Philip J. Holmes - US grants

This project supports resear5ch in dynamical systems with significant emphasis on computation. The specific projects discussed in the proposal are: 1. Hilbert's 16th Problem: Numerical and symbolic computations to obtain bounds on the number of limit cycles possessed by polynomial vector fields in the plane. 2. Dynamical studies of vector fields derived from equations of fluid boundary layers. 3. The implementation of perturbation methods for ordinary differential equations problems using MACSYMA. 4. The study of multiparameter systems of differential equations near points of multiple bifurcation. 5. Numerical studies of the complex Henon mapping, holomorphic diffeomorphism of complex two space. 6. Symbolic computation of polynomial knot invariants associated to periodic orbits of three dimensional flows. More broadly, the project will support the development of algorithms and efficient computing environments for the studies of dynamical systems.

This research program introduces new methods into the study of dynamical systems and represents a continuation of a program established by the principal investigator. The work is comprised of three major components: (1) Global perturbation methods in dynamical systems related to nonlinear elasticity and fluid mechanics; (2) Studies of knotted period orbits in forced nonlinear oscillators and of the chaotic attracting sets exhibited by such systems; (3) Analysis and predictions of the power spectra of strange attractors and analyses of the effect of small noise on such systems. These studies represent the use of applied mathematics in understanding properties and behavior of mechanical systems.

In addition to the mathematical aspects of the work, the project has the potential for wide impact on the scientific community through the development of a comprehensive computer environment for the exploration of dynamical systems. The intention is to provide the user of this environment with interactive, efficient access to all currently available algorithms for studying the dynamics of general systems together with the tools for effectively managing the data produced in these studies. A prototype for such an environment has been developed in the context of the first problem listed above, but further work is required to produce a package that meets the needs of a broad spectrum of applications and users. These further developments will occur in conjunction with studies of problems of significant theoretical interest.

The principal investigators undertake studies of a variety of problems in mechanics and biology, using techniques of nonlinear dynamical systems. The topics include the dynamics of dissipative partial differential equations and pattern formation in fluid systems, global dynamics of symmetric systems, knots and orbit genealogy in three-dimensional flows, dynamics of oscillators with slowly varying potentials, modelling of neurobiological oscillators, iterations of one-dimensional mappings, nonlinear stability and bifurcation in Hamiltonian and weakly damped systems. Arnold diffusion in four-dimensional symplectic maps, and dynamics of the growth of certain plants. These topics have applications in the design and development of "smart" materials, in aerospace (particularly satellites and spacecraft), in the control of turbulent boundary layers,and in biology.

9318637 Maxwell The American Mathematical Society requests a grant in the amount of $62,371 for the twenty-fourth AMS-SIAM Summer Seminar in Applied Mathematics, to be held in the summer of 1994. Previous seminars were held annually or at two or three year intervals since 1957 for a total or 23 seminars. The 1994 topic, "Dynamical Systems and Probabilistic Methods for Nonlinear Waves", was selected by the 1992 AMS-SIAM Committee on Applied Mathematics. Members of the Organizing Committee for the 1994 Summer seminar are: David W. MCLaughlin (co-chair), Philip Holmes (co-chair), Percy Deift, James M. Hyman, C. David Levermore, Y. Sinai, and C. Eugene Wayne. The organizers propose and AMS-SIAM Summer Seminar in 1994 which would focus on the use of probabilistic and dynamical systems methods in the study of nonlinear waves. The conference program will be specifically geared toward advanced graduate students and recent Ph.D recipients who will soon be participating in these fast developing and important fields. The 1994 Summer Seminar will be held during a two-week period in June/July 1994, at the Mathematical Sciences Research Institute in Berkeley,California. The Society will be responsible for making suitable arrangements for lecture and seminar rooms, and for coordinating local administrative matters with MSRI staff.

9508634 Holmes The University-Industry Cooperative Research Programs in the Mathematical Sciences are programs designed to establish a mechanism for exchange of mathematical scientists between academia and industry at different stages of their careers. These programs will strengthen the links between industry and academia and identify and encourage new avenues of research in the mathematical sciences. The University-Industry Postdoctoral Research Fellowships provide two years of support for recent recipients of doctoral degrees in the mathematical sciences. These awards are a means of contributing to the future vitality of the scientific effort of the nation. The postdoctoral fellow will work closely with scientific advisors from both academia and industry. Jose Nathan Kutz received his doctoral degree from Northwestern University and will work under the direction of Philip Holmes at Princeton and Paul Wright and Pierre Hohenberg at AT&T. The research will deal with optical fiber technology and nonlinear wave propagation and pattern formation in spatially extended systems. The common mathematical ground linking these areas is the use of nonlinear partial differential equations in modeling wave and pulse propagation in active and disordered media. In particular, equations of nonlinear Schroedinger (NLS) and Ginsburg-Landau type frequently arise.

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The investigators model and analyze a mechano-chemical instability recently discovered in the study of catalytic chemical reactions on very thin crystals: spatiotemporal variation in the reaction rate coupled with heat release causes spatially varying thermal expansion and buckling of the catalyst surface. This changes the catalytic properties, closing the loop between reaction, heat release and deformation; it has been observed to lead to sustained mechano-chemical oscillations. The project builds on experience in studying pattern formation and instability independently in mechanical and in chemically reacting systems. Theoretical and computational studies of nonlinear buckling of beams under constraints are extended to account for static as well as dynamic variations of the beam properties, including thermally-induced strains. Earlier computer-assisted studies of pattern formation on catalytic surfaces are used to drive, through the temperature fields thereby generated, mechanical deformations of the catalyst. Finally, the loop is closed by allowing the deformation to change the surface catalytic properties, thus attempting to reproduce the oscillations observed in experiments. The modeling, stability analysis and computer-assisted work are performed in constant dialogue with experimental groups in Princeton (on inhomogeneous beams) and in Berlin (on deforming catalytic crystals).

Heterogeneous catalytic reactions constitute the backbone of chemical processes of industrial and environmental relevance: the CO oxidation reaction the project addresses occurs, for example, in every automotive catalytic converter. The development of techniques for analyzing in fine spatial and temporal detail what occurs on the catalyst in situ under reaction conditions is revolutionizing the way modeling of these processes is done and enabling the discovery of new phenomena. This will in turn improve the design of catalytic processes. Specifically, this project aims at understanding a phenomenon only recently observed in CO oxidation: oscillations involving mechanical deformations of the catalyst coupled to temperature and chemical concentrations. Computer-aided analyses of the process lead to better, predictive models, which can be exploited in optimizing both the material properties and the operating conditions of the process. Funding for the project is provided by the program of Computational Mathematics and the Office of Multidisciplinary Activities in MPS and by the Chemical Reaction Processes program in ENG.

Drug abuse and depression frequently coexist. The proposed experiments will employ a well-established model of depression to study neurochemical and behavioral changes associated with both drug abuse and depression. The olfactory bulbectomy (OBX) model of depression produces such changes, thus providing a unique opportunity to study the common neurobiological bases of these disorders. OBX produces a behavioral syndrome independent of anosmia that resembles human depression, including loss of circadian activity rhythms, hyperreactivity to stress, decreased meal size, decreased weight gain, and decreased sexual behavior. The OBX syndrome is reversed by chronic but not acute antidepressant treatment. Some of the behavioral changes caused by OBX suggest anhedonia. Lesion-induced plasticity in mesolimbic dopamine systems may mediate this anhedonia. Furthermore, the nature of the dopamine plasticity caused by OBX predicts that bulbectomized rats would be hypersensitive to the stimulant and reinforcing effects of amphetamine. Previous research reveals that OBX increases D1 and D2 receptor density in the olfactory tubercle. Preliminary in situ hybridization experiments from this laboratory indicate that OBX increases levels of D2 receptor and prepro-enkephalin mRNAs in the olfactory tubercle. The olfactory tubercle has previously been implicated in behavioral reinforcement and the actions of drugs of abuse. Experiments will test the hypothesis that bulbectomized rats are more sensitive than intact rats to the stimulant and reinforcing effects of amphetamine because of postsynaptic receptor supersensitivity in the olfactory tubercle. Experiments will also determine whether bulbectomized rats will self-administer amphetamine at higher rates. Brains from all rats in these behavioral studies will be analyzed by in situ hybridization histochemistry, and increases in dopamine receptor and prepro-enkephalin mRNA levels will serve as markers for OBX- induced plasticity in the olfactory tubercle. Preliminary data reveal wide individual differences in amphetamine sensitivity and prepro-enkephalin mRNA plasticity in bulbectomized rats. The ultimate objective of this proposal will therefore be to test the relationship between individual differences in amphetamine-induced behaviors and individual differences in OBX-induced plasticity in the olfactory tubercle. The proposed experiments may reveal a novel mechanism for individual differences in amphetamine sensitivity. The studies may also provide the basis for the first animal model of drug self-administration as a compensatory response to anhedonia.

