Martin Golubitsky - US grants

Bifurcations in differential equations with symmetry will be studied from a theoretical and applied standpoint. In this context the term "bifurcation" refers to a radical change in the solution set of an equation as parameters in the equation pass through certain critical values. A particular system under consideration is a model of Conette-Taylor flow in which fluid systems with symmetries other than rotational are also studied.

9403624 Golubitsky In this project we study both the theory and application of symmetric dynamical systems. In many applications, symmetry is introduced either by the supposition of a regular geometry or by the interchangability of parts in a complex system. It has been well established that the existence of regular patterns in experiments and in computer simulations are often due to symmetry. Only recently has it been shown that the existence of intermittency in the dynamics and patterns in the time-average of solutions or experiments can also be forced by symmetry. It is the purpose of this proposal to investigate these and related phenomena. In particular we will study how symmetry and chaotic dynamics intertwine in dynamical systems and how symmetry affects the types of solutions found in systems of coupled cells. In this project we will study iteration of symmetric maps (symmetric chaos), the investigation of coupled systems of ODEs where each subsystem or cell has its own internal symmetry, the stability of heteroclinic cycles (which occur naturally in symmetric systems of differential equations), and the effect of forced symmetry breaking (where perturbations of a symmetric equation to one with less symmetry are considered). Applications to patterns on average in systems of PDEs and experiments (particularly in the Taylor-Couette system and the Faraday surface wave experiment) will be considered as will the existence of intermittency (heteroclinic cycles) in an experiment modeling convection in a porous media. In addition, the dynamics of a number of differential equations with symmetry will be studied by direct computer simulation and the results compared with current theory.

9704980 Golubitsky Bifurcations to spiral waves and to cellular patterns in partial differential equations posed on a circular disk will be investigated. In unbounded domains, the bifurcation and meandering of spirals and scroll waves and the Ginzburg-Landau theory of spatially extended systems will be considered -- - with application to spatially aperiodic solutions in spatially extended systems. In addition, part of the effort will be devoted to studying the dynamics present in ordinary differential equations with symmetry. Stable ergodicity of chaotic attractors in problems with continuous symmetry, and the existence, stability and bifurcations of robust heteroclinic cycles will be studied. Some of these ideas are relevant to intermittent magnetic dynamos in rotating convection. Patterns appear in physical, chemical, and biological systems and are characteristically striking and reproducible. Consequently, scientists and mathematicians have developed theories to explain the origins of these patterns. There are several approaches to the study of pattern; this one is based on symmetry and bifurcation. It is proposed to investigate the theory and application of symmetric dynamical systems and its relation to pattern formation. There are fundamental differences in the analyses depending on whether the patterns being studied fit neatly into a bounded domain or whether boundaries are unimportant and the equations are posed on infinite domains. Aspects of both cases will be studied.

A system of differential equations is called a cell. A coupled cell network is a collection of cells that are coupled together and the architecture of that network is a graph that indicates which cells are identical and which cells are coupled to which. The primary question addressed by this proposal is: What part of the dynamics of coupled systems is due to network architecture? The theory of these systems was developed with recent NSF support: we will continue its development by focusing on bifurcations and forcing of feedforward networks, stability of solutions obtained by bifurcation, and bursting in coupled systems. We will also focus more directly on applications including the vestibular system (in particular, the network of connections between neurons of the six semicircular canals in the ears and eight neck muscle groups), the development of a frequency filter/amplifier associated to synchrony breaking Hopf bifurcation in a simple feedforward network, and `cortical songs' obtained from different dynamical patterns in coupled phase oscillators.

The biologist J.B.S. Haldane, when asked what we can learn about the Creator by examining the world, replied that God seemed to have an inordinate fondness for beetles. Today's biologists could be forgiven for pointing to the deity's inordinate fondness for networks. Networks are ubiquitous in biology: examples include gene expression, neural circuitry, ecological food webs, and disease transmission. Networks are also common in many other branches of science, and there has been a recent explosion of interest in the topic. The research literature, including applications, now extends to many thousands of papers. We have been developing a theory of network dynamics, where the nodes in the network are systems of differential equations. In this proposal we focus on some generalizations of the theory and on specific applications including the dynamics associated to a network in the vestibular system called the canal-neck projection.

Proposal: DMS-0635561
Principal Investigator: Friedman, Avner
Institution: Ohio State University
Proposal Title: Mathematical Biosciences Institute

ABSTRACT

This is a renewal proposal for the Mathematical Biosciences Institute (MBI) at Ohio State University (OSU). The mathematical biosciences include the biological, medical, and environmental sciences where the application of mathematical, statistical, and computational science may lead to significant progress in understanding key processes in the application area. The mission of the MBI is (i) to develop mathematical theories, statistical methods, and computational algorithms for the solution of fundamental problems in the biosciences; (ii) to involve mathematical scientists and bioscientists in the solution of these problems; and (iii) to nurture a community of scholars through education and support of students and researchers in the mathematical biosciences. The Institute organizes a yearly program consisting of 6-8 workshops and several tutorials around selected themes. Workshops emphasize discussions over formal talks and offer an environment that encourages developing collaborations. The themes selected for the next three years are Bioengineering (2007-'08), Developmental Biology (2008-'09), and From Genes to Cells: Networks, Scales, and Complexity (2009-'10). A unique postdoc program is designed to develop young researchers into truly interdisciplinary scientists. They are immersed in a broad range of topics and interact with workshop participants, visiting scientists, and local researchers, while being co-mentored by a mathematical scientist and a life scientist.

The Mathematical Biosciences Institute (MBI) at Ohio State University (OSU) is one of seven Mathematical Sciences Institutes (http://mathinstitutes.org/) supported by the NSF Division of Mathematical Sciences. The MBI supports programs that promote multidisciplinary research at the interface of the mathematical and life sciences. It hosts scientific programs on long (academic year) and short (one week or less) time scales. The programs are organized around selected themes, which range from Mathematical Neurosciences (2002-'03) to Systems Physiology (2006-'07) and From Genes to Cells: Networks, Scales, and Complexity (2009-'10). The MBI provides opportunities for researchers in the mathematical and life sciences to interact and form new collaborations. Most of the Institute's programs are conducted on the OSU campus, but the MBI also sponsors conferences and workshops at its academic Institute Partners. These Institute Partners and MBI's Corporate Partners contribute to the Institute's programming and planning through dedicated governance committees. The MBI website (http://www.mbi.ohio-state.edu/) describes upcoming and past programs, solicits ideas for future programs, and offers application forms for Institute activities.

The principal investigator and his students and collaborators study the dynamics of networks of systems of differential equations and their applications. Previous work identified when network architecture forces synchrony to be present in such systems (generalizing network symmetry to a combinatorial notion of a balanced coloring). Balanced colorings also lead to quotient networks and previous work also showed that robust phase shift synchrony can be associated to symmetry in a quotient network. These theories were motivated by studies of locomotor central pattern generators of animal gaits. The studies have progressed and several general questions are now studied like: Are all robust phase shifts present in periodic solutions of network equations the result of symmetry (using quotient networks), what are the patterns of oscillations associated with network architecture, and do these patterns always appear through bifurcation? (Surprisingly the answer to the last question is no; additionally, network architecture can change Hopf bifurcation in ways that transcend symmetry.) How network dynamics affects Takens reconstruction is studied (is it possible to reconstruct network dynamics from the output of one node) and the dynamics of small networks (can general network studies lead to a better understanding of motifs in Systems Biology) is studied.

The study of networks of differential equations is central to much of modern biology (from biochemical networks through protein interaction networks and gene transcription networks to ecology through food webs). Recent work by many research groups (usually based on chemical kinetics equations) has shown that network architecture does affect the kind of solutions that appear in coupled systems. Moreover, these solutions lead to function and to conjectures about how large biological networks work. General mathematical studies of network dynamics are providing both the tools needed to investigate more specific networks and an understanding of the kinds of phenomena that network models can produce. The project involves studying general theories for network dynamics, specific examples of networks that promise to yield new (mathematical) phenomena, and specific application areas (such as the ways in which sound waves are processed in the cochlear region of the inner ear, which in part is based on a network model of inner hair bundles).

The goal of this project is to establish a multi-institution research experience for undergraduate students (REU) in the mathematical biosciences, facilitated by the Mathematical Biosciences Institute (MBI) at The Ohio State University. The objectives of the program are: (1) to introduce undergraduate students at all levels to the field of mathematical biology; (2) to encourage students to pursue graduate study in the mathematical biosciences; and (3) to increase the number of students who enter the workforce with training in this field. Students will work on applying mathematics and statistics to research in areas such as molecular evolution, neuronal patterns, cancer modeling, epidemic models and vaccination strategies, and models of sensory systems, such as vision and smell. The infrastructure provided by the MBI gives the necessary resources to accomplish these goals by providing both an authentic research experience as well as exposure to the field of mathematical biology broadly.

The program consists of three components. The first is a week-long Overview of Mathematical Biosciences, which consists of lectures by experts in the field, laboratory tours and field trips, and computer exercises using the Matlab software. This is held at the MBI in Columbus, OH annually in the second week of June. Following completion of the one-week overview session, the students will travel to one of MBI's Institute Partners to participate in an 8-week mentored research experience at the host institution. Finally, the participants will join other students doing research in the mathematical biosciences and present their results at a Capstone Conference hosted by the MBI. Throughout the program, a cohort will be maintained by weekly virtual seminars and discussions among participating institutions. They will work in pairs under the guidance of one or more mentors to make genuine research contributions in these areas, often leading to publications in peer-reviewed journals and presentations at conferences. The student participants will develop skills needed to continue work in these areas and to pursue graduate study. Their involvement in this research will expand the group of students trained to work in the field of mathematical biology.