CTW: New Developments in Dynamical Systems Arising from the Biosciences

(March 22,2011 - March 26,2011 )

Organizers


Tasso Kaper
Mathematics, Boston University
Bernd Krauskopf
Dept of Engineering Mathematics, University of Bristol
Hinke Osinga
Department of Engineering Mathematics, University of Bristol
Martin Wechselberger
School of Mathematics and Statistics, University of Sydney

The biosciences provide rich grounds for mathematical problems, and many questions require the development of new mathematical theory and algorithms. With this workshop we give particular attention to new ideas and developments in dynamical systems. We have chosen four themes to showcase how the biosciences inspired recent progress: systems with delays, systems with multiple scales, dynamics of networks, and stochastic bifurcation theory. The meeting will highlight and discuss new directions of fundamental research in each of the themes, how they are connected, and how they contribute to the understanding of specific questions in bioscience applications.

Accepted Speakers

Dimitri Breda
Department of Mathematics and Computer Science, University of Udine
Evelyn Buckwar
Dept. of Mathematics, Heriot-Watt University
Sue Ann Campbell
Applied Mathematics, University of Waterloo
Steve Coombes
School of Mathematical Sciences, University of Nottingham
Bard Ermentrout
Department of Mathematics, University of Pittsburgh
Ingo Fischer
IFISC, University of the Balearic Islands and the Spanish National Research Council
Barbara Gentz
Faculty of Mathematics, University of Bielefeld
John Guckenheimer
Mathematics Department, Cornell University
Ron Harris-Warrick
Neurobiology and Behavior, Cornell University
Phil Holmes
Program in Applied & Computational Mathematics, Princeton University
Ale Jan Homburg
KdV Institute for Mathematics, University of Amsterdam
Tony Humphries
Mathematics and Statistics, McGill University
Vivien Kirk
Mathematics, The University of Auckland
Dan Koditschek
Electrical & Systems Engineering, University of Pennsylvania
Rachel Kuske
Mathematics , University of British Columbia
Adilson Motter
Physics and Astronomy, Northwestern University
John Rinzel
Center for Neural Science, New York University
Arthur Sherman
National Institutes of Health
Jan Sieber
Dept. of Mathematics, University of Portsmouth
Peter Szmolyan
Institut for Analysis and Scientific Computing, Vienna University of Technology
Lev Tsimring
BioCircuits Institute, University of California, San Diego
John White
Department of Bioengineering, University of Utah
Lai-Sang Young
Courant Institute of Mathematical Sciences, New York University
Tuesday, March 22, 2011
Time Session
09:00 AM
09:30 AM
John Guckenheimer - Math to Bio and Bio to Math
The interchange between dynamical systems theory with biology has had lasting impact upon both. As biology becomes increasingly quantitative, this relationship is likely to strengthen still further. This lecture will review my experience as a mathematician working at the interface with biology, emphasizing the role of multiple time scales in biological models. It will also look discuss why the solution of outstanding mathematical questions is essential to progress within biology.
09:45 AM
10:15 AM
Ingo Fischer - Synchronization phenomena in delay-coupled network motifs
Delayed coupling of oscillators can give rise to dynamical instabilities and the onset of characteristic synchronization phenomena. We show that two elements coupled with long delay will never synchronize isochronously. Nevertheless, isochronous synchronization of the chaotic dynamics can occur via coupling through a relay element. This has been demonstrated for lasers in experiments and modeling. This mechanisms has been extended to neuronal systems, where isochronously synchronized activity is assumed to underlie cognitive processes. Therefore, a major question is the stability of this state and the limits of its stability. In this talk we will present results from a stability analysis of delay-coupled lasers using either an additional laser or a semitransparent mirror as relay element. We discuss the occurrence of destabilizing mechanisms, including blow-out bifurcations and bubbling. Finally, we present that isochronous identical synchronization between distant elements can be even obtained when the relay element via which they are coupled exhibits uncorrelated dynamics. Even mutual information between the synchronized dynamics and the relay dynamics can vanish. We discuss the implications for neuronal networks.
11:00 AM
11:30 AM
Adilson Motter - Identifying Compensatory Perturbations in Biological Networks
Our recent research shows that a faulty or sub-optimally operating metabolic network can often be rescued by the targeted removal of enzyme-coding genes. Predictions go as far as to assert that certain gene knockouts can restore the growth of otherwise nonviable gene-deficient cells. In this talk, I will discuss how the theory of dynamical systems can be combined with network modeling to develop computational methods for the systematic identification of compensatory perturbations and rescue interactions in a range of biological contexts. The proposed problem is mathematically challenging and has the potential to illuminate biological and medical research.

Main references: A.E. Motter, Improved network performance via antagonism: From synthetic rescues to multi-drug combinations, BioEssays 32, 236 (2010); A.E. Motter, N. Gulbahce, E. Almaas, A.-L. Barabasi, Predicting synthetic rescues in metabolic networks, Molecular Systems Biology 4, 168 (2008).
11:45 AM
12:15 PM
Arthur Sherman - Cross-currents between Biology and Mathematics on Models of Bursting
I will trace the history of models for bursting, concentrating on square-wave bursters descended from the Chay-Keizer model for pancreatic beta cells. The model was originally developed on a biophysical and intutive basis but was put into a mathematical context by John Rinzel's fast-slow analysis. Rinzel also began the process of classifying bursting oscillations based on the bifurcations undergone by the fast subsystem, which led to important mathematical generalization by others. Further mathematical work, notably by Terman, Mosekilde and others, focused rather on bifurcations of the full bursting system, which showed a fundamental role for chaos in mediating transitions between bursting and spiking and between bursts with different numbers of spikes. The development of mathematical theory was in turn both a blessing and a curse for those interested in modeling the biological phenomena - having a template of what to expect made it easy to construct a plethora of models that were superficially different but mathematically redundant. This may also have steered modelers away from alternative ways of achieving bursting, but instructive examples exist in which unbiased adherence to the data led to discovery of new bursting patterns. Some of these had been anticipated by the general theory but not previously instantiated by Hodgkin-Huxley-based examples. A final level of generalization has been the addition of multiple slow variables. While often mathematically reducible to models with a one-variable slow subsystem, such models also exhibit novel resetting properties and enhanced dynamic range. Analysis of the dynamics of such models remains a current challenge for mathematicians.
03:30 PM
04:00 PM
Rachel Kuske - Noise sensitivities in systems with delays and multiple time scales
Dynamical systems with delayed feedback often exhibit complex oscillations not observed in analogous systems without delay. Stochastic effects can change the picture dramatically, particularly if multiple time scales are present. Then transients ignored in the deterministic system can dominate the long range behavior. This talk will contrast the effects of different noise sources in certain systems with delayed feedback. We show how ideas from canonical physical and mechanical systems can be applied in biological models for disease and balance. The approaches we consider capture the effects of noise and delay in the contexts of piecewise smooth systems, nonlinearities, and discontinuities.
04:15 PM
04:45 PM
Lev Tsimring - Generation and synchronization of oscillations in synthetic gene networks
In this talk, I will describe our recent experimental and theoretical work on small synthetic gene networks exhibiting oscillatory behavior. Most living organisms use internal genetic "clocks" to govern fundamental cellular behavior. While the gene networks that produce oscillatory expression signals are typically quite complicated, certain recurring network motifs are often found at the core of these biological clocks. One common motif which may lead to oscillations is delayed auto-repression. We constructed a synthetic two-gene oscillator based on this design principle, and observed robust and tunable oscillations in bacteria. Computational and theoretical modeling suggests that the key mechanism of oscillations is a small time delay in the negative feedback loop. In a strongly nonlinear regime, this time delay can lead to long-period oscillations that can be characterized by "degrade and fire" dynamics. We also demonstrated synchronization of synthetic gene oscillators across cell population using a variant of the same design in which oscillators are synchronized by a chemical signal freely diffusing through cell membranes.
Wednesday, March 23, 2011
Time Session
09:00 AM
09:30 AM
Steve Coombes - Waves in random neural media
The propagation of waves of neural activity across the surface of the brain is known to subserve both natural and pathological neurobiological phenomena. An example of the former is spreading excitation associated with sensory processing, whilst waves in epilepsy are a classic example of the latter. There is now a long history of using integro-differential neural field models to understand the properties of such waves. For mathematical convenience these models are often assumed to be spatially translationally-invariant. However, it is hard even at a first approximation to view the brain as a homogeneous system and so there is a pressing need to develop a set of mathematical tools for the study of waves in heterogeneous media that can be used in brain modeling. Homogenization is one natural multi-scale approach that can be utilized in this regard, though as a perturbation technique it requires that modulation on the micro-scale be both small in amplitude and rapidly varying in space. In this talk I will present novel techniques that improve upon this standard approach and can further tackle cases where the inhomogeneous environment is modeled as a random process.
09:45 AM
10:15 AM
Peter Szmolyan - Geometric singular perturbation theory beyond the standard form
In many biological models multiple time scale dynamics occurs due to the presence of variables and parameters of very different orders of magnitudes. Situations with a clear "global" separation into fast and slow variables governed by singularly perturbed ordinary differential equations in standard form have been investigated in great detail.

For multi-scale problems depending on several parameters it can already be a nontrivial task to identify meaningful scalings. Typically these scalings and the corresponding asymptotic regimes are valid only in certain regions in phase-space or parameter-space. Another issue is how to match these asymptotic regimes to understand the global dynamics. In this talk I will show in the context of examples from enzyme kinetics that geometric methods based on the blow-up method provide a systematic approach to problems of this type.

(Joint work with Ilona Kosiuk, MPI MIS Leipzig)
11:00 AM
11:30 AM
Vivien Kirk - Understanding intracellular calcium dynamics: modelling and mathematics
Change in the concentration of free intracellular calcium is a crucial control mechanism in almost every cell type, with oscillations of calcium concentration being thought to play an important role in muscle contraction, secretion, cardiac electrophysiology and many other aspects of cell physiology. Experiments have been done in a number of different physiological settings to investigate intracellular calcium dynamics, with the results used to construct mathematical models of intracellular calcium dynamics. A main aim of experimental and modeling work is to identify the mechanisms underlying calcium oscillations.

In this talk, I will show how attempts to understand the oscillatory dynamics of calcium models has given rise to new results in bifurcation theory and geometric singular perturbation theory. I will also briefly outline some areas where current gaps in theory are delaying our understanding of the models.
11:45 AM
12:15 AM
Jan Sieber - Periodic orbits in problems with state-dependent delays
Delays in feedback loops tend to destabilize dynamical systems, inducing self-sustained oscillations or chaos. I will show some typical examples in my presentation. I will also show how one can reduce the study of periodic oscillations in systems with delay to low-dimensional smooth algebraic systems of equations. The approach works also when the delay depends on the state, a case in which it is not clear in general if the underlying differential equations are smooth dynamical systems.
Thursday, March 24, 2011
Time Session
09:00 AM
09:30 AM
Evelyn Buckwar - Stability analysis for stochastic delay differential equations
Stochastic delay differential equations often arise in biosciences as models involving, e.g., negative feedback terms and intrinsic or extrinsic noise. Examples of applications range from stochastic models of human immune response systems, neural networks or neural fields to genetic regulatory systems. Stability theory for stochastic delay differential equations is quite well established and we will provide a brief review of available methods and results. Stochastic dynamical systems theory for stochastic delay differential equations beyond the stability analysis of equilibria is much less developed and we will report on some open problems in this area.
09:45 AM
10:15 AM
Dimitri Breda - Numerics for stability analysis of delay systems and population dynamics
The plan is to divide the talk in three distinct but related parts.

First, the question of asymptotic stability for equilibria of delay differential equations is addressed numerically. The proposed method, based on the discretization of the infinitesimal generator of the solution operator semigroup via pseudospectral differentiation, allows to approximate the stability determining eigenvalue with spectral accuracy. Hence it is fast and suitable for robust analysis.

Second, the numerical scheme is extended for investigating the stability of steady states of population dynamics, where the study of the associated transcendental characteristic equations is often too difficult to be approached analytically. The fruitful interplay between theoretical and numerical analysis is highlighted through examples taken from age- and physiologically-structured models, as well as delayed epidemics.

Third, recent advances in the numerical stability analysis of delay systems are illustrated, showing how equilibria (characteristic roots), periodic orbits (Floquet multipliers) and chaotic motion (Lyapunov exponents) can be faced under the same discretization framework. Examples arising in the populations context are discussed which demand for adapting such treatment.
11:00 AM
11:30 AM
Ale Jan Homburg - Bounded noise: bifurcations of random dynamical systems
Random dynamical systems with bounded noise can have multiple stationary measures with different supports. Under variation of a parameter, such as the amplitude of the noise, bifurcations of these measures may occur. We discuss such bifurcations both in a context of random diffeomorphisms and of random differential equations.

References:

* A.J. Homburg, T. Young. Bifurcations for random differential equations with bounded noise on surfaces Topol. Methods Nonlinear Anal. 35 (2010), 77-98.
* H. Zmarrou, A.J. Homburg. Dynamics and bifurcations of random circle diffeomorphisms Discrete Contin. Dyn. Syst. Ser. B 10 (2008), 719-731.
* H. Zmarrou, A.J. Homburg. Bifurcations of stationary measures of random diffeomorphisms Ergod. Th. and Dynam. Sys. 27 (2007), 1651-1692.
11:45 AM
12:30 PM
Sue Ann Campbell - Phase Models for Oscillators with Time Delayed Coupling
We consider a network of inherently oscillatory neurons with time delayed connections. We reduce the system of delay differential equations to a phase model representation and show how the time delay enters into the reduced model. For the case of two neurons, we show how the time delay may affect the stability of the periodic solution leading to stability switching between synchronous and antiphase solutions as the delay is increased. Numerical bifurcation analysis of the full system of delay differential equations is used determine constraints on the coupling strength such that the phase model is valid. Both type I and type II oscillators are considered.
Friday, March 25, 2011
Time Session
09:00 AM
09:30 AM
Bard Ermentrout - Noisy oscillators
Noise interacts with oscillators in often counterintuitive ways. In this talk, I will discuss the interactions of noise with coupling in systems of limit cycle oscillators.
09:45 AM
10:15 AM
John White - Dynamic-clamp studies of neuronal synchronization
Coherent neuronal activity is ubiquitous and presumably important in brain function. I will review my group's experimental studies of the mechanisms underlying coherent activity using dynamic clamp technology, which allows us to perform virtual-reality-inspired experiments in neurons in vitro. Using these techniques and mathematical tools from dynamical systems theory, we are trying to understand which factors give rise to stable neuronal synchronization in the presence of heterogeneity, noise, and conduction delays.
11:00 AM
11:30 AM
Barbara Gentz - The effect of noise on mixed-mode oscillations
Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behaviour. In this talk, we will focus on MMOs in a slow-fast dynamical system with one fast and two slow variables, containing a folded-node singularity. The main question we will address is whether and how noise may change the dynamics.

We will first outline a general approach to stochastic slow-fast systems which allows

1. to construct small sets in which the sample paths are typically concentrated, and
2. to give precise bounds on the exponentially small probability to observe atypical behaviour.

Applying this method to our model system shows the existence of a critical noise intensity beyond which the small-amplitude oscillations become hidden by noise. Furthermore, we will show that in the presence of noise sample paths are likely to jump away from so-called canard solutions earlier than the corresponding deterministic orbits. This early-jump mechanism can drastically change the mixed-mode patterns, even for rather small noise intensities.

Joint work with Nils Berglund (Orleans) and Christian Kuehn (Dresden).
11:45 AM
12:15 PM
Tony Humphries - Dynamics of Differential Equations with Multiple State Dependent Delays
The Mackey-Glass equation is a seemingly simple delay differential equation (DDE) with one fixed delay which can exhibit the full gamut of dynamics from a trivial stable steady state to fully chaotic dynamics, and has inspired decades of mathematical research into DDEs. However, much of that research has focused on equations with fixed or prescribed delays, whereas many biological delays would be more naturally modelled as state-dependent delays. Before incorporating state-dependent delays in complex biochemical network models, it is desirable to understand the dynamics which result from including state-dependent delays in simpler model problems. Accordingly, in this talk we will consider a simple model problem with multiple state-dependent delays, and show that it can exhibit a wide range of dynamical behaviour, including stable periodic solutions and bi-stable periodic solutions, to stable tori, together with the associated bifurcation structures.
03:30 PM
04:00 PM
Dan Koditschek - Gaits, Gait Obstacles and Gait Assays
Over the last decade, inspired by several key animal studies, my collaborators and I have extended the domain of dynamically dexterous legged robots to include running over rough natural terrain, quasi-static climbing of exterior vertical walls and trees, and more recently, dynamical ascents of more structured vertical surfaces. In so doing we have found it advantageous to engineer gait generators through the interconnection of internal coupled oscillators with the mechanical body and limbs through various proprioceptive and vestibular feedback channels. When construed in this manner as attracting limit cycles on the torus, the organization of these resulting gaits takes the form of a cell complex whose adjacency relations structure the design of robust steady behaviors and safe transitions between them.

Often, the imperatives of locomotion impose additional constraints (e.g., keep at least two limbs in contact with the wall at all times) manifest as combinatorial obstacles in the gait space, complicating any refinement of the gait complex that would effect their excision. We have been developing gait generators and gait transition mechanisms that respect these obstacles and I will give some examples of work in progress. The gait complex and its combinatorially punctured variants have characteristic topological signatures that constrain the manner in which the basins of distinct attracting limit cycles achieved by smooth controllers can fit together. Such constraints impact the engineering of dynamical gait controllers for robots and to the extent that animal motion controllers target smooth dynamical systems, they must impact the animals' designs as well. I will close with some speculation about the possibility for developing gait assays that probe those designs. If we hypothesize that biological preflexes (i.e., those animal motion controllers implemented by the tuned musculoskeletal system) must be smooth, then observations of animal gait transitions may help gain greater insight into the boundary between neural and mechanical control.
04:15 PM
04:45 PM
Lai-Sang Young - Dynamics of neuronal networks as models of visual cortex
I will discuss joint work with Aaditya Rangan in which we model a small patch of layer 2 of the primary visual cortex (V1) as a large network of point neurons. Network architecture is chosen to reflect a few coarse structures of V1. Our aim is to understand macroscopic observations from dynamics on the neuronal level. Using biological data to constrain parameters, we arrive at models which exhibit a number of empirically observed V1 phenomena (including e.g. localized receptive fields and spontaneous pattern formation in background). In this talk, I will discuss dynamical mechanisms behind a phenomenon called surround suppression.
Saturday, March 26, 2011
Time Session
09:45 AM
10:15 AM
Hinke Osinga - Computing 2D invariant manifolds: Can you do this?
The Lorenz system is the classical example of a seemingly simple dynamical system that exhibits chaotic dynamics. In fact, there are numerous studies to characterize the complicated dynamics on the famous butterfly attractor. This talk addresses how the dynamics is organized more globally. An important role in this regard is played by the stable manifold of the origin, also known as the Lorenz manifold. In 1992 John Guckenheimer suggested this manifold as a bench-mark challenge for developing computational methods in dynamical systems. We show how the numerical continuation of orbit segments can be used to investigate and characterize the transition to chaos in the Lorenz system.

Joint work with Eusebius Doedel (Concordia University, Montreal) and Bernd Krauskopf (University of Bristol).
10:30 AM
11:30 AM
John Rinzel - John Rinzel Lecture
John Rinzel Lecture
11:00 AM
11:30 AM
- Animal gaits and symmetries of periodic solutions
In the first part of this talk I will briefly describe previous work on quadruped gaits (which distinguishing gaits by their spatio-temporal symmetries). In the second part, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations. This work is joint with David Romano, Yunjiao Wang, and Ian Stewart.
11:45 AM
12:15 PM
Ron Harris-Warrick - Modeling Neural Networks for Rhythmic Movements
Central Pattern Generators (CPGs) are limited neural networks that drive rhythmic behaviors such as locomotion, respiration and mastication. We have been studying the structure, function, and modulation of CPGs, with an emphasis on neuronal and ionic mechanisms that allow flexibility in the output from an anatomically defined network. Both biological and modeling studies show that individual oscillatory neurons can be modulated to generate bursting activity by a variety of independent ionic mechanisms, allowing flexibility in the frequency and output properties of these important neurons. The phasing of neuronal activity in the rhythmic pattern is not determined only by the pattern of synaptic connections; the intrinsic electrophysiological properties of the neurons also play a major role. These points raise issues with regard to the appropriate level of complexity in models of neural networks. I will discuss these issues based on work done in collaboration with John Guckenheimer on the pyloric network in the crustacean stomatogastric ganglion and the rodent spinal locomotor CPG. Supported by NIH grants NS17323, NS050943 and NSF grant IOS-0749467
Name Email Affiliation
Ambike, Satyajit ambike.1@buckeyemail.osu.edu
Armbruster, Dieter armbruster@asu.edu Mathematics, Arizona State University
Baesens, Claude Claude.Baesens@warwick.ac.uk Mathematics Institute, The University of Warwick
Barreiro, Andrea akb6@washington.edu Applied Mathematics, University of Washington
Bertram, Richard bertram@math.fsu.edu Mathematics Department, Florida State University
Breda, Dimitri dimitri.breda@uniud.it Department of Mathematics and Computer Science, University of Udine
Buckwar, Evelyn e.buckwar@hw.ac.uk Dept. of Mathematics, Heriot-Watt University
Buzzard, Greg buzzard@math.purdue.edu Dept. of Mathematics, Purdue University
Campbell, Sue Ann sacampbell@uwaterloo.ca Applied Mathematics, University of Waterloo
Cannon, Jonathan cannon@math.bu.edu Mathematics and Statistics, Boston University
Cipra, Barry bcipra@rconnect.com Freelance Mathematics Writer
Clewley, Robert rclewley@gsu.edu Neuroscience Institute and Department of Mathematics and Statistics , Georgia State University
Coombes, Stephen stephen.coombes@nottingham.ac.uk School of Mathematical Sciences, University of Nottingham
Cortez, Michael mhc37@cornell.edu School of Biology , Georgia Institute of Technology
Diniz Behn, Cecilia cdbehn@umich.edu Mathematics , Gettysburg College
Ermentrout, Bard bard@pitt.edu Department of Mathematics, University of Pittsburgh
Fernandez, Bastien Bastien.Fernandez@cpt.univ-mrs.fr Centre de Physique Theorique , Centre National de la Recherche Scientifique (CNRS)
Fischer, Ingo ingo@ifisc.uib-csic.es IFISC, University of the Balearic Islands and the Spanish National Research Council
Gentz, Barbara gentz@math.uni-bielefeld.de Faculty of Mathematics, University of Bielefeld
Govinder, Kesh govinder@ukzn.ac.za Mathematics, Statistics and Computer Science, University of KwaZulu-Natal
Guckenheimer, John jmg16@cornell.edu Mathematics Department, Cornell University
Haiduc, Radu radu.haiduc@credit-suisse.com Department of Mathematics, Cornell University
Harris-Warrick, Ron rmh4@cornell.edu Neurobiology and Behavior, Cornell University
Hoffman, Kathleen khoffman@math.umbc.edu Mathematics and Statistics , University of Maryland Baltimore County
Holmes, Phil pholmes@princeton.edu Program in Applied & Computational Mathematics, Princeton University
Homburg, Ale Jan a.j.homburg@uva.nl KdV Institute for Mathematics, University of Amsterdam
Huguet, Gemma gemma.huguet@upc.edu Center for Neural Science, New York University
Humphries, Antony tony.humphries@mcgill.ca Mathematics and Statistics, McGill University
Iacob, Andrei axi@ams.org Mathematical Reviews, American Mathematical Society
Johnson, Stewart sjohnson@williams.edu Mathematics and Statistics, Williams College
Johnston, Matthew mdjohnst@math.uwaterloo.ca Applied Mathematics, University of Waterloo
Joo, Jaewook jjoo1@utk.edu Physics, University of Tennessee
Kaper, Tasso tasso@math.bu.edu Mathematics, Boston University
Khibnik, Alexander alexander.i.khibnik@pw.utc.com Control & Diagnostic Systems, Pratt & Whitney, United Technologies Corporation
Kirk, Vivien v.kirk@auckland.ac.nz Mathematics, The University of Auckland
Knobloch, Edgar knobloch@berkeley.edu Physics, University of California, Berkeley
Koditschek, Daniel johnruss@seas.upenn.edu Electrical & Systems Engineering, University of Pennsylvania
Kramer, Mark mak@bu.edu Math and Stats, Boston University
Krauskopf, Bernd b.krauskopf@bristol.ac.uk Dept of Engineering Mathematics, University of Bristol
Krupa, Martin M.Krupa@donders.ru.nl Donders Institute, Radboud Universiteit
Kuske, Rachel rachel@math.ubc.ca Mathematics , University of British Columbia
Lajoie, Guillaume guillaume.lajoie@gmail.com Applied Mathematics, University of Washington
Li, Yao yli@math.gatech.edu Mathematics , Georgia Institute of Technology
Lin, Kevin klin@math.arizona.edu Department of Mathematics, University of Arizona
Mahdi, Adam adam.mahdi@uncc.edu Mathematics , University of North Carolina, Charlotte
Meerkamp, Philipp pm329@cornell.edu Mathematics , Cornell University
Misiurewicz, Michal mmisiure@math.iupui.edu Department of Mathematical Sciences, Indiana University--Purdue University
Motter, Adilson motter@northwestern.edu Physics and Astronomy, Northwestern University
Munther, Dan munther@math.ohio-state.edu Mathematics , The Ohio State University
Murrugarra, David davidmur@vbi.vt.edu Virginia Bioinformatics Institute, Virginia Polytechnic Institute and State University
Ngonghala, Calistus cngongha@mix.wvu.edu Mathematics , West Virginia University
Osinga, Hinke h.m.osinga@bristol.ac.uk Department of Engineering Mathematics, University of Bristol
Pasour, Virginia virginia.pasour@us.army.mil Mathematical Sciences, U.S. Army Research Office
Prokopiou, Sotiris pmxsp@nottingham.ac.uk School of Biosciences, University of Nottingham
Rankin, James James.Rankin@inria.fr NeuroMathComp, INRIA Sophia Antipolis, INRIA
Revzen, Shai shrevzen@seas.upenn.edu School of Engineering and Applied Science, University of Pennsylvania
Riess, Thorsten thorsten.riess@uni-konstanz.de INCIDE, Universitat Konstanz
Rinzel, John rinzel@cns.nyu.edu Center for Neural Science, New York University
Rubin, Jonathan rubin.math.pitt.edu Mathematics, University of Pittsburgh
Ruina, Andy ruina@cornell.edu Theoretical and Applied Mechanics, Cornell University
Schaeffer, David dgs@math.duke.edu Mathematics, Duke University
Scheper, Christopher cjs73@cornell.edu Center for Applied Mathematics, Cornell University
Sheets , Alison sheets.203@osu.edu Mechanical Engineering , The Ohio State University
Sherman, Arthur asherman@nih.gov National Institutes of Health
Sherwood, William wesher@bu.edu Center for Biodynamics, Boston University
Shiau, LieJune shiau@uhcl.edu Mathematics, University of Houston--Clear Lake
Shih, Chih-Wen cwshih@math.nctu.edu.tw Applied Mathematics, National Chiao Tung University
Shlizerman, Eli shlizee@uw.edu Applied Mathematics, University of Washington
Sieber, Jan jan.sieber@port.ac.uk Dept. of Mathematics, University of Portsmouth
Smith, Ruth pmxrs3@nottingham.ac.uk University of Nottingham
Spardy, Lucy les65@pitt.edu Mathematics, University of Pittsburgh
Srinivasan, Manoj srinivasan.88@osu.edu Mechanical and Aerospace Engineering , The Ohio State University
Szmolyan, Peter szmolyan@tuwien.ac.at Institut for Analysis and Scientific Computing, Vienna University of Technology
Thomas, Peter pjthomas@case.edu neuroscience, Oberlin College
Thul, Ruediger ruediger.thul@nottingham.ac.uk School of Mathematical Sciences, University of Nottingham
Tien, Joe jtien@math.ohio-state.edu Department of Mathematics, The Ohio State University
Tsimring, Lev ltsimring@ucsd.edu BioCircuits Institute, University of California, San Diego
Tsygankov, Denis dtsygank@med.unc.edu Department of Pharmacology, University of North Carolina, Chapel Hill
Vladimirsky, Alexander vlad@math.cornell.edu Department of Mathematics, Cornell University
Wang, Ying wang@math.umn.edu Mathematics, University of Minnesota
Wang, Yang wang.1513@osu.edu Mechanical Engineering, The Ohio State University
Wechselberger, Martin wm@maths.usyd.edu.au School of Mathematics and Statistics, University of Sydney
Wedgwood, Kyle pmxkw2@nottingham.ac.uk Mathematical Sciences, University of Nottingham
Wedgwood, Kyle pmxkw2@nottingham.ac.uk School of Mathematical Science, University of Nottingham
White, John john.white@utah.edu Department of Bioengineering, University of Utah
Williams, Robert bob@math.utexas.edu Dept of Mathematics, University of Texas
Wiser, Justin jwiser84@gmail.com Department of Mathematics, The Ohio State University
Yang, Dennis Guang gyang@math.drexel.edu Department of Mathematics, Drexel University
Yang, Lixiang yang.1130@buckeyemail.osu.edu Mechanical Engineering, The Ohio State University
Yarahmadian, Shantia syarahmadian@math.msstate.edu Department of Mathematics, Mississippi State University
Young, Todd youngt@ohio.edu Department of Mathematics, Ohio University
Young , Lai-Sang lsy@cims.nyu.edu Courant Institute of Mathematical Sciences, New York University
Numerics for stability analysis of delay systems and population dynamics
The plan is to divide the talk in three distinct but related parts.