This project supports an NSF-Industrial Postdoctoral Fellow, Deborah Alterman, to carry out research jointly in the Program in Applied and Computational Mathematics at Princeton University and Lucent-Bell Laboratories, Murray Hill, NJ. The Faculty Mentor will be Philip Holmes, Professor of Mechanics and Applied Mathematics, and the Supervisor at Lucent Technologies will be Dr Michael Weinstein who has recently joined the Mathematical Sciences Research Center (Fundamental Mathematics Department). We envisage two principal areas of research: (1) Nonlinear wave theory applied to problems of optical communications, and (2) Coupled oscillators as models of neural networks. The mathematical tools to be used and developed include dynamical systems and bifurcation theory, ordinary and partial differential equations, asymptotics, and nonlinear wave and stability theory. A pervasive theme is the reduction of dynamical systems with infinite or high dimensional phase spaces, describing distributed phenomena, to simpler ``amplitude'' partial differential or ``modal'' ordinary differential equations. Such reduced models lead to improved understanding of the underlying physics, and offer shortcuts in predictions and design. This GOALI project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

stress; depression; neurobiology; gene expression; neuropeptide Y; antisense nucleic acid; olfactory lobe; neural plasticity; disease /disorder model; hormone regulation /control mechanism; neurochemistry; laboratory rat;

DESCRIPTION (provided by applicant): One of the most challenging aspects of treating drug addiction is preventing relapse due to daily challenges such as stress or exposure to drugs or drug-associated cues. Despite many years of research, no generally accepted pharmacotherapy exists. Aerobic exercise has beneficial effects on both physical and mental health, suggesting that it may also be an effective therapy for the treatment of drug dependence. We and others have shown that activation of the central noradrenergic system is essential for stress-induced and drug-primed reinstatement in the rat model of drug self- administration. We have also discovered that chronic exercise increases expression of the neuropeptide galanin in noradrenergic neurons and impairs stress-induced norepinephrine release. The purpose of this proposal is to test the hypothesis that voluntary exercise can attenuate stress-induced and drug-primed reinstatement of cocaine seeking. In Aim 1 of this proposal, we will determine whether chronic voluntary exercise (wheel running) blunts stress-induced or drug-primed reinstatement of cocaine seeking in rats. In Aim 2, we will determine whether a negative correlation exists between the magnitude of wheel running-induced galanin expression in the noradrenergic locus coeruleus and reinstatement, and whether blockade of galanin signaling reverses the beneficial effects of exercise. In Aim 3, we will further investigate the interaction between exercise-induced galanin expression and norepinephrine release. PUBLIC HEALTH RELEVANCE: Drug addiction is a chronic, relapsing disease that is very difficult to treat and places enormous social and economic stress on society. Aerobic exercise is beneficial for many aspects of physical and mental health, and may be beneficial for the treatment of drug dependence. The purpose of this proposal is to assess the effects of aerobic exercise in a rat model of drug relapse, and to investigate potential underlying mechanisms. Completion of these experiments may indicate a new therapy for the treatment of drug addiction.

Animal locomotion begins in the central nervous system and results in quantifiable mechanical activity; it therefore provides an excellent window into the neural computations that create intentional behaviors and respond to environmental conditions. Locomotion originates in central pattern generators: neural networks in the spines of vertebrates and thoracic ganglia of insects that produce rhythmic movements. Some preparations (e.g. lamprey, crayfish) can produce stable, near-periodic rhythms in isolation, others either require sensory feedback to produce functional gaits or are significantly stabilized by it.

Cockroaches and stick insects exemplify these two extremes. They share the same basic neural and biomechanical architecture, but the former run rapidly over rough ground, while the latter are adapted for slow walking on twigs and leaves with varied orientations to gravity. Using existing information and collecting new data, this project will compare these species and address questions such as:

(1) What is the functional organization of feedforward motor coordination: How are neural circuits that drive individual legs and joints coupled to achieve inter-limb coordination?

(2) What is the role of sensory input in coordination: How does feedback affect motor patterns, and how does locomotion modulate incoming sensory information?

(3) How are pattern generators and sensory feedback systems modulated to create appropriate actions as animals change speed, face unexpected perturbations, and maneuver to negotiate complex terrain?

The development of integrated mathematical and computational models is central to answering such questions. New data will improve existing models, and new models will be created to span the morphological and behavioral ranges from stick insects to cockroaches, allowing us to illuminate their adaptive strategies to different environments. This research will deepen our understanding of the generation and control of locomotion, with general relevance to animals and humans.

The US component of this project is jointly funded by the Mathematical Biology program in the Division of Mathematical Sciences and the Neural Systems Cluster in the Division of Integrative Organismal Systems. The German component is funded by the German Ministry of Education and Research (BMBF).