The research funded by this grant will contribute to the understanding of complex spatio-temporal structures in two types of dynamical systems with many degrees of freedom. 1) The impact of near-resonant forcing with multiple frequencies on spatially extended continuous oscillatory media will be investigated using analytical and computational methods. Multi-frequency forcing affords substantially greater control of the system and may give access to dynamical labyrinthine patterns and may allow the annihilation of spiral waves; by allowing to tune the interaction between spatially periodic modes it may stabilize and select spatial structures exhibiting multiple length scales such as superlattices or quasipatterns. In contrast to the well-studied Faraday waves in vertically vibrated fluids, the oscillators may loose their phase-locking relative to the forcing, which may add temporal complexity to the superlattices and quasipatterns. Multi-frequency forcing of oscillatory media can be implemented in the light-sensitive chemical reactions currently investigated experimentally. 2) The importance of networks comprised of discrete elements and exhibiting complex topology has been increasingly appreciated. Most attention has been given to their geometrical properties. The second project will elucidate how the dynamics of a network of locally coupled discrete excitable elements is impacted by the addition of random long-range connections, which transform the network into a small-world network. The research is inspired by studies of cortical brain tissue in which neural networks with local and non-local connectivity exhibit persistent activity without external input. Can the long-range connections induce the persistent activity in the absence of external input? How does it depend on the network topology? How robust is the bistability between the active and the quiescent state with respect to noise?

The mathematical theory of dynamical systems provides powerful tools to understand and predict the dynamical behavior of systems in many areas of science and engineering. The work of the PI and his collaborators will focus on two distinct classes of systems that are comprised of a large number of interacting dynamical elements. 1) Spontaneous oscillations occur in many spatially extended natural systems, e.g. chemical systems. They can lead to the propagation of waves that have important biological functions. For instance, cAMP-waves provide the signaling between Dictyostelium cells when they aggregate to form a multi-cellular organism, and calcium waves provide communication inside a wide variety of cells. It is important to understand how such oscillations respond to a variability of their environment. The variation has strongest impact when its frequency is close to a multiple of the natural frequency of the oscillation. The first project will identify various consequences resulting from such near-resonant variations. It is expected that the results on spiral dynamics will also be relevant for excitable media like heart muscle. In view of the significance of spirals during life-threatening ventricular fibrillation the question whether multi-frequency forcing can annihilate spirals is of particular interest. 2) Self-sustained activity of networks of neurons is essential for various tasks of the brain such as its ability to quickly store information for a brief duration while performing a task based on that information (e.g. when dialing a phone number) and then deleting it. The second project will shed light on what kind of connectivity between the neurons is favorable for tasks like that. - Teaching graduate students analytical and computational methods and their application as well as communication skills is an integral part of both research projects.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0309657
Program Officer
Henry A. Warchall
Project Start
Project End
Budget Start
2003-07-15
Budget End
2008-06-30
Support Year
Fiscal Year
2003
Total Cost
$239,951
Indirect Cost
Name
Northwestern University at Chicago
Department
Type
DUNS #
City
Evanston
State
IL
Country
United States
Zip Code
60201