First, the question of asymptotic stability for equilibria of delay differential equations is addressed numerically. The proposed method, based on the discretization of the infinitesimal generator of the solution operator semigroup via pseudospectral differentiation, allows to approximate the stability determining eigenvalue with spectral accuracy. Hence it is fast and suitable for robust analysis.

Second, the numerical scheme is extended for investigating the stability of steady states of population dynamics, where the study of the associated transcendental characteristic equations is often too difficult to be approached analytically. The fruitful interplay between theoretical and numerical analysis is highlighted through examples taken from age- and physiologically-structured models, as well as delayed epidemics.

Third, recent advances in the numerical stability analysis of delay systems are illustrated, showing how equilibria (characteristic roots), periodic orbits (Floquet multipliers) and chaotic motion (Lyapunov exponents) can be faced under the same discretization framework. Examples arising in the populations context are discussed which demand for adapting such treatment.
Stability analysis for stochastic delay differential equations
Stochastic delay differential equations often arise in biosciences as models involving, e.g., negative feedback terms and intrinsic or extrinsic noise. Examples of applications range from stochastic models of human immune response systems, neural networks or neural fields to genetic regulatory systems. Stability theory for stochastic delay differential equations is quite well established and we will provide a brief review of available methods and results. Stochastic dynamical systems theory for stochastic delay differential equations beyond the stability analysis of equilibria is much less developed and we will report on some open problems in this area.
Phase Models for Oscillators with Time Delayed Coupling
We consider a network of inherently oscillatory neurons with time delayed connections. We reduce the system of delay differential equations to a phase model representation and show how the time delay enters into the reduced model. For the case of two neurons, we show how the time delay may affect the stability of the periodic solution leading to stability switching between synchronous and antiphase solutions as the delay is increased. Numerical bifurcation analysis of the full system of delay differential equations is used determine constraints on the coupling strength such that the phase model is valid. Both type I and type II oscillators are considered.
Waves in random neural media
The propagation of waves of neural activity across the surface of the brain is known to subserve both natural and pathological neurobiological phenomena. An example of the former is spreading excitation associated with sensory processing, whilst waves in epilepsy are a classic example of the latter. There is now a long history of using integro-differential neural field models to understand the properties of such waves. For mathematical convenience these models are often assumed to be spatially translationally-invariant. However, it is hard even at a first approximation to view the brain as a homogeneous system and so there is a pressing need to develop a set of mathematical tools for the study of waves in heterogeneous media that can be used in brain modeling. Homogenization is one natural multi-scale approach that can be utilized in this regard, though as a perturbation technique it requires that modulation on the micro-scale be both small in amplitude and rapidly varying in space. In this talk I will present novel techniques that improve upon this standard approach and can further tackle cases where the inhomogeneous environment is modeled as a random process.
Noisy oscillators
Noise interacts with oscillators in often counterintuitive ways. In this talk, I will discuss the interactions of noise with coupling in systems of limit cycle oscillators.
Synchronization phenomena in delay-coupled network motifs
Delayed coupling of oscillators can give rise to dynamical instabilities and the onset of characteristic synchronization phenomena. We show that two elements coupled with long delay will never synchronize isochronously. Nevertheless, isochronous synchronization of the chaotic dynamics can occur via coupling through a relay element. This has been demonstrated for lasers in experiments and modeling. This mechanisms has been extended to neuronal systems, where isochronously synchronized activity is assumed to underlie cognitive processes. Therefore, a major question is the stability of this state and the limits of its stability. In this talk we will present results from a stability analysis of delay-coupled lasers using either an additional laser or a semitransparent mirror as relay element. We discuss the occurrence of destabilizing mechanisms, including blow-out bifurcations and bubbling. Finally, we present that isochronous identical synchronization between distant elements can be even obtained when the relay element via which they are coupled exhibits uncorrelated dynamics. Even mutual information between the synchronized dynamics and the relay dynamics can vanish. We discuss the implications for neuronal networks.
The effect of noise on mixed-mode oscillations
Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behaviour. In this talk, we will focus on MMOs in a slow-fast dynamical system with one fast and two slow variables, containing a folded-node singularity. The main question we will address is whether and how noise may change the dynamics.

We will first outline a general approach to stochastic slow-fast systems which allows

1. to construct small sets in which the sample paths are typically concentrated, and
2. to give precise bounds on the exponentially small probability to observe atypical behaviour.

Applying this method to our model system shows the existence of a critical noise intensity beyond which the small-amplitude oscillations become hidden by noise. Furthermore, we will show that in the presence of noise sample paths are likely to jump away from so-called canard solutions earlier than the corresponding deterministic orbits. This early-jump mechanism can drastically change the mixed-mode patterns, even for rather small noise intensities.

Joint work with Nils Berglund (Orleans) and Christian Kuehn (Dresden).
Math to Bio and Bio to Math
The interchange between dynamical systems theory with biology has had lasting impact upon both. As biology becomes increasingly quantitative, this relationship is likely to strengthen still further. This lecture will review my experience as a mathematician working at the interface with biology, emphasizing the role of multiple time scales in biological models. It will also look discuss why the solution of outstanding mathematical questions is essential to progress within biology.
Modeling Neural Networks for Rhythmic Movements
Central Pattern Generators (CPGs) are limited neural networks that drive rhythmic behaviors such as locomotion, respiration and mastication. We have been studying the structure, function, and modulation of CPGs, with an emphasis on neuronal and ionic mechanisms that allow flexibility in the output from an anatomically defined network. Both biological and modeling studies show that individual oscillatory neurons can be modulated to generate bursting activity by a variety of independent ionic mechanisms, allowing flexibility in the frequency and output properties of these important neurons. The phasing of neuronal activity in the rhythmic pattern is not determined only by the pattern of synaptic connections; the intrinsic electrophysiological properties of the neurons also play a major role. These points raise issues with regard to the appropriate level of complexity in models of neural networks. I will discuss these issues based on work done in collaboration with John Guckenheimer on the pyloric network in the crustacean stomatogastric ganglion and the rodent spinal locomotor CPG. Supported by NIH grants NS17323, NS050943 and NSF grant IOS-0749467
Bounded noise: bifurcations of random dynamical systems
Random dynamical systems with bounded noise can have multiple stationary measures with different supports. Under variation of a parameter, such as the amplitude of the noise, bifurcations of these measures may occur. We discuss such bifurcations both in a context of random diffeomorphisms and of random differential equations.

References:

* A.J. Homburg, T. Young. Bifurcations for random differential equations with bounded noise on surfaces Topol. Methods Nonlinear Anal. 35 (2010), 77-98.
* H. Zmarrou, A.J. Homburg. Dynamics and bifurcations of random circle diffeomorphisms Discrete Contin. Dyn. Syst. Ser. B 10 (2008), 719-731.
* H. Zmarrou, A.J. Homburg. Bifurcations of stationary measures of random diffeomorphisms Ergod. Th. and Dynam. Sys. 27 (2007), 1651-1692.
Dynamics of Differential Equations with Multiple State Dependent Delays
The Mackey-Glass equation is a seemingly simple delay differential equation (DDE) with one fixed delay which can exhibit the full gamut of dynamics from a trivial stable steady state to fully chaotic dynamics, and has inspired decades of mathematical research into DDEs. However, much of that research has focused on equations with fixed or prescribed delays, whereas many biological delays would be more naturally modelled as state-dependent delays. Before incorporating state-dependent delays in complex biochemical network models, it is desirable to understand the dynamics which result from including state-dependent delays in simpler model problems. Accordingly, in this talk we will consider a simple model problem with multiple state-dependent delays, and show that it can exhibit a wide range of dynamical behaviour, including stable periodic solutions and bi-stable periodic solutions, to stable tori, together with the associated bifurcation structures.
Understanding intracellular calcium dynamics: modelling and mathematics
Change in the concentration of free intracellular calcium is a crucial control mechanism in almost every cell type, with oscillations of calcium concentration being thought to play an important role in muscle contraction, secretion, cardiac electrophysiology and many other aspects of cell physiology. Experiments have been done in a number of different physiological settings to investigate intracellular calcium dynamics, with the results used to construct mathematical models of intracellular calcium dynamics. A main aim of experimental and modeling work is to identify the mechanisms underlying calcium oscillations.

In this talk, I will show how attempts to understand the oscillatory dynamics of calcium models has given rise to new results in bifurcation theory and geometric singular perturbation theory. I will also briefly outline some areas where current gaps in theory are delaying our understanding of the models.
Gaits, Gait Obstacles and Gait Assays
Over the last decade, inspired by several key animal studies, my collaborators and I have extended the domain of dynamically dexterous legged robots to include running over rough natural terrain, quasi-static climbing of exterior vertical walls and trees, and more recently, dynamical ascents of more structured vertical surfaces. In so doing we have found it advantageous to engineer gait generators through the interconnection of internal coupled oscillators with the mechanical body and limbs through various proprioceptive and vestibular feedback channels. When construed in this manner as attracting limit cycles on the torus, the organization of these resulting gaits takes the form of a cell complex whose adjacency relations structure the design of robust steady behaviors and safe transitions between them.

Often, the imperatives of locomotion impose additional constraints (e.g., keep at least two limbs in contact with the wall at all times) manifest as combinatorial obstacles in the gait space, complicating any refinement of the gait complex that would effect their excision. We have been developing gait generators and gait transition mechanisms that respect these obstacles and I will give some examples of work in progress. The gait complex and its combinatorially punctured variants have characteristic topological signatures that constrain the manner in which the basins of distinct attracting limit cycles achieved by smooth controllers can fit together. Such constraints impact the engineering of dynamical gait controllers for robots and to the extent that animal motion controllers target smooth dynamical systems, they must impact the animals' designs as well. I will close with some speculation about the possibility for developing gait assays that probe those designs. If we hypothesize that biological preflexes (i.e., those animal motion controllers implemented by the tuned musculoskeletal system) must be smooth, then observations of animal gait transitions may help gain greater insight into the boundary between neural and mechanical control.
Noise sensitivities in systems with delays and multiple time scales
Dynamical systems with delayed feedback often exhibit complex oscillations not observed in analogous systems without delay. Stochastic effects can change the picture dramatically, particularly if multiple time scales are present. Then transients ignored in the deterministic system can dominate the long range behavior. This talk will contrast the effects of different noise sources in certain systems with delayed feedback. We show how ideas from canonical physical and mechanical systems can be applied in biological models for disease and balance. The approaches we consider capture the effects of noise and delay in the contexts of piecewise smooth systems, nonlinearities, and discontinuities.
Identifying Compensatory Perturbations in Biological Networks
Our recent research shows that a faulty or sub-optimally operating metabolic network can often be rescued by the targeted removal of enzyme-coding genes. Predictions go as far as to assert that certain gene knockouts can restore the growth of otherwise nonviable gene-deficient cells. In this talk, I will discuss how the theory of dynamical systems can be combined with network modeling to develop computational methods for the systematic identification of compensatory perturbations and rescue interactions in a range of biological contexts. The proposed problem is mathematically challenging and has the potential to illuminate biological and medical research.

Main references: A.E. Motter, Improved network performance via antagonism: From synthetic rescues to multi-drug combinations, BioEssays 32, 236 (2010); A.E. Motter, N. Gulbahce, E. Almaas, A.-L. Barabasi, Predicting synthetic rescues in metabolic networks, Molecular Systems Biology 4, 168 (2008).
Computing 2D invariant manifolds: Can you do this?
The Lorenz system is the classical example of a seemingly simple dynamical system that exhibits chaotic dynamics. In fact, there are numerous studies to characterize the complicated dynamics on the famous butterfly attractor. This talk addresses how the dynamics is organized more globally. An important role in this regard is played by the stable manifold of the origin, also known as the Lorenz manifold. In 1992 John Guckenheimer suggested this manifold as a bench-mark challenge for developing computational methods in dynamical systems. We show how the numerical continuation of orbit segments can be used to investigate and characterize the transition to chaos in the Lorenz system.

Joint work with Eusebius Doedel (Concordia University, Montreal) and Bernd Krauskopf (University of Bristol).
John Rinzel Lecture
John Rinzel Lecture
Cross-currents between Biology and Mathematics on Models of Bursting
I will trace the history of models for bursting, concentrating on square-wave bursters descended from the Chay-Keizer model for pancreatic beta cells. The model was originally developed on a biophysical and intutive basis but was put into a mathematical context by John Rinzel's fast-slow analysis. Rinzel also began the process of classifying bursting oscillations based on the bifurcations undergone by the fast subsystem, which led to important mathematical generalization by others. Further mathematical work, notably by Terman, Mosekilde and others, focused rather on bifurcations of the full bursting system, which showed a fundamental role for chaos in mediating transitions between bursting and spiking and between bursts with different numbers of spikes. The development of mathematical theory was in turn both a blessing and a curse for those interested in modeling the biological phenomena - having a template of what to expect made it easy to construct a plethora of models that were superficially different but mathematically redundant. This may also have steered modelers away from alternative ways of achieving bursting, but instructive examples exist in which unbiased adherence to the data led to discovery of new bursting patterns. Some of these had been anticipated by the general theory but not previously instantiated by Hodgkin-Huxley-based examples. A final level of generalization has been the addition of multiple slow variables. While often mathematically reducible to models with a one-variable slow subsystem, such models also exhibit novel resetting properties and enhanced dynamic range. Analysis of the dynamics of such models remains a current challenge for mathematicians.
Periodic orbits in problems with state-dependent delays
Delays in feedback loops tend to destabilize dynamical systems, inducing self-sustained oscillations or chaos. I will show some typical examples in my presentation. I will also show how one can reduce the study of periodic oscillations in systems with delay to low-dimensional smooth algebraic systems of equations. The approach works also when the delay depends on the state, a case in which it is not clear in general if the underlying differential equations are smooth dynamical systems.
Geometric singular perturbation theory beyond the standard form
In many biological models multiple time scale dynamics occurs due to the presence of variables and parameters of very different orders of magnitudes. Situations with a clear "global" separation into fast and slow variables governed by singularly perturbed ordinary differential equations in standard form have been investigated in great detail.

For multi-scale problems depending on several parameters it can already be a nontrivial task to identify meaningful scalings. Typically these scalings and the corresponding asymptotic regimes are valid only in certain regions in phase-space or parameter-space. Another issue is how to match these asymptotic regimes to understand the global dynamics. In this talk I will show in the context of examples from enzyme kinetics that geometric methods based on the blow-up method provide a systematic approach to problems of this type.

(Joint work with Ilona Kosiuk, MPI MIS Leipzig)
Generation and synchronization of oscillations in synthetic gene networks
In this talk, I will describe our recent experimental and theoretical work on small synthetic gene networks exhibiting oscillatory behavior. Most living organisms use internal genetic "clocks" to govern fundamental cellular behavior. While the gene networks that produce oscillatory expression signals are typically quite complicated, certain recurring network motifs are often found at the core of these biological clocks. One common motif which may lead to oscillations is delayed auto-repression. We constructed a synthetic two-gene oscillator based on this design principle, and observed robust and tunable oscillations in bacteria. Computational and theoretical modeling suggests that the key mechanism of oscillations is a small time delay in the negative feedback loop. In a strongly nonlinear regime, this time delay can lead to long-period oscillations that can be characterized by "degrade and fire" dynamics. We also demonstrated synchronization of synthetic gene oscillators across cell population using a variant of the same design in which oscillators are synchronized by a chemical signal freely diffusing through cell membranes.
Dynamic-clamp studies of neuronal synchronization
Coherent neuronal activity is ubiquitous and presumably important in brain function. I will review my group's experimental studies of the mechanisms underlying coherent activity using dynamic clamp technology, which allows us to perform virtual-reality-inspired experiments in neurons in vitro. Using these techniques and mathematical tools from dynamical systems theory, we are trying to understand which factors give rise to stable neuronal synchronization in the presence of heterogeneity, noise, and conduction delays.
Dynamics of neuronal networks as models of visual cortex
I will discuss joint work with Aaditya Rangan in which we model a small patch of layer 2 of the primary visual cortex (V1) as a large network of point neurons. Network architecture is chosen to reflect a few coarse structures of V1. Our aim is to understand macroscopic observations from dynamics on the neuronal level. Using biological data to constrain parameters, we arrive at models which exhibit a number of empirically observed V1 phenomena (including e.g. localized receptive fields and spontaneous pattern formation in background). In this talk, I will discuss dynamical mechanisms behind a phenomenon called surround suppression.
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The effect of noise on mixed-mode oscillations
Barbara Gentz Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behavio

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Dynamic-clamp studies of neuronal synchronization
John White Coherent neuronal activity is ubiquitous and presumably important in brain function. I will review my group's experimental studies of the mechanisms underlying coherent activity using dynamic clamp technology, which allows us to perform virtual-

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Phase Models for Oscillators with Time Delayed Coupling
Sue Ann Campbell We consider a network of inherently oscillatory neurons with time delayed connections. We reduce the system of delay differential equations to a phase model representation and show how the time delay enters into the reduced model. For the case of two

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Bounded noise: bifurcations of random dynamical systems
Ale Jan Homburg Random dynamical systems with bounded noise can have multiple stationary measures with different supports. Under variation of a parameter, such as the amplitude of the noise, bifurcations of these measures may occur. We discuss such bifurcations both

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Numerics for stability analysis of delay systems and population dynamics
Dimitri Breda The plan is to divide the talk in three distinct but related parts.

First, the question of asymptotic stability for equilibria of delay differential equations is addressed numerically. The proposed method, based on the discretization o

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Stability analysis for stochastic delay differential equations
Evelyn Buckwar Stochastic delay differential equations often arise in biosciences as models involving, e.g., negative feedback terms and intrinsic or extrinsic noise. Examples of applications range from stochastic models of human immune response systems, neural net

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Periodic orbits in problems with state-dependent delays
Jan Sieber Delays in feedback loops tend to destabilize dynamical systems, inducing self-sustained oscillations or chaos. I will show some typical examples in my presentation. I will also show how one can reduce the study of periodic oscillations in systems wit

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Geometric singular perturbation theory beyond the standard form
Peter Szmolyan In many biological models multiple time scale dynamics occurs due to the presence of variables and parameters of very different orders of magnitudes. Situations with a clear "global" separation into fast and slow variables governed by singu

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Waves in random neural media
Stephen Coombes The propagation of waves of neural activity across the surface of the brain is known to subserve both natural and pathological neurobiological phenomena. An example of the former is spreading excitation associated with sensory processing, whilst wave

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Noise sensitivities in systems with delays and multiple time scales
Rachel Kuske Dynamical systems with delayed feedback often exhibit complex oscillations not observed in analogous systems without delay. Stochastic effects can change the picture dramatically, particularly if multiple time scales are present. Then transients igno

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Cross-currents between Biology and Mathematics on Models of Bursting
Arthur Sherman I will trace the history of models for bursting, concentrating on square-wave bursters descended from the Chay-Keizer model for pancreatic beta cells. The model was originally developed on a biophysical and intutive basis but was put into a mathemati

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Math to Bio and Bio to Math
John Guckenheimer The interchange between dynamical systems theory with biology has had lasting impact upon both. As biology becomes increasingly quantitative, this relationship is likely to strengthen still further. This lecture will review my experience as a mathema

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Modeling Neural Networks for Rhythmic Movements
Ron Harris-Warrick Central Pattern Generators (CPGs) are limited neural networks that drive rhythmic behaviors such as locomotion, respiration and mastication. We have been studying the structure, function, and modulation of CPGs, with an emphasis on neuronal and ionic

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Animal gaits and symmetries of periodic solutions
Marty Golubitsky In the first part of this talk I will briefly describe previous work on quadruped gaits (which distinguishing gaits by their spatio-temporal symmetries). In the second part, I will discuss how the application to gaits has led to results about phase-s

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Computing 2D invariant manifolds: Can you do this?
Hinke Osinga The Lorenz system is the classical example of a seemingly simple dynamical system that exhibits chaotic dynamics. In fact, there are numerous studies to characterize the complicated dynamics on the famous butterfly attractor. This talk addresses how

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Dynamics of Differential Equations with Multiple State Dependent Delays
Antony Humphries The Mackey-Glass equation is a seemingly simple delay differential equation (DDE) with one fixed delay which can exhibit the full gamut of dynamics from a trivial stable steady state to fully chaotic dynamics, and has inspired decades of mathematical