Hermann Riecke - US grants

In this project the principal investigator will study the structure of waves that arise in a number of physical systems through the action of temporally periodic driving forces. The research will follow two directions. First of all, the principal investigator will study how the effect of temporal driving in exciting the system is balanced by the influence of spatial symmetries. This will entail using some recent methods of bifurcation theory with symmetry to derive and analyze the amplitude equations. Secondly, the principal investigator will study the dynamics of parametrically driven waves in large aspect-ratio systems. Such waves exhibit different types of instabilities that can lead to spatio-temporal chaos. The transitions to this chaotic behavior will be studied analytically as well as numerically.

The Department of Engineering Sciences and Applied Mathematics of Northwestern University will purchase workstations which will be dedicated to support research in the mathematical sciences. The equipment will be used for several research projects concerning the dynamics of nonlinear partial differential equations and applications, including in particular: . Advection by a dipolar vortex couple in the presence of small diffusion. . The numerical computation of pulse propagation in nonlinear optical fibers. . The influence of noise and quenched disorder on the growth of spatial structures.

9804673
Riecke

The general goal of this project is to contribute to the understanding of
spatio-temporal structures that arise spontaneously in many driven
dissipative systems. Mathematically, these systems represent dynamical
systems with many degrees of freedom. The PI will focus on waves in
two-dimensional systems with axial anisotropy. The immediate motivation
is given by recent experiments by G. Ahlers et al. on electroconvection in
nematic liquid crystals, where three interesting regimes have been
observed: 1) extended waves exhibiting spatio-temporally chaotic
dynamics, 2) waves localized into narrow, long patches (`worms'), and 3)
temporally irregular, spatially localized bursting of the wave amplitude.
These phenomena are found immediately above onset at small amplitudes
of the waves. They therefore afford an excellent opportunity to develop
and test mathematical ideas that will be relevant quite generally in weakly
nonlinear theories of dissipative structures. Furthermore, due to the
anisotropy this two-dimensional system can be described systematically
by Ginzburg-Landau-type equations. No recourse to phenomenological
equations of the Swift-Hohenberg-type is needed. The main part of the
project is devoted to the worms. They cannot be described by the
Ginzburg-Landau equations derived by straightforward asymptotic
analysis. The analysis has to be extended to include certain additional
modes that are strictly speaking not on the center manifold but become
relevant already very close to threshold. The main questions to be
addressed are: the localization mechanism of the worms and suitable one-
dimensional reductions (standing- and traveling-wave pulses), the stability
of long worms, and the nucleation of worms from small noise. The project
will demonstrate how in cases like the one described here the qualitative
and quantitative relevance of the weakly nonlinear theory can be
drastically increased by including a single additional mode. This aspect
carries over to a wide range of other systems. On a more specific level, the
advection of a slow mode will be important in other wave systems as well.
At the same time, for the electroconvection system detailed qualitative and
quantitative comparisons with ongoing experiments will be achieved
through close cooperation with the experimentalists.

In wide range of systems spatial structures appear spontaneously. These
patterns can be time-independent or may have the form of traveling or
standing waves. Just a few examples are animal coat markings, waves on
the surface of vibrated fluids, cloud streets, waves in chemical reactions
and in nerve conduction. If the properties of a system are the same over its
whole extent one would usually expect that the patterns also cover the
whole system. However, in quite a number of cases a spontaneous
localization of the pattern in patches or even single wave crests has been
observed. In repeated runs of the same experiment the patches occur in
randomly varying locations. This indicates that their localization is not due
to special properties of the system at that location. Such localized waves
have been observed, e.g., in vibrated granular material, fluid convection in
mixtures, and in nematic liquid crystals. The funded research will
contribute to the understanding of the causes that lead to localization
focussing on the `worms' observed in liquid crystals. It will elucidate the
mechanism in detail and investigate the chaotic bursting of such structures
and how they are generated (nucleated) from weak noise that is due to the
thermal motion in the fluid. Using advanced mathematical methods, the
research will bring out the central features that are relevant more generally
in other systems as well.

The Department of Engineering Sciences and Applied Mathematics
of Northwestern University propose to purchase a Hewlett Packard
multiprocessor graphics workstation and two X-terminal stations. The
equipment will be used for scientific computation for the following
research projects:

* Numerical simulation of viscous sintering
* Numerical studies of frequency conversion and light propagation
in optical systems
* Complex Structures in Pattern-Forming Systems
* Superlattice wave patterns

The principal investigator is D. Chopp. Co-principal investigators are
G. G. Luther, H. Riecke, and M. Silber. Northwestern University has
agreed to contribute 50% of the cost of the equipment.

