William L. Kath - US grants

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Science Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of basic research in the mathematical sciences. This equipment will be used to support two research projects in the Department of Engineering Sciences and Applied Mathematics at Northwestern University: Solitary Wave Pulse Propagation in Nonlinear Optical Fibers and Microwave Heating of Materials. Both projects will be directed by William Kath and Gregory Kriegsmann.

In the work supported by this award the principal investigator will extend some of his work on the infinite-line Kaup-Newell perturbation theory for a single soliton solution of a nonlinear Schroedinger system to the multiple soliton case. This work has applications to the transmission of data in the form of wave packets through nonlinear optical fibers, and so such it is highly relevant to the design of modern communications' systems that utilize optical fiber technology. The principal investigator will employ methods based upon asymptotics in order to study the behavior of solutions of nonlinear systems of partial differential equations. Every day, it seems, there are articles in the newspapers about the use of fiber optics technologies in the design of modern communications systems. Indeed, telephone companies and computer companies are very much interested in designing "lossless" ways of transmitting data. One possible solution is the emerging technologies based upon fiber optics, since fiber optics affords the potential of transmitting data without significant dispersion or distortion. With this award the principal investigator will study solutions of the nonlinear system of equations that describes the propagation of waves through optical fibers.

The principal investigator will develop and improve methods for studying the dynamics of pulses in nonlinear optical fibers, providing a description of the behavior of soliton-like pulses in several applications. This study has the potential of providing new technological applications for nonlinear optics. The qualitative analysis of nonlinear waves for such systems is of significant scientific interest in view of the wide spread occurrence of such phenomena in real world situations.

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.

DMS-9500615 Kath The goal this project is to analyze the effect various types of perturbations have upon the stability and dynamics of soliton solutions of the nonlinear Schroedinger equation. These perturbations include: periodic parametric forcing, stochastic noise, and higher-order linear and nonlinear derivatives. The intent is to develop and use approximate methods which allow both a simple qualitative understanding of the resulting pulse dynamics and an accurate quantitative description that is as concise as possible. The nonlinear Schroedinger equation models pulse propagation in nonlinear optical fibers, and in particular, optical solitons. As a result, an additional goal of the project is to investigate mathematically whether it is possible to use new optical fiber technologies to significantly increase the transmission speed of optical communication systems, or to improve upon the performance of these systems. One application that is being examined along these lines is the use of novel optical amplifier devices to control solitons, such as the use of phase-sensitive parametric amplifiers.

Dispersion management is the technique of concatenating two or
more optical fibers with different dispersion parameters to form a system
with periodically varying dispersion. This technique makes it is
possible to have both high local and low average dispersion in
the system. The high local dispersion reduces four-wave mixing
(an effect that distorts signals and produces intersymbol
interference in transmissions), while the low average dispersion
reduces the net cumulative effects of dispersion over long
optical fiber spans. Overall, therefore, dispersion management
reduces effects which are detrimental to the performance of
optical-fiber based communication systems, thus allowing transmission
capacity to be increased. In mathematical terms, what must be
studied are nonlinear wave equations with rapidly varying
coefficients. The theory of these equations is currently not well
understood and will be advanced with support from this award.

The theme of this research is the development of new mathematical
techniques with which to model the propagation of optical pulses
in nonlinear optical fibers. It is important to properly
understand the dynamics of such pulses in optical communication
and information processing systems so that new techniques can be
devised to increase their overall capacity. In particular, new
mathematical methods for dealing with dispersion management
will be investigated, in order to predict and improve the performance
of such systems.

