Featured minisymposia at DS15

By Peter van Heijster
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Title: Applications of Algebraic Topology to Neuroscience
By: Chad Giusti, Robert W. Ghrist and Danielle Bassett
When: Sunday, May 17, 2 – 4 pm
Where: Ballroom I
Intro: The rapid advance of laboratory technology and a steady increase in scientific and public interest in basic neuroscience research has produced a deluge of large data sets. These data are usually high-dimensional, noisy and subject to unknown nonlinearities both within the dynamics of the system being recorded and at the level of the recording equipment itself. Extracting meaningful information from the data and drawing conclusions about the underlying biology is extremely challenging, and in many cases requires the development of new tools and methodologies.

We believe the emerging field of applied algebraic topology has a great deal to offer in the study of these problems by adapting the powerful machinery of algebraic topology to the specific needs of mathematical biology. On the other hand, mathematicians and biologists studying dynamics in neuroscience can provide their extensive domain experience and ask the right questions to guide the development of these ideas into a useful toolkit for the community. By hosting this minisymposium at the SIAM Dynamical Systems meeting, our hope is to put interested members of these often disjoint groups in a position to make connections and establish productive collaborations.

The talks will highlight a cross-section of new algebraic and topological tools created to address problems in neuroscience, dynamic networks and big data. These include a tool for the detection of structure (or lack thereof) in symmetric matrices, with applications to correlation of recorded activity in neural populations, a technique for classification of dynamic community structure in time-varying networks, an application of the Mapper algorithm for clustering across filtrations to the classification of clinical subjects, and an algebraic construction for extracting structure in neural codes.

The order complex of a (symmetric) matrix is a filtered family of graphs which encodes the data in the matrix invariant under entry-wise monotonic transformations. (Giusti et al., arXiv:1502.06172 and Curto et al., arXiv:1502.06173.)


Title: Data Assimilation in the Life Sciences
By: Elizabeth M. Cherry and Timothy Sauer
When: Sunday, May 17, 2 – 4 pm
Where: Ballroom II
Intro: Biological processes pose distinct modeling challenges due to complex factors including technological limitations on the number and spatiotemporal resolution of observed variables, system nonstationarity, noise in output and measurement, and variability of properties both within individuals and across populations. Assimilation of data with models of biological processes is a crucial component of obtaining more reliable information about these systems. In recent years, nonlinear versions of Kalman filtering have been developed, in addition to methods that estimate model parameters in parallel with the system state. This minisymposium highlights the state of the art in applying these techniques to nonlinear dynamical systems in the life sciences to gain understanding from biological data, especially with regard to constructing estimates of unobserved variables and assessing model fidelity. The talks will illustrate the use of data assimilation to analyze the dynamics of several prototypical systems, such as cardiac tissue, hippocampal neurons and neural cultures.

Voltage on a one-dimensional ring of cardiac tissue (blue: low voltages; orange: high voltages). (a) A two-second simulation of the Fenton-Karma model on a ring while in a state of discordant alternans, as evident by the oscillations in wavelength. This is used as truth for data assimilation using the Local Ensemble Transform Kalman Filter (LETKF). Observations are generated by sub-sampling the true state and adding random Gaussian error. (b) When the system is initialized from imperfect initial conditions, the LETKF initially struggles to synchronize with the truth but after approximately 500 ms the LETKF is able to recover the true alternans state.


Title: Data-Driven Modeling of Dynamical Processes in Spatially-Embedded Random Networks
By: Gyorgy Korniss
When: Sunday, May 17, 2 – 4 pm
Where: Maybird
Intro: Infrastructure, transportation, and brain networks are embedded in strongly heterogeneous spatial environments, posing formidable challenges in understanding the vulnerability or ''plasticity'' of these systems. There exists a large gap between some universal features of simplified dynamical processes on non-spatial complex networks and their applicability to real-world problems. This minisymposium will present recent progress by data-driven approaches in diverse areas. Topics will include the lack of self-averaging and predictability of cascading failures in power grids, optimization in smart grids, the applicability of scaling laws of human travel, and the impact of wiring constraints and weight-based heterogeneity in the cortical networks.

Visualizing the surviving component of the UCTE power grid after cascading failures, using identically allocated relative excess capacities. See http://dx.doi.org/10.1371/journal.pone.0084563.s004 for a movie of the progression of the cascade in the UCTE power grid. (A. Asztalos et al., PLoS One 9(1): e84563 (2014); http://dx.doi.org/10.1371/journal.pone.0084563.)


Title: Stochastic Delayed Networks
By: Gabor Orosz
When: Sunday, May 17, 2 – 4 pm
Where: Primrose A
Intro: Complex systems of the 21st century may exhibit rich dynamical behaviors due to complicated network structures, time delays and stochastic effects. Understanding the dynamics of such infinite dimensional dynamical systems requires new approaches and novel mathematical tools. This sections highlights the latest research result on the dynamics of networks where time delays arise due to finite-time information propagation between the nodes and stochasticity appears both in the state as well as in the time delays. For example, in connected vehicle systems, where vehicles communicate via wireless vehicle-to-vehicle (V2V) communication, time delays vary stochastically due to intermittencies and packet loss. By analyzing the mean and the covariance dynamics one can ensure the stability of traffic flow using connected cruise control despite the stochastic delay variations. Similar techniques may be used to understand the dynamics of gene regulatory networks where stochastic delays appear since the completion time of biochemical processes vary stochastically. This may lead to counter-intuitive behavior where increasing uncertainty can improve stability of steady states. Stochasticity and delays may also change the pattern formation mechanisms in neurosystems leading cluster states and traveling waves that do not emerge in traditional neural network models.

A connected vehicles system where vehicles communicate with wireless vehicle-to-vehicle communication. (Adapted from: J. I. Ge and G. Orosz. Dynamics of connected vehicle systems with delayed acceleration feedback. Transportation Research Part C, 46 (2014) http://dx.doi.org/10.1016/j.trc.2014.04.014)


Title: Localized Pattern Formation in Reaction-diffusion Equations
By: Arik Yochelis
When: Sunday, May 17, 2 – 4 pm
Where: Primrose B
Intro: Spatially localized states arise in many physical systems, examples of which include solitary waves (solitons) in fluids, bounded gas discharge spots, cavity solitons in optics, and oscillons in vibrating granular media. In addition, spatially localized states play a paramount role in a variety of biological systems. While the most famous example is related to insights that emerged out of the seminal work on excitable pulses by Hodgkin and Huxley, which is central today to neuroscience and cardiac arrhythmia, there are many other significant examples. These include resonant responses in the cochlea, aggregations by molecular motors inside cellular protrusions, and vegetation patches in semi-arid climates.

Localized states arise in both conservative and dissipative systems and impose fundamental questions and challenges in the field of nonlinear partial differential equations. For example, extensive studies of dissipative solitons, i.e., localized states that obey a reflective spatial symmetry, have led to identification of a novel bifurcation structure that is known today as homoclinic snaking. While the basic mathematical understanding of homoclinic snaking exploits the gradient Swift-Hohenberg model through a Maxwell construction, it was shown that reaction-diffusion (RD) models are capable of capturing this phenomenon as well. In fact, both cases show that homoclinic snaking arises through the subcritical finite wave-number instability, which is also referred to as a Turing instability in the RD context. Further details on homoclinic snaking and dissipative solitons can be found in a recent review by Knobloch in Annu. Rev. Condens. Matter Phys. (2015).

Localized patterns can also occur for RD systems in the singularly perturbed limit corresponding to a large diffusivity ratio between two solution components in the system. In this context the physical dimensions introduce a further complexity: the singularly perturbed RD system (e.g., Brusselator model), posed on a surface of a sphere, allows for the existence of quasi steady-state localized spot patterns that can exhibit self-replication and other instabilities in various parameter regimes. Prior analytical studies have focused on using weakly nonlinear and equivariant bifurcation theory to derive normal form equations characterizing the development of small amplitude spatial patterns that bifurcate from a spatially uniform steady-state. However, due to the typical high degree of degeneracy of the eigenspace associated with spherical harmonics of large mode number, these amplitude equations typically consist of a large coupled set of nonlinear ODEs. The latter are known to have an intricate subcritical bifurcation structure. As a result, the preferred spatial pattern that emerges from an interaction of these weakly nonlinear modes is difficult to predict theoretically. Alternatively, asymptotic analysis can be used to derive a differential algebraic system (DAE) of ODEs characterizing the time-dependent spatial locations of a collection of localized spots. The DAE system is vaguely similar to that for the motion of Eulerian point vortices on the sphere. In certain cases, the equilibria of this DAE system for spot dynamics have a close connection to the classical problem of determining a set of elliptic Fekete points, corresponding to globally minimizing the discrete logarithmic energy for N points on the sphere. Similar DAE systems for spot dynamics can be derived for other RD models.

A common unifying theme of the talks in this Featured Minisymposium is that, due to the non-gradient nature of RD models, localized states can exhibit more complex behavior than energy driven systems, since they are not restricted by the gradient dynamics of an associated variational structure. Consequently, the objective of this Featured Minisymposium is two-fold: (i) to report on novel mathematical phenomena and methods that are related to localized states in RD models and (ii) to link between the mathematics and applications that cross a range of problems in chemistry, biology, and approximation theory (Fekete point problem). The minisymposium constitutes four talks on distinct aspects of dynamical systems and bifurcation theory that are related to recent advances on the subject. The organizer, Arik Yochelis (Ben-Gurion University), will reveal the conditions at which homoclinic snaking may arise in excitable media using extended FitzHugh-Nagumo model and show why these conditions are connected to the generation of finite pulse trains out of a single localized perturbation. Justin Tzou (Dalhousie University) will show the organization of time-oscillatory patterns in a snaking region near a codimension-two Turing-Hopf bifurcation in the Brusselator model. Michael Ward (University of British Columbia) will present a singular perturbation theory characterizing the slow dynamics of localized spots on the surface of the sphere for the Brusselator model, and the connection between this theory and the elliptic Fekete Point problem of distributing point charges on the surface of the sphere. Finally, Nicolas Verschueren Van Rees (University of Bristol) will demonstrate the connection between the snaking scenario and front dynamics in a model for cell polarization. Notably, talks in this Featured Minisymposium are carefully designed to provide both accessibility to a general mathematical audience and to elaborate on the rapidly evolving field of spatially localized states.


Top left: Slanted homoclinic snaking structure in a model for excitable medium. (Arik Yochelis, from Phys. Rev. E 91, 032924 (2015)).
Top right: Localized Turing state embedded in uniform Hopf oscillations. (Justin Tzou, from Phys. Rev. E 87, 022908 (2013)).
Bottom left: Self-replicating spot patterns on a slowly growing sphere. (Michael Ward, from SIADS 13, 564 (2014)).
Bottom right: Several snaking regions in the prototype model for cell polarization. (Nicolas Verschueren Van Rees, unpublished)


Title: Non-Autonomous Instabilities
By: Sebastian M. Wieczorek
When: Wednesday, May 20, 4 – 6 pm
Where: Ballroom I
Intro: In many applications from the natural sciences and engineering, aperiodic external forcing often leads to interesting nonlinear phenomena that can be characterised as non-autonomous instabilities. Such instabilities are mathematically challenging and often puzzle scientists because they cannot, owing to their transient or finite-time nature, be explained using traditional bifurcation theory.

This minisymposium highlights recent techniques to analyse stability of non-autonomous ODEs and PDEs, with applications to rate-induced tipping points, failure boundaries in earthquake engineering and finite-time transport in (ocean) surface flows.

An example of a rate-induced bifurcation and non-obvious thresholds in slow-fast systems. (a) Below a critical rate, all trajectories started on the stable slow manifold track the moving stable equilibrium x. (b) Above the critical rate, some trajectories fail to track the moving stable equilibrium, cross the fold F of the slow manifold and destabilise along the direction. (c-d) Above the critical rate, initial states that track the moving stable equilibrium (destabilise) are shown in blue (red) to reveal instability thresholds. These thresholds can be explained in terms of folded singularities and canard trajectories, including composite canards due to folded saddle-node.


Title: Random Walks, First Passage Time and Applications
By: Theodore Kolokolnikov and Sidney Redner
When: Wednesday, May 20, 4 – 6 pm
Where: Ballroom II
Intro: Numerous problems — from the molecular to macroscopic scales — involve strategies based on random (possibly biased) walks. Examples range from fast-folding DNA strands, to predator-prey interactions and search and rescue operations. For this symposium, we have a line-up of speakers who present novel results that also show the diversity and beauty of the subject. The topics discussed range from cellular biology, chemistry and statistical mechanics, to neuroscience and even sports. The topics of this session unify various approaches used, offering unique perspectives to reach and inform a wide target audience.

Comparison of NBA basketball game data with random walk prediction. Top: Empirical probability that a scoring event occurs at time t, with the game-average scoring rate shown as a horizontal line. The data are aggregated in bins of 10 seconds each. Bottom: Distribution of times for the last lead change. (A. Clauset et al., arXiv:1503.03509).


Title: Time-Delayed Feedback
By: Andreas Amann
When: Wednesday, May 20, 4 – 6 pm
Where: Wasatch B
Intro: Time-delayed feedback is a simple and powerful method for influencing and controlling complex nonlinear systems. In its simplest form, the method has been successfully used to control unstable periodic orbits in dynamical models arising for instance in engineering, physics or biology. More recently, the scope of time-delayed feedback has been expanded to systems with multiple delay and the connection with fixed point problems has been clarified. In this minisymposium we will cover these new developments and discuss applications of time delay to climate systems and cluster synchronisation.

Title: Emerging Strategies for Stability Analysis of Electrical Power Grids
By: Lewis G. Roberts and F. Dörfler
When: Wednesday, May 20, 4 – 6 pm
Where: Maybird
Intro: The future of electric power systems is uncertain: alternative generation sources (such as renewables), an increasing demand and the constraints of the physical network can all change the dynamics of a power system.

These dynamics can be modelled using equations that describe synchronisation and this attribute has caught the attention of interdisciplinary researchers in physics, mathematics and engineering. As a result, creative techniques are being developed with the intention to improve the monitoring, operation or control of power systems.

In our Featured mini-symposium we have four speakers: Paul Hines from the University of Vermont will be talking about statistical indicators for stability in stochastic power systems, Konstantin Turitsyn from the Massachusetts Institute of Technology will be talking about modern Lyapunov techniques as certificates for system stability, and Florian Dörfler from ETH Zürich will be talking about his research on voltage stability and an analytical method for testing the existence and uniqueness of solutions to the power flow problem in microgrids. Finally, Lewis Roberts from the University of Bristol will be talking about a new algebraic metric that can be used to survey stability trends of many different power networks.

Power system dynamics can be captured by models for synchronisation. (Credit: F. Dörfler)


Title: Invariant Manifolds Unravelling Complicated Dynamics
By: Stefanie Hittmeyer and Pablo Aguirre
When: Wednesday, May 20, 4 – 6 pm
Where: Primrose A
Intro: Higher-dimensional invariant manifolds of maps and vector fields play a key role in the organisation of a phase space. Whether they act as basin boundaries of the attractors or, in general, as separatrices for different qualitative behaviours, there is a growing need for understanding the mathematical features of global invariant manifolds. In most cases, these manifolds can not be obtained analytically but recent advanced numerical techniques for the accurate computation of invariant manifolds have emerged to obtain and render them as global objects.

Global (un)stable manifolds may undergo critical re-arrangements under parameter variation which, effectively, give rise to bifurcations of the invariant manifolds. As a consequence, these topological and geometric transitions may result in drastic changes of the global dynamics. Typically, major changes to invariant manifolds may trigger the onset of chaos, transformation or creation of basins of attraction, the formation of homo- and heteroclinic orbits and, ultimately, the reorganisation of the overall structure of the phase space. This is of special interest to understand the nature of systems near global bifurcations in many applications, such as in laser dynamics, nerve impulses in neurons, electrochemical reactions, communication systems based on chaos, food chains in predation models, etc.

The featured talks in this session will cover a wide range of topics: from invariant manifolds in the interplay with complex dynamics to numerical methods for the computation of local invariant manifolds of periodic orbits, as well as global bifurcations in the context of slow-fast systems, and the role of the centre manifold in the spatially restricted three-body problem. In this way, we aim to shed light on how the study of global invariant manifolds allows one to obtain deeper insight into the nature of complicated global dynamics in both theoretical and applied contexts.

Stable and unstable sets interacting with the Julia set in a nonanalytic perturbation of the complex quadratic family. (Credit: S. Hittmeyer, B. Krauskpf, H.M. Osinga)


Title: Medical Applications
By: Dan D. Wilson and Jeff Moehlis
When: Wednesday, May 20, 4 – 6 pm
Where: Primrose B
Intro: As technology continues to evolve at a rapid pace, scientists are able to observe and record more and more of the underlying dynamical behavior that can contribute to diseases; with these advances in technology comes a growing interest in employing ideas from dynamical systems and control theory to medical applications. The goal of this minisymposium is to provide in-depth examples of how understanding the underlying dynamics associated with debilitating medical disorders can give insight into new therapies and treatment options.

We include a broad sampling of speakers representing the subfields of cardiology, neuroscience and diabetes. The first talk (by Dan Wilson from the University of California, Santa Barbara) describes a dynamical instability in heart cells that can lead to cardiac arrest and presents a new method of state reduction that allows for the design of an efficient strategy for stabilizing the system. The next talk (by Steve Schiff from Pennsylvania State University) presents a recent advance in seizure modelling which allows for a wide range of neuronal activities and may yield new strategies for control of pathological seizure states. The next speaker (Sridevi Sarma from Johns Hopkins University) investigates and gives new insight into the functional mechanisms of high frequency deep brain stimulation in Parkinