component of the stress the neuroendocrine response to stress. Stress activates neurons in the hypothalamus to release corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP). These neurohormones act on corticotrophs in the anterior pituitary gland, which secrete another hormone, adrenocorticotropic hormone (ACTH). ACTH enters the general circulation and stimulates the adrenal cortex to secrete corticosteroid (cortisol in humans). Corticotrophs display single spike activity under basal conditions which can be converted to complex bursting behavior after stimulation by the combination of CRH and AVP. Bursting is much more effective at releasing ACTH than is spiking, so this transition is physiologically important. We investigated the underlying mechanisms controlling this transition to bursting by mathematical modelling combined with the experimental data. The significance of the work in this dissertation is that it provides a very good example of how experiments and modelling can complement each other and how the right mathematical tools can increase our understanding of even a very old and much studied model.