We propose to build a cache storage buffer based upon parametric amplification that will be capable of operating in the Gb/s range, permitting kbits of data to be stored. We will carry out experiments to explore various ways of reading, writing, and erasing the stored data patterns. Our work has shown that the ultrafast parametric nonlinearity can be exploited either to provide broad-band tunable gain or dynamic gain modulation for clock-recovery. We propose to combine the two to demonstrate an optical phase-lock loop, which in principle can be extremely fast as it relies on the Kerr nonlinearity for envelope-phase discrimination. Such a phase-lock loop will also be wavelength tunable-a feature that is quite important as it adds the function of wavelength conversion to the standard 3R all-optical regenerator. Simultaneous to the above experimental studies we will also develop numerical models of the various optical systems. This will provide a design tool to determine the parameter values allowing the most efficient operation of the experimental setups. We have previously demonstrated the possibility of stably propagating sub-picosecond pulses in fiber lines in which conjugating gain is used to compensate the linear loss. We propose to assemble a re-circulating loop experiment in which linear loss will be compensated by a pair of non-degenerate parametric (conjugating) amplifiers. The location of the two amplifiers will be chosen based upon further theoretical/numerical results. We will experimentally and theoretically study the stability properties of the sub-picosecond pulses by making various signal and noise measurements, and will compare the experimental results directly with numerical simulations. The proposed experiments will be performed in both the 1550 and 1300 nm wavelength regions in order to demonstrate the wideband capability of parametric amplification.

This grant supports an interdisciplinary research project in applied mathematics and neurobiology. Specifically, the PI will spend an academic year leave visiting the Department of Neurobiology and Physiology at Northwestern University. The goal of this interdisciplinary research program is: 1) to participate
in and assist with the modeling of ongoing experiments on hippocampal neurons; 2) to use the focused learning environment to develop a sufficient understanding of neurobiology and the skills necessary for continued research in this area; 3) to create materials that can be used as part of the graduate and
undergraduate curricula in the PI's home department. Specific problems to be examined include the modeling of slow, cumulative sodium channel inactivation in hippocampal CA1 pyramidal neurons, and a study of the causes of bursting in subicular hippocampal neurons. To help ensure the project's success, the PI will be
externally advised throughout the duration of the work by other researchers familiar with the application of mathematics to biology. This IGMS project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

The high demand likely to be placed on the capacity of telecommunication networks in the near future urges
conversion of hybrid electro-optical signal processing to all-optical processing, exploiting the largest bandwidth available in the optical domain. One way of catering to this demand is by multiplexing in time as
well as wavelength domains. Using picosecond-duration optical pulses, which could be soliton-like over
portions of the network, one can first perform multiplexing in the time domain (i.e., time-division multiplexing, or TDM) for local to metropolitan-area network applications and then in the wavelength domain (wavelength-division multiplexing, or WDM) for wide area coverage. This scenario leads one to conclude that the key issue to be addressed is how to take advantage of the powerful digital-processing techniques in the pure-optical domain, that minimize the detrimental effects of noise at a very fundamental level. The idea is that, for digitally encoded data [1's (0's) represented by the presence (absence) of pulses], instead of using linear amplifiers which act on signals in an analog fashion and inevitably introduce 3 dB of noise one can employ digital-switching amplifiers or optical regenerators. At the same time, pure-optical digital switching is potentially much more reliable and faster than electro-optical switching. Furthermore, optical switching will also be needed to implement other networking functions, such as demultiplexing to process at very high speed the header of a data packet used for addressing to different users on the network.
Our preliminary experiments show that the parametric nonlinearity of optical fibers can be exploited
to perform functionalities that will be needed in packet-switched all-optical networks, such as fiber-optic
cache-memory buffers, picosecond-pulse all-optical regenerators, all-optical limiters, and tunable clock re-covery modules. These devices take advantage of the ultrafast parametric nonlinearity of glass fiber and
hence are capable of operating at speeds in excess of 100 Gb/s. Moreover, they will be essential for deploying
packet-switched, ultrahigh-speed time-division and wavelength-division multiplexed all-optical networks.
In all of our experiments thus far, standard dispersion-shifted fiber (DSF) has been used. Fiber lengths
on the order of 100's of meters are required for used with ps-duration pulses of a few watts peak power to
achieve the data processing functions. Here we propose to explore the use of high-nonlinearity fiber, such
as microstructure fiber (MF, which is only now becoming commercially available), to perform essential
functions in high-speed all-optical processing. Because of their strongly guiding behavior, the MFs can
be wound into very tight loops, suggesting that they could potentially fit into a compact modular switching
package. Specifically, we propose to utilize the high-nonlinearity microstructure fibers to develop all-optical data processing modules. These include a cache storage buffer based upon parametric amplification that will be capable of operating in the 10's of Gb/s range. With use of the high-nonlinearity fiber, the average pump power requirement can be met with commercially-available watt-class optical amplifiers. We will carry out experiments to explore various ways of reading, writing, and erasing the stored data patterns.
Our work has shown that the ultrafast parametric nonlinearity can be exploited either to provide broadband
tunable gain or dynamic gain modulation for clock-recovery. We propose to combine the two to demonstrate
optical phase-lock loops, which in principle can be extremely fast as they rely on the Kerr nonlinearity
for envelope-phase discrimination. Simultaneous to the above experimental studies we will also develop
numerical models of the various optical systems. This will provide a design tool to determine the parameter
values allowing the most efficient operation of the experimental setups. We have previously demonstrated
the possibility of stably propagating sub-picosecond pulses in fiber lines in which conjugating gain is used
to compensate the linear loss. We propose to assemble a re-circulating loop experiment in which linear loss
will be compensated by a pair of non-degenerate parametric (conjugating) amplifiers. The location of the
two amplifiers will be chosen based upon further theoretical/numerical results. We will experimentally and
theoretically study the stability properties of the sub-picosecond pulses by making various signal and noise
measurements, and will compare the experimental results directly with numerical simulations.

NSF Award Abstract - DMS-0101476

Mathematical Sciences: FRG: Mathematical and Computational Methods for High-Data-Rate Optical Fiber Communications

Abstract
DMS-0101476
Kath


The goal of this research project is to develop new methods that can be used to determine the behavior of optical transmission systems under realistic circumstances. This will be accomplished by a combination of various techniques. One approach will exploit the mathematical structure of fiber transmission models in order to eliminate unessential degrees of freedom. The reduced models that will result will be more tractable mathematically and also much more computationally efficient. Another approach that will be used is the application of linearization and importance sampling techniques to enable the simulation of systems at realistic data error rates. These methods will be combined to study the main sources of impairment in optical fibers in order to achieve an accurate evaluation of system performance. All the techniques to be developed will be carefully validated by comparison to more computationally time-consuming models and to experiments.

The development of high-data-rate optical fiber communications is one of the great technological achievements of the late 20th century; in the last decade alone, data rates have increased by four orders of magnitude. This enormous increase has made possible the growth of the global Internet that promises to continue to revolutionize day-to-day communications. Because demand for further growth continues unabated, however, system capacity is becoming limited by fiber transmission effects. It has therefore become crucial to accurately model and calculate the impairments due to non-ideal fiber properties when designing systems. Due to the tremendous data capacity that will be required of future transmission systems (terabits per second of aggregate capacity) and the need for extremely small transmission error rates (less than one error per trillion bits), realistic attempts to model and predict the effects of these impairments as they appear in practical systems present a number of difficult mathematical and computational challenges. The techniques that will be developed in this collaborative research project are expected to yield large reductions in the computational time required to model optical communication systems, and at the same time produce new insights into system behavior. Because these methods will be capable of providing detailed information about system performance at realistic data error rates, we believe they will lead to significant changes in the way in which optical transmission systems are modeled, and, ultimately, in the way that they are built.

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.

Abstract

DMS-046513
William Kath, Northwestern University
Title: Analytical/Computational Methods for Rare Events in Lightwave Systems


The goal of this project is to construct new methods based upon variance reduction techniques, such as importance sampling, that can be used to efficiently simulate rare events caused by noise in lightwave systems. Importance sampling is a method of biasing (or, altering) the probability distributions used to generate random Monte-Carlo trials so that simulated errors occur more frequently than would be the case otherwise. The aim here is to exploit the mathematical structure of the equations governing the propagation of signals to allow importance sampling to be performed. For example, in optical fibers, the governing equation is the nonlinear Schroedinger (NLS) equation, which is a completely integrable Hamiltonian system. The inverse scattering solution of the NLS equation shows that each pulse has a set of modes associated with it; these modes correspond to changes in the pulse's amplitude, phase, position and frequency. Since any
value of the pulse parameters yields a valid solution of the NLS equation, no resistance is encountered if any of them is changed by noise. These four modes thus provide a natural basis upon which to construct methods by which the noise is intentionally biased to produce large signal fluctuations.

The development of high-bit-rate data transmission over optical fibers is one of the major technological achievements of the late 20th century. Optical fibers have fueled the growth of the global internet and are revolutionizing the ways in which information is communicated and processed. Because of the enhanced levels of performance demanded of modern lightwave systems, traditional analytical or computational methods are by themselves insufficient to accurately model the rare events that determine the overall
performance of these highly complex systems. At the same time, their often large development costs makes the accurate prediction of their behavior and performance essential. Recent work has demonstrated, however, that hybrid analytical/computational approaches can make accomplishing this task not only possible, but also practical. Proof-of-concept examples have shown that error probabilities in such systems as small as one part in a trillion should no longer be considered beyond the reach of estimation. The methods that will be developed as part of this project are expected to provide the basis for computational tools that can yield large reductions in the time required to determine the performance of such systems.

The development of high-speed optical systems to transmit and process information is a major technological achievement of the late twentieth century. The goal of this project is to develop new hybrid analytical/computational techniques capable of identifying the errors that limit the performance of such systems, and the probabilities with which they occur. The first component of the project is the application of the singular value decomposition to the equations governing the combination of optical pulse propagation and signal detection at the end of transmission that converts light into electrical energy, which will determine the perturbations that produce the largest changes at the output. The second component is the application of the cross-entropy method, an adaptive variance reduction technique which assesses the relative importance, in terms of probability, of each type of perturbation.

Experimental testing and laboratory or field measurements of optical systems can be quite costly and require months of time. A goal of this project is to produce simulation methods that can provide, in an efficent manner, detailed information about system performance, thus reducing the need for testing. The methods will be used to model both high-speed optical communication systems and new classes of extremely stable short-pulse lasers. These lasers provide ultra-precise frequency references for communication, radar, and remote chemical sensing, and are an essential component in optical atomic clocks, next generation timekeeping devices that are predicted to have accuracies several orders of magnitude better than current atomic clocks. Improved accuracies in global position systems and other technologies are expected to result from these devices.

Increasing evidence indicates that cancer results not only from changes in genetic information;in the form of point mutations, chromosomal rearrangements, gene segment amplification and deletion, but also from changes in epigenetic information. Recent theoretical analyses of epigenetic regulation in a model organism (yeast) show that ideas from the dynamical systems field may explain the stability of epigenetic states and the dynamics of spontaneous changes between them. This project will experimentally define the epigenetic changes resulting from the action of (1) a mutant kinase JAK2V617F, a causative factor in myeloproliferative disease;(2) a defined set of oncogenes that transform normal human cells to cancer cells (T antigen, ras, shRNA PP2a, telomerase);and (3) the changes in state of the MLH1 promoter that accompany gene silencing associated with promoter methylation, and reactivation that is induced by 5-aza-2'-deoxycytidine and is associated with DNA demethylation. We will use this information to develop quantitative predictive models of epigenetic inheritance and switching in these systems, based on a dynamical systems framework.

This conference support award is for the Annual International Conference on Chaos and Nonlinear Dynamics "Dynamics Days" to be held in Evanston, Illinois, on January 4-7, 2010. The conference will bring together a diverse group of experts in complementary areas of nonlinear phenomena, including chaos, complex networks, granular materials, time series analysis, non-equilibrium statistical physics, pattern formation, self-organization, and fluid dynamics. The conference participants will be comprised of experts in theory and applications as well as students and postdoctoral researchers. There will be substantial emphasis on topics that cut across disciplinary boundaries, such as dynamical processes in complex systems.

Dynamics Days is an annual conference organized with the specific purpose of providing an interface for researchers from diverse scientific disciplines but with common interests in chaos and nonlinear dynamics. This meeting has become internationally known as a very effective venue for researchers to share their most recent results in understanding, modeling, and controlling nonlinear dynamical systems.
The proposed conference will follow this tradition and create an interdisciplinary environment that will facilitate interactions between researchers from related but different fields. This is expected to generate new insights and lead to potentially transformative interdisciplinary research. The inclusion of students and early-career researchers, including those from underrepresented groups, is a central part of the conference. This will promote the transfer of expertise to the next generation of engineers, physicists, and mathematicians.

