Christian Sevenheck
Lehrstuhl für Mathematik VI
Universität Mannheim

In diesem Projekt sollen Bernstein-Polynome verschiedener nicht-isolierter Singularitäten bestimmt werden. Insbesondere sind dabei die sogenannten linear freien Divisoren, welche z.B. als Diskriminanten von Köcher-Darstellungen auftreten, interessant. Diese hängen mit Gauss-Manin-Systemen, welche in der Spiegelsymmetrie betrachtet werden zusammen. Da die entsprechenden Darstellungsräume auch für einfache Beispiele sehr schnell hohe Dimensionen haben, sind die entsprechenden Berechnungen sehr ressourceaufwending.

Software
Singular

Publications

• Christian Sevenheck; Bernstein polynomials and spectral numbers for linear free divisors pdf

Markus Knodel
Goethe Center for Scientific Computing (GCSC)
Frankfurt University

Daniel Buchner
Department of Neurobiology, IZN
Heidelberg University

Romina Geiger
Department of Neurobiology, IZN
Heidelberg University

Alfio Grillo
Goethe Center for Scientific Computing (GCSC)
Frankfurt University

Guillian Queisser
Goethe Center for Scientific Computing (GCSC)
Frankfurt University

Christoph Schuster
Department of Neurobiology, IZN
Heidelberg University

Gabriel Wittum
Goethe Center for Scientific Computing (GCSC)
Frankfurt University

One aim of the burgeoning field of computational neuroscience is to produce highly realistic, quantitative models which accurately reproduce of the physical processes which underlie synaptic transmission. To truly appreciate the myriad of events which relate synaptic function and vesicle dynamics, simulations must be done in a spatially realistic environment. This holded true in particular in order to explain the rather astonishing motor patterns presented here which we observed within in vivo recordings which underlie peristaltic contractions at a well characterized synapse, the neuromuscular junction (NMJ) of the Drosophila larva. To this end, we have employed a reductionist approach and generated three dimensional models of single presynaptic boutons at the Drosophila larval NMJ. Vesicle dynamics are described by diffusion-like partial differential equations which are solved numerically on unstructured grids using the UG platform. In our model we varied parameters such as bouton-size, vesicle output probability (Po), stimulation frequency and number of synapses, to observe how altering these parameters effected bouton function. Hence we demonstrate that the morphologic and physiologic specialization maybe a convergent evolutionary adaptation to regulate the trade off between sustained, low output, and short term, high output, synaptic signals. There seems to be a biologically meaningful explanation for the co-existence of the two different bouton types as previously observed at the NMJ (characterized especially by the relation between size and Po), the assigning of two different tasks with respect to short- and long-time behaviour could allow for an optimized interplay of different synapse types. As a side product, we demonstrate how advanced methods from numerical mathematics could help in future to resolve also other difficult experimental neurobiological issues.

• Knodel M., Bucher D., Schuster C. and Wittum G. (2008): Mathematical modeling of the Drosophila neuromuscular junction. Frontiers in Computational Neuroscience. Conference Abstract: Bernstein Symposium 2008. online

Area: Biology/Mathematics

Software:

• UG

Links:

• http://www.dmspin.org/