Speaker
Description
Cell biological systems are characterized by complex relationships and non-linear processes. The modelling of these processes improves the understanding and represents a significant enrichment of the experimental investigation. An example of such a system is the regulation of blood glucose concentration by pancreatic β-cells through the secretion of insulin. β-cells are electrical active and insulin secretion is regulated by an interplay of metabolic and electrophysiological components, resulting in a membrane potential change between silent and active burst phases.
Mathematically, this behavior can be described by a set of ordinary non-linear differential equations. In the electrical system, different types of bifurcations occur as the ATP concentration varies, which links the metabolic and electrical activity. The state of the system changes from a stable equilibrium to a limit cycle and back again. The transition from the limit cycle to the equilibrium point is characterized by an increase in period duration, which is due to the type of bifurcation, the merging of a limit cycle with a saddle point.
The change in the period duration can be approximated comparatively by the eigenvalue analysis of the saddle point as well as by the harmonic balance method with respect to the limit cycle. Small changes in the bifurcation parameter, i.e., the ATP concentration, have a very strong effect on the system behavior. To gain a better understanding, it is useful to separate the metabolic and electrical activity in experiments and thus in simulations. However, the ATP concentration in an experimentally investigated cell can also vary greatly during a measurement or cannot be accurately determined by the measurement method.
In order to take this variation into account, an uncertainty-based view of the problem is recommended. The uncertainty of the amplitude and period of the limit cycle is quantified by combining the harmonic balance method with the generalized polynomial chaos expansion. This spectral stochastic method achieves a significant reduction in computational time compared to the most commonly used Monte Carlo method. The intrusive and non-intrusive approach of the combined method as well as the approximation of the period duration are compared regarding their efficiency and accuracy on the example of the β-cell.
