DE Seminars: Spring 2013

Monday March 11

Abstract:
Computational models are an important tool for determing dendritic properties as well as for understanding their functional roles. However, these models are limited by simulation time and storage requirements, particularly when modeling neuronal networks. We review reduced models of the neuron that accurately report the transmembrane potential at a few specified locations while retaining dendritic properties, including the spatial distribution of synaptic inputs throughout the dendritic tree. These models are rooted in two classes of methods from linear algebra: methods based on the singular value decomposition and moment-matching methods. The reduced models can be used to further elucidate dendritic function as they greatly reduce the computational cost associated with simulating networks of morphologically accurate neurons. We demonstrate this capability by simulating a network of hippocampal pyramidal cells and interneurons coupled through chemical synapses and electrical gap junctions.

Monday April 1

Abstract:
We consider a hyperbolic/parabolic rod model introduced by Antman and Seidman (’96). The original hypotheses for this model provided more regularity than is consistent with impact, which implies a discontinuous boundary condition for the velocity. In the context of the weakened regularity, the needed compactness is obtained by use of the Nash/Moser estimates. The increased generality allowed by the new hypotheses can be expected to permit not only treatment of impact, as here, but also modeling of colliding rods and improved results on optimal control.

Monday April 8

Abstract:
Glucose homeostasis is largely controlled by alpha and beta cells located in pancreatic islets. Both of these cell types respond to blood glucose levels, albeit in an opposite manner. While beta cells secrete insulin when blood glucose is elevated, alpha cells secrete glucagon when blood glucose levels are low. The mechanism by which glucose induces insulin secretion in beta cells is fairly well understood. Despite years of research, however, the mechanism of glucagon secretion is still not well established. It has been proposed that glucose can affect the alpha cell through effects on the conductance of ATP-dependent potassium channels (K (ATP)). An alternative theory asserts that glucose regulates glucagon secretion through a store-operated current (SOC). We developed a mathematical model to test these hypotheses and found that both mechanisms are possible. Glucose can suppress glucagon secretion by closing K(ATP) channels, which depolarizes the cells but paradoxically reduces calcium influx by inactivating voltage-dependent calcium channels. Glucose can also suppress glucagon secretion by activating SERCA pumps, which fills the endoplasmic reticulum and hyperpolarizing the cells by reducing the inward current through SOC channels. We suggest that both mechanisms contribute and that they can combine to account for the non-monotonic dependence of secretion on glucose observed in some studies.

Monday April 15

Abstract:
The goal of this talk is to provide an introduction to finite element methods for partial differential equations with input data affected by uncertainty, described by random variables and random fields. Specifically, I will present the basic theoretical and algorithmic aspects of the Stochastic Galerkin and Stochastic Collocation finite element methods for an elliptic model problem.

Monday April 22

Abstract:
We consider the Restricted Planar Elliptic 3 Body Problem, which models the Sun, Jupiter and an Asteroid (which we assume that has negligible mass). We take a realistic value of the mass ratio between Jupiter and the Sun and their eccentricity arbitrarily small and we study instability mechanisms that arise in the mean motion resonances, namely when the period of the Asteroid is approximately resonant with the period of Jupiter. It is well known that if one neglects the influence of Jupiter on the Asteroid, the orbit of the latter is an ellipse. In this talk we will show how the influence of Jupiter may cause a substantial change on the shape of Asteriod’s orbit. This instability mechanism may give an explanation of the existence of the Kirkwood gaps in the Asteroid belt. This is a joint work with J. Fejoz, V. Kaloshin and P. Roldan.

Monday April 29

Abstract:
We consider a parallel matrix-free implicit finite volume scheme for the solution of three-dimensional advection-diffusion-reaction equations with smooth and Dirac-Delta source terms. The scheme is formally second order in space and a Newton-Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix-vector product required is hardcoded without any approximations, obtaining a matrix-free method that needs little storage and is well suited for parallel implementation. We give numerical evidence of its second order convergence in the presence of smooth source terms. For non-smooth source terms the convergence order drops to one half. Furthermore, we demonstrate the method s applicability for the long time simulation of calcium flow in heart cells and show its parallel scaling.

Monday May 13

Abstract:
Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as well as the need to develop efficient, scalable, stable and convergent computational methods for solving differential equations with random inputs. Stochastic Galerkin methods based on polynomial chaos expansions have shown superiority to other non-sampling and many sampling techniques. However, for complicated governing equations numerical implementations of stochastic Galerkin methods can become non-trivial. On the other hand, Monte Carlo and other traditional sampling methods, are straightforward to implement. However, they do not offer as fast convergence rates as stochastic Galerkin. Other numerical approaches are the stochastic collocation (SC) methods, which inherit both, the ease of implementation of Monte Carlo and the robustness of stochastic Galerkin to a great deal. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient.