MBI Videos
Workshop on Topics in Applied Dynamical Systems: Equivariance and Beyond

Pete AshwinThe Endoplasmic Reticulum (ER) is an important and geometrically complex organelle within most living cells that can form highly dynamic networklike structures within the cell. This talk will discuss a study of some of these dynamic network structures within plant cells based on live cell imaging of movies showing dynamic ER networks. We hypothesise that the structures observed are minimal geometric networks perturbed by deterministic (cytoplasmic streaming) and random (Brownian) forces. We also try to quantify the biophysics behind the structures (joint work with Imogen Sparkes, Congping Lin and Yiwei Zhang).

Harry Swinney
Bacillus subtilis bacteria in a growing colony are observed to exhibit giant fluctuations in the number N in a given volume: the rms fluctuation is proportional to N^(3/4) rather than N^(1/2) as in systems in thermodynamic equilibrium. Measurements of the speed and orientational correlations of bacteria within a growing colony yield an unexpected scale invariance. Studies of another rodshaped motile bacterium common in soil, Paenibacillus dendritiformis, reveal that neighboring colonies secrete a previously unknown toxic protein, Slf, which is not secreted by an isolated colony. Models help in understanding the growth inhibition and why the competition between neighboring colonies is deadly. Some bacteria within a colony survive Slf by switching to an immotile Slfresistant coccus (spherical) form. If cocci encounter sustained favorable conditions, they switch back to the rodshaped form. Genes that encode components of this shapechanging transition are widespread among bacterial species, suggesting that this survival mechanism is not unique to P. dentritiformis.

Bob RinkNetworks of coupled nonlinear dynamical systems often display unexpected phenomena. They may for example synchronise. This form of collective behaviour occurs when the agents of the network behave in unison. An example is the simultaneous firing of neurons. An elaborate theory for synchrony was developed by Golubitsky and coworkers. This theory was recently reformulated by DeVille and Lerman. They show that the patterns of synchrony of network systems are determined by socalled graph fibrations.
In this talk I will show how graph fibrations also impact the global dynamics of networks. They are for example responsible for the unusual character of certain synchronybreaking bifurcations. These bifurcations are forced by selffibrations of a high dimensional lift of the network. This observation implies that networks are nothing but unusual examples of equivariant dynamical systems, and can be understood with the help of semigroup theory, representation theory and techniques from equivariant dynamical systems theory.
This is joint work with Jan Sanders and Eddie Nijholt. 
Lee DeVille
The stability of any fixed point of a dynamical system defined on a network is determined by the spectrum of the Jacobian at that point.
For a wide variety of networked dynamical systems, this Jacobian takes the form of a "graph Laplacian". In contrast to the classical Laplacian, for many fixed points, the network configuration will be such that we need to consider negativelyweighted edges, i.e. configurations with repelling pairs. We present the spectral theory of such operators, give a natural description using the language of social networks, and show that many of the dynamical properties of such networks can be intuited using this conceptual framework.

Reiner LauterbachIn 1984 Ihrig & Golubitsky generalized previous work by Sattinger and Busse et.al. to classify generic bifurcations in the absolutely irreducible representations of O(3). This classification is based on the Equivariant Branching Lemma. Later on Becker & KrÂ¨amer used this classification to do a similar work for the representations of SO(4). While in the O(3)â€“case this gives a reasonable complete picture of the possible bifurcation scenario, this is not the case in the SO(4)â€“ case and similarly in the SO(8)â€“case. In both cases Lauterbach & Matthews [5], respectively Lauterbach constructed infinite families of groups, where the set of group orders in each family forms an arithmetic progression, where in each case the Equivariant Branching Lemma does not apply and the generic bifurcation behavior is a priori unknown. In [5, this behavior was investigated. However there are many more cases, where the Equivariant Branching Lemma does not apply. In this talk we investigate cases of infinite families of groups which have a rich structure (among other propertes their orders do not form an arithmetic progression) and where the bifurcation analysis faces new challenges.

Ian StewartIn nonlinear dynamics, the most significant phenomena are those that remain invariant under appropriate changes of coordinates â€” diffeo(morphism)s. In equivariant dynamics, for example, appropriate means preserving the symmetry, leading to the usual concept of an equivariant diffeo. What about network dynamics? The most sensible definition would seem to be preserving admissible maps. So what are the diffeos with this property? There are at least three reasonable ways for a diffeo to act on a map: composition on the right, composition on the left, and a vector fieldchange in which the inverse of the derivative of the diffeo acts on the left and the diffeo also acts on the right. In joint work with Marty Golubitsky, we have classified diffeos that preserve admissibility in all three cases, for networks in which no two arrows are equivalent. (This is an important, though somewhat neglected, condition: it is very common in realworld networks). The answer is different in each case, although the three are closely related. It depends on three special subnetworks: the left core, right core, and their intersection (core). Being admissible for the appropriate core is almost the answer, but not in the vector field case. The proofs are easy for left and right actions, and require some abstract algebra (notably the WedderburnMalcev Theorem for associative algebras) in the vector field case. We describe the problem, define the cores, outline the proofs, and make a guess about one potential application.

Laurette TuckermanThe von Karman vortex street is one of the most striking visual images in fluid dynamics. Immersed in a uniform flow of sufficient strength, a circular cylinder periodically sheds propagating vortices of alternating sign on either side of the "street". Although the von Karman vortex street can be simulated numerically with great accuracy, predicting its properties from general theoretical principles has proved elusive. It has been shown that the vortexshedding frequency can be obtained by carrying out a linear stability analysis about the temporal mean, but there is no understanding of why the correct answer emerges from such an unorthodox procedure.
We have carried out a similar analysis of thermosolutal convection, which is driven by opposing thermal and solutal gradients. In a spatially periodic domain, branches of traveling waves and standing waves are created simultaneously by a Hopf bifurcation. We find that linearization about the mean fields of the traveling waves yields an eigenvalue whose real part is almost zero and whose imaginary part corresponds very closely to the nonlinear frequency, consistent with the cylinder wake. In marked contrast, linearization about the mean field of the standing waves yields neither zero growth nor the nonlinear frequency. It is shown that this difference can be attributed to the fact that the temporal power spectrum for the traveling waves is peaked, while that of the standing waves is broad. We give a general demonstration that the frequency of any quasimonochromatic oscillation can be predicted from its temporal mean. 
Bjorn SandstedeMany planar spatially extended systems exhibit localized radial patterns such as spots and oscillons. In particular, such structures can emerge at Turing and forced Hopf bifurcations. In this talk, I will give an overview of these mechanisms and show how geometric blowup techniques can be used to analyse them: among the findings is the bifurcation of localized structures that have significantly larger amplitudes than expected from amplitude equations.

Edgar KnoblochChimera states consisting of domains of coherently and incoherently oscillating nonlocallycoupled phase oscillators on a ring are studied. Systems of both identical and heterogeneous oscillators are considered. In the former several classes of chimera states have been found: (a) stationary multicluster states with evenly or unevenly distributed coherent clusters, and (b) traveling chimera states. Single coherent clusters traveling with a constant speed across the system are also found. In the presence of spatial heterogeneity in the oscillator frequencies these traveling states undergo a variety of pinning and depinning transitions. In this talk I will describe these results, provide a selfconsistent continuum description of many of these states, and will use this description to study transitions between them. This is joint work with J Xie and HC Kao.

Dwight BarkleyI will discuss a surprising analogy between the subcritical transition to turbulence and the dynamics action potentials in neurons. Laminar flow plays the role of the rest state of a neuron, while turbulent flow plays the role of the excited state. From a simple model motivated by electrophysiology, I will show that it is possible to explain many largescale patterns observed in transitional pipe flow.

Mary SilberEarly investigations of pattern formation in fluid mechanical systems inspired some of the development of equivariant bifurcation theory. Recently, these pattern formation perspectives have been applied to modeling the vegetation in dryland ecosystems, where satellite images have revealed strikingly regular spatial patterns on large scales. Ecologists have proposed that characteristics of vegetation pattern formation in these waterlimited ecosystems may serve as an early warning sign of impending desertification. We use the framework of equivariant bifurcation theory to investigate the mathematical robustness of this approach to probing an ecosystems robustness.

Eric SheaBrownExperimental breakthroughs are yielding an unprecedented view of the brain's connectivity and of its coherent dynamics. But how does the former lead to the latter? We use graphical and point process methods to reveal the contribution of successively morecomplex network features to coherent spiking.

Pascal ChossatLearning or memory in the cortex are associated with the strengthening of the synaptic connections between neurons according to a pattern reflected by the input. According to this theory a retained memory sequence corresponds to a dynamic pattern of the associated neural circuit. I present a recent work with M. Krupa in which we consider neuronal network models known as Hopfield networks, with a learning rule which consists of transforming an information string to a coupling pattern, Within this class of models we study dynamic patterns defined by robust heteroclinic cycles, and establish a tight connection between their existence and the structure of the coupling.

Phil HolmesI will describe coupled oscillator models of hexapedal locomotion and attempt to show that essentially the same central pattern generator, with appropriate parameter choices can produce the ranges of gait patterns exhibited by animals as diverse as stick insects, cockroaches and fruit flies.

JeanJacques SlotineComputation, synchronization, and measurement are key issues in complex networks. Vast nonlinear networks are encountered in biology, for instance, and in neuroscience, where for most tasks the human brain grossly outperforms engineered algorithms using computational elements 7 orders of magnitude slower than their artificial counterparts. We show that nonlinear dynamical systems analysis tools yield simple but highly nonintuitive insights about such issues, and that they also suggest systematic mechanisms to build progressively more refined networks and novel algorithms through stable accumulation of functional building blocks and motifs.

John GuckenheimerThis talk describes joint work with HorstMoritz Maus, Shai Revzen, Christian Ludwig, Johann Reger and Andre Seyfarth. We utilize motion capture movies of individuals to construct low dimensional models of treadmill running. Our starting point was to assume that the data could be represented as a noisy limit cycle in a hybrid (piecewise smooth) dynamical system. Springmass (SLIP) models are concrete physical systems that have long been used as caricatures of running. Feedback that determines foot placement is an essential ingredient of the SLIP models. Our primary goal was to find still low dimensional extensions of SLIP that predict center of mass location for one and two steps ahead almost as well as is possible using all of the motion capture data.

Paul Bressloff

Jack Cowan