The models that describe endocrine cells bursting activity typically involve different time scales, but a standard bifurcation analysis of the fast time limit does not capture some phenomena. In particular termination of the active phase and spike adding are still not very well understood. Furthermore, dynamical systems tools are designed to explain the long-term bahaviour of a system, that is, what happens after transients have died out. In many situations, however, it may be important to understand the transient rather than sustained (asymptotic) behaviour. In this talk we employ standard tools from dynamical systems in order to analyse sustained and transient bursting behaviour. We show how manifolds of the fast subsystem are involved in the termination of the active phase and investigate these models further by performing bifurcation analysis of the full fast-slow system. We take a geometric approach to illustrate how the underlying fast subsystem organises the spike adding in much the same way as for spike adding in sustained periodic bursts, but the bifurcation analysis for spike onset is entirely different. Our results highlight the similarities between spike adding as a transient phenomenon and spike adding for periodic bursting, because the transients are organised by the same underlying geometric structure.
Beta cells in pancreatic islets release the hormone insulin in response to elevated glucose. Gap junctional coupling between these cells tends to synchronize bursting, which drives release. However, heterogeneity within the islet leads to counterintuitive emergent bursting behavior. We have developed the computational islet, a three-dimensional network, to capture the impact of variation in cells and their connectivity, glucose exposure, and coupling strength on bursting. Numerical advances have sped up computation of 1000-cell islet by up to two orders of magnitude. We exhibit these for a 7-variable and a 3-variable model. We show that the peak of a non-monotonic average burst period vs. coupling strength curve depends monotonically on a connectivity measure of slow and fast cells in each model. In certain situations we can get emergent bursting that is slower than the slowest cells bursting alone, which has been observed in coupled cell models, and is connectivity dependent. Finally, we show that the computational islet supports waves of calcium in a model of high glucose flow on the edge of an islet reminiscent of experiments.
This study presents systems of delayed differential equations which predict serum concentrations of hormones important for regulation of the menstrual cycle. Parameters for the systems are fit to two different data sets for normally cycling women. For these best-fit parameter sets, simulations for the two models agree well with the data but one model also has a stable periodic solution representing an abnormal menstrual cycle. Differences in model behavior are explained by studying hysteresis curves in bifurcation diagrams with respect to sensitive model parameters. For instance, one sensitive parameter is indicative of the estradiol concentration that promotes pituitary synthesis of a large amount of luteinizing hormone, which is required for ovulation. The model may be extended to normally cycling women from age 20 to age 50 by including the pool of primordial follicles that a woman is born with and its natural decline with age. Model simulations show that this decline may be slowed by the administration of exogenous antimüllerian hormone resulting in a delay in the onset of menopause as measured by the number of primordial follicles remaining in the ovaries.
The magnocellular vasopressin neurons of the hypothalamus form part of the homeostatic systems that maintain osmotic pressure. Experiments show a robust linear relationship between osmotic pressure and vasopressin hormone secretion despite the very non-linear properties of spike generation and stimulus-secretion coupling in the neurons. Using simulated synaptic inputs to encode osmotic pressure, we have coupled an integrate and fire based spiking model to a model of the secretion mechanism in order to investigate how the properties of the spiking and secretion mechanisms combine to shape the secretion response, and how this combines with the neurons acting as a heterogeneous population.