9987577
Stephen Davis - Northwestern University
IGERT: Dynamics of Complex Systems in Science and Engineering

This Integrative Graduate Education and Research Training (IGERT) award supports the establishment of a multidisciplinary graduate training program of education and research on the dynamics of complex systems. Nonlinear science has increasing impact on many disciplines in the natural sciences, engineering, and medicine, and requires training that bridges traditional departmental and school boundaries. This graduate training program emphasizes the unity of fundamental concepts underlying a broad range of scientific research areas: nonlinear optics, computational neuroscience, pattern formation, chaos, ergodic theory; and applications to engineering and materials science problems: interface motion, combustion, and mixing. The goal is to prepare the students for today's rapidly evolving professional environment. Students will be equipped with the tools and intuition needed to tackle complex nonlinear problems arising in various guises and technical fields. Cross-disciplinary research and communication skills will be developed through an intensive year-long course, in which small graduate-student teams will investigate a topic guided by two faculty members with complementary perspectives. This early research experience will be followed by a thesis on a different topic; in the case of IGERT fellows the thesis project will be co-advised and cross-disciplinary. Internships will provide additional training experience. Cross-departmental research seminars and student-run seminars, regional workshops, yearly retreats, and an active visitor program will foster a highly cooperative, diverse, cross-disciplinary training environment.

IGERT is an NSF-wide program intended to meet the challenges of educating Ph.D. scientists and engineers with the multidisciplinary backgrounds and the technical, professional, and personal skills needed for the career demands of the future. The program is intended to catalyze a cultural change in graduate education by establishing new, innovative models for graduate education and training in a fertile environment for collaborative research that transcends traditional disciplinary boundaries. In the third year of the program, awards are being made to nineteen institutions for programs that collectively span all areas of science and engineering supported by NSF. The intellectual foci of this specific award reside in the Directorates for Mathematical and Physical Sciences; Engineering; Computer and Information Science and Engineering; and Education and Human Resources.

Proposal: DMS-0309657
PI: Hermann Riecke [h-riecke@northwestern.edu]
Institution: Northwestern University
Title: Complex Structures in Spatially Extended Dynamical Systems

ABSTRACT

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.

This project incorporates four different applications, which are
related through their common need for high-speed computational
capabilities. The first project involves the simulation of growing
bacterial biofilms. The simulation will couple the extended finite
element method with the level set method in order to capture the
interaction between the growing biofilm and the surrounding fluid
flow. The second project will solve a low Mach number model for
deflagrations in stellar envelopes. By filtering out sound waves from
the model, significantly larger time steps can be achieved than are
possible for full hydrodynamical models, thus making long time
computations feasible. The third project will study the role of
defects in the break-down of ordered structures. Parallelized
Monte-Carlo and Fourier spectral methods will be employed to
investigate defect trajectories in two paradigmatic systems, a
magnetic system and a dynamical system of coupled oscillators in two
dimensions. They exhibit spatially ordered as well as disordered
states. The goal is to identify to what extent various statistical
measures of the trajectories follow universal power laws. The fourth
project will use recently developed importance-sampling methods for
simulating rare events that set the performance of optical fiber
communication systems.

The impact of these projects will be felt across a broad spectrum of
disciplines, and will aid in the training and education of several
graduate students and postdoctoral research associates. In addition
to the scientific areas described above, the training will also
include the efficient use of high-performance parallel computing
architectures. The study of biofilms will improve our understanding
of quourum sensing organisms, and how to treat them. Such organisms
are responsible for a number of diseases, including diseases
associated with cystic fibrosis and deep burn wounds. The study of
deflagrations in stellar envelopes will improve our understanding of
the dynamics of novae of white dwarfs and the nature of X-ray bursts
from neutron stars. Transitions from ordered patterns to disordered
states are observed in many physical systems undergoing phase
transitions as well as in dynamical systems like arrays of coupled
oscillators, various kinds of fluid flow, and optical systems. Often
defects in the patterns are striking features of the disordered
states. The research will elucidate the role the defect dynamics play
in the break-down of the ordered states. The study of rare events in
optical fiber communication systems will lead to the construction of a
set of simulation tools capable of predicting the performance of
lightwave communication systems.

[unreadable] DESCRIPTION (provided by applicant): How does the brain encode sensory information using action potentials? Is only the mean firing rate relevant or is the precise timing of the spikes significant? Since near-synchrony of spikes is effective in driving downstream neurons, it could be an important feature of the neural code. Possibly, the brain makes use of both types of encoding, emphasizing one or the other, depending on the task at hand. In recent experiments on the olfactory bulb in zebrafish, complex temporal behavior involving near-synchrony and oscillations has been observed that suggests a multiplexing of the information transmission, with near-synchronous and asynchronous spikes carrying different information about the sensory input. In the proposed project, a biophysically detailed computational model of the circuitry of the olfactory bulb will be developed in close collaboration with experiments, in order to uncover the mechanisms underlying the multiplexing and the near-synchronization of neural sub-populations. The results may also shed light on other brain functions where temporal correlations and oscillations appear to be relevant (e.g., attention, perception, sleep, memory consolidation/retrieval). [unreadable] [unreadable]

The goal of this project is to develop a mathematical model of the initial processing of olfactory information in the brain, which is performed in the olfactory bulb. General information-theoretic considerations as well as experiments indicate that the main task of the olfactory bulb is to decorrelate odor stimuli. Its ability to do so depends sensitively on a match between the connectivity of the neural network and the odor environment. The project will focus on the ability of the bulbar neural network to adapt its connectivity to changing olfactory environments of the animal, while retaining the ability to discriminate and recognize significant previous odors. To model the plasticity of the network Hebbian long-term potentiation of the synapses will be included as well as the substantial neurogenesis that is an almost unique feature of the olfactory bulb. First the research will aim to establish biophysically plausible synaptic plasticity rules that enable the resulting network to decorrelate its odor inputs effectively. Then networks resulting from neurogenesis and cell death alone will be studied and their ability to decorrelate inputs will be assessed. Finally, synaptic and neurogenetic plasticity will be combined. It is expected that in the full model synaptic plasticity will provide fast fine-tuning of the adaptation within the connectivity framework set by neurogenesis, which in turn will retain significant information about past environments. Mathematically, the model will consist of a co-evolving bipartite network of weighted nodes and links. Computational and asymptotic solution methods will be developed to deal efficiently with the combination of discrete, stochastic evolution of the network structure and continuous deterministic evolution of its weights, involving three different time scales associated with neural activity, synaptic plasticity, and network connectivity, respectively.

A central function of the brain is to extract relevant features from the information it receives from the sensory organs and to make decisions based on new and previously stored information. The brain areas that perform the first information processing play an essential role by providing an internal representation of the environment that facilitates the extraction of relevant information. An important goal in neuroscience is to understand these internal representations and how they depend on the task at hand. In contrast to vision, where great progress has been made in understanding the neural encoding of the visual scenes that an animal is seeing in its natural environment, the processing of odor information in the olfactory system is much less understood. Unlike the visual world the olfactory world is high-dimensional and its characteristic features change significantly over time (seasons, migration, etc.). As the animal adapts to these changes the internal representation of a given odor is likely to change, as well. If this were the case the same rose would be perceived as smelling like one flower in spring and like another flower in fall, say, and the animal would have difficulties to identify odors. The goal of this project is to elucidate how the olfactory system copes with the conflicting tasks of adapting to changing environments and recognizing significant odors reliably in different environments. A central component of the model will be the substantial birth of new neurons (`neurogenesis') that is observed in the olfactory system throughout the life of the animal. The insights gained in this project are likely to be useful for the design of `artificial noses'. The project will also shed light on the role of neurogenesis in the hippocampus, where altered neurogenesis has been associated with mood disorders and neurodegenerative diseases.

Two pervasive aspects of the brain motivate this project: brain rhythms and feedback connectivity. Rhythms indicating partial synchronization, neurons acting in unison, are ubiquitously observed in the brain by EEG and other techniques. But their function is still not well understood; certain rhythms tend to be enhanced when an animal pays attention to specific aspects of its environment. A second characteristic aspect of brain networks is their extensive feedback connectivity, which allows the processing of information in a brain area to be modulated by the very brain area that receives that information. This project will study rhythms in neuronal networks with feedback connectivity, investigating whether such targeted feedback allows selective synchronization of specific neurons and what functionalities emerge from it. It is expected that selective synchronization contributes to solving tasks similar to the cocktail-party problem of following the voice of a specific speaker among many people talking simultaneously. The research will focus on the olfactory system. There, a corresponding challenging task is, e.g., for a dog to track the faint smell of a drug or landmine amidst many other smells. The project will inform efforts to enhance the processing of the outputs of biologically motivated, event-driven technological sensors for complex stimulus scenes. The training of graduate and undergraduate students in applied mathematics, neuroscience, and interdisciplinary communication skills is an integral part of this project.

The research will develop a combined neuronal network model for the olfactory bulb and the piriform cortex. Incorporating aspects of the extensive structural plasticity of the olfactory system, the network will be adaptive, allowing the feedback connections to reflect previous experience. Biophysically detailed neuron models as well as highly simplified neuronal networks will be employed. Mathematically, the research will use and further various analytical and computational dynamical systems methods: weak coupling of mixed-mode oscillators and of delay-induced oscillators; weakly nonlinear analysis of neuronal populations interacting with multiple delays; bifurcation analysis of resonant, delay-induced Hopf bifurcations; numerical branch continuation. These approaches will be complemented by direct simulations.

? DESCRIPTION (provided by applicant): An ubiquitous feature of the mammalian brain, which is likely a major factor contributing to its formidable computational power, is the preponderance of extensive top-down feedback. Depending on the tasks and demands at hand, this top-down feedback allows a brain area to control and enrich the processing of information in the very brain areas from which it receives the information. Such feedback is not restricted to high-order brain areas. In fact, in the olfactory system the valence of an odor stimulus and its context affec the processing of odor stimuli already in the olfactory bulb, which is only a single synapse away from the sensory space. So far, it is only poorly understood what functionalities the olfactory system may gain through this top-down influence. The impact of the top-down projections depends critically on their connectivity. Strikingly, in the olfactory system the feedback network exhibits extraordinary plasticity, which includes the cortically controlled persistent turnover of he dominant neuronal population of the olfactory bulb through adult neurogenesis and strongly fluctuating spine dynamics. The functional benefits of this structural plasticity still remain unclear. Intellectual Merit (provided by applicant): The proposed project will combine electrophysiology, optogenetics, imaging, and behavioral experiments with computational modeling * to characterize experimentally the dependence of spine fluctuations and cell survival on feedforward odor * input and cortical feedback and to develop in parallel a computational model for the network evolution incorporating the experimental results, * to analyze computationally the resulting network connectivities and the mechanisms determining them and, guided by the computational model, to assess the functional connectivities experimentally in awake animals by driving conditionally tagged cortical cells, * to exploit the model to suggest functionalities that may emerge from the cortically controlled connectivity. Possibilities indicated by preliminary results using a highly simplified model include * context-enhanced background subtraction and stimulus discrimination, * rapid cortical switching between previously learned bulbar circuits. Functionalities predicted by the newly developed model will be investigated in behavioral experiments in which the top-down feedback will be controlled optogenetically. With this effort the PIs aim to contribute to the understanding of what functions top-down feedback can support, how they may be implemented, and what special features adult neurogenesis contributes to these functions in the olfactory system. Broader Impact (provided by applicant): Imbalances between bottom-up and top-down pathways have been associated with psychological disorders like schizophrenia, which manifest themselves also in olfactory dysfunction. Insights into the mechanisms at work in the interaction between these two pathways may aid in understanding such psychological disorders. Chemical sensor technology may benefit from a better understanding of how the brain uses active topdown processes to extract the relevant information from the complex activity patterns generated by the chemical sensor array of the nose. The computational model developed in the project will be made publicly available via the ModelDB database, a public repository of neuronal models. A graduate student and a postdoc will be part of an international and interdisciplinary research team. Through extensive lab visits the student will gain interdisciplinary communication skills and international experience. The computational project will involve undergraduates. The French and the U.S. researchers will continue to participate in outreach efforts.

This project is an interdisciplinary Research Training Grant (RTG) on Quantitative Biological Modeling. Postdoctoral fellows, graduate students, and undergraduates will gain experience with mathematical modeling, numerical methods, and modern data analysis/statistical tools relevant to biological problems. Trainees will perform individual research in an interdisciplinary environment focusing on the mathematical modeling of current problems in experimental and computational biology. Each trainee will benefit from having two mentors, one from applied mathematics and one from a biological discipline. Trainees will gain knowledge and experience beyond that of their own research areas via regular group meetings organized around specific biological themes, and research internships outside Northwestern University will broaden trainee perspectives. Graduate students and postdoctoral fellows will gain pedagogical experience through creation of new course materials in collaboration with participating faculty. Trainees and undergraduates will also receive coaching in written and oral communication as part of the program. A major goal of this project is to expand the preparation of applied mathematics students and postdoctoral fellows for interdisciplinary research in the life sciences and for their subsequent careers.

Research performed as part of this project will focus on mathematical models in fields including neuroscience, developmental biology, evolutionary biology, and ecology. Models will encompass state-of-the art mathematical, computational, and statistical data-analysis methods appropriate for these application areas. New high-throughput experimental techniques have produced a wealth of data from biological systems in recent years, and major efforts are now underway to integrate these datasets to understand fundamental biological mechanisms and functionality. Within an interdisciplinary environment, trainees participating in this project will analyze imaging, sequencing and other data to address pressing biological questions. Co-mentorships and programmatic oversight of students by interdisciplinary faculty, creation of new courses, and development of a new vertical and peer teaching structure will greatly expand undergraduate, graduate, and postdoctoral training in applied mathematics in the high impact area of biological modeling. This will lead to a new generation of mathematicians, cross-trained and attuned to the particular needs of the field. The project will further connect faculty across Northwestern University and other research institutions, strengthening research progress in the focus areas.