DESCRIPTION (provided by applicant): This is a collaborative project between Northwestern University (Nelson Spruston and Bill Kath), Stanford University (Stephen Smith), and the University of Bonn (Stefan Remy). The project will lead to an improved understanding of neurons in the hippocampus, which were selected because of their roles in learning and memory as well as a number of cognitive disorders. Studies of these neurons will also offer insight into neurons in other areas of the brain, many of which have shared structural and functional properties. The goals of the project are as follows: We will collect functional data from hippocampal CA1 pyramidal neurons using patch-clamp recording in brain slices combined with two-photon uncaging of glutamate and two-photon calcium imaging. We will also collect structural and molecular data from the same dendritic branches using array tomography, which provides the highest possible resolution using light microscopy. We will examine the distribution of excitatory synaptic weights, as well as the distribution of inhibitory synapses from different interneuron subtypes. All experiments will be performed for dendrites in different dendritic compartments (e.g., basal versus apical dendrites). By performing both functional and structural experiments in the same neurons, we will be able to correlate and integrate the data sets. We will construct compartmental models of CA1 pyramidal neurons, using the data from the experiments to inform improvements on our existing models of these neurons. The models will be used to generate experimentally testable predictions concerning the integration of synaptic inputs. These predictions will extend beyond the range of experiments performed to constrain the model, so they will constitute predictions designed to inform future work on these neurons. Spruston and Kath have a record of using such predictions to design and perform experiments that lead to new discoveries. We will use the models developed in Aim 2 to examine whether stochastic activation of thousands of excitatory and inhibitory synaptic inputs, combined with the excitable properties of the dendrites and synaptic plasticity rules based on the resulting dendritic voltage changes, can lead to non-uniform gradients of excitatory synaptic weights in CA1 pyramidal neurons. Our working hypothesis is that the natural gradients of voltage that exist in CA1 dendrites can contribute to the development of non-uniform synaptic weights. We will compare the results of these simulations to the results from array tomography studies as a means of determining which activity patterns and synaptic plasticity rules best explain the observed distribution of synaptic weights. Collaboration: All team members will exchange data and interact on a regular basis. The Spruston and Remy labs will perform experiments using patch-clamp recording and two-photon uncaging and imaging. Filled cells from these experiments will be sent to Stanford for array tomography in the Smith lab. Spruston, Kath, Smith and Remy will supervise the integration of array tomography data with functional data, working together with the postdoc and student supported by this project. All members of the group will meet regularly to discuss progress and future plans. Intellectual Merit: The project will provide critical data concerning the structure and function of pyramidal neurons in the hippocampus, which will be used to generate computational models of unprecedented detail. The models will be used to advance our understanding of synaptic integration in dendrites and the contribution of excitable dendrites to synaptic plasticity and the distribution of excitatory synaptic weights in the dendritic tree. The underlying philosophy is that the function of neural circuits, as well as diseases that affect them, cannot be understood without an accurate understanding of the structure and function of the component parts in the circuit. Broader Impacts: The broader impacts of this work include international collaboration and international and multi-disciplinary training of students and postdocs. In addition, our experimental data and computational models will be shared with the larger research community. We will also work with Michael Kennedy, Director of Northwestern's "Science in Society" program, to use our data to generate interactive, web-based educational tools targeting high-school students as well as post-secondary students. Our goal will be to develop visually exciting tools that appeal to a teenage audience. The tools will be promoted through the Science in Society website and through Kennedy's personal interactions with Chicago Public Schools and the Boys &Girls club of Chicago, both of which have large populations of under-served minorities. Stefan Remy will promote these educational tools in Germany. Long term, we believe that these tools could reach national and international audiences.

DESCRIPTION (provided by applicant): Neuronal circuits exhibit specialization at many levels;in particular, neuronal diversity and differences in connectivity are thought to be crucial. Significant advances have been made in understanding the types of plasticity that may contribute to learning and memory in the hippocampus, but much less has been discovered about how the circuit stores and extracts information. A key aspect of all circuit function is the diverse population of inhibitory interneurons, which differ in physiological properties, dendritic morphology and axon targeting. Understanding of circuit function can be significantly enhanced by capturing the complexity inherent in neuronal diversity and connectivity in detailed computer models of the system. Here we propose to study and model specific populations on interneurons in the hippocampus identified in BAG transgenic mice generated by the NINDS-Gensat project. We propose to record from identified neurons in these mice in order to determine their physiological properties. The recorded cells will also be stained so that dendritic morphology can be determined and quantified. Computational models will then be generated of the interneurons and pyramidal neurons to which they project for the purpose of making experimentally testable predictions concerning hippocampal circuit function. This is a collaborative project that brings together investigators, students, and postdocs to take a multidisciplinary approach to the study of hippocampal circuit function. The project has five specific aims: 1) Physiological investigation of hippocampal interneurons in BAG transgenic mice. 2) Anatomical investigation of hippocampal interneurons in BAG transgenic mice. 3) Studies of modulation of interneurons in BAG transgenic mice. 4) Modeling hippocampal interneurons from BAG transgenic mice. 5) Developing advanced computational methods for microcircuit modeling. The proposed work has important implications for several neurological disorders, including Alzheimer's disease, epilepsy, and schizophrenia.

The goal of this project is to develop new mathematical models of mode-locked lasers. Models will take into account descriptions of the optical and electronic components, and the coupling between them, on multiple time scales. The different time scales will be exploited to provide analytical methods and computational tools by which the nonlinear dynamics of the laser can be analyzed. These mathematical techniques will allow a careful study of the fundamental factors governing the existence and stability of pulse solutions, including a determination as to how noise affects these lasers' precision.

Recent advances have ushered in a new era of extremely stable ultra-short modelocked lasers, with typical pulse durations of just a few femtoseconds. Such advanced devices are important as precise sources for ultraviolet and infrared spectroscopy (e.g., for trace gas sensing) and optical and microwave waveform generation (e.g., for low noise microwave sources). They are also an essential component in optical atomic clocks, next-generation timekeeping devices with accuracies several orders of magnitude better than current microwave atomic clocks. A goal of this project is to provide results capable of predicting the performance of femtosecond lasers and that will assist scientists and engineers who design and build them. Another important component of this project is the training of students. As part of this component, the PI will visit Delaware State University to assist in the mentoring of applied mathematics graduate students being advised in the field of nonlinear optics.

A fundamental question in neuroscience is to understand how different sets of neurons in a network combine or integrate their various inputs, and how both the pattern of inputs and the resulting outputs are related to behavior. Novel, even unprecedented, experimental methods have been and are being developed with which to image or record from large numbers of distributed neurons but understanding the integration step can be difficult since inputs and outputs are generally distributed over many millimeters, and it is very difficult to record from both simultaneously, especially so in a behaving animal. This project will combine experimental and computational methods to elucidate such synaptic integration in pyramidal neurons associated with mammalian spatial navigation in awake, behaving animals. The imaging data acquired from awake behaving mice will be made available to other groups and the full results will serve as a model for other research concerned with the integration of inputs in networks of neurons. The computational models will be made available on the ModelDB database and will be a resource to others working to understand other aspects of functionality in this brain region. Furthermore, the work will involve a close collaboration between experimental and computational research groups, thus giving postdoctoral fellows and graduate students cross-disciplinary research training.

In pyramidal neurons of the hippocampus, the large dendritic tree constitutes an elaborate network of branching processes involving tens of thousands of excitatory synapses containing a variety of voltage-gated ion channels. The pattern of synaptic inputs impinging upon the dendritic arbor and the degree to which these inputs are processed by it to drive place field firing (i.e., firing correlated with spatial location) during behavior are currently unknown. The goals of the project are first to 1) develop improved computational models of dendritic place cell firing constrained by current imaging data and 2) establish new experimental techniques to image the inputs to pyramidal cells in the dendritic tree, at single spine resolution, during place field firing. Together the experiments and models will be used to 3) determine the degree to which local dendritic processing is involved in place cell firing. The proposed experiments will allow for the construction of significantly improved models of hippocampal function and the models will provide a framework within which to understand activity recorded at a local level in the dendritic tree and assemble a comprehensive picture of dendritic processing across the whole arbor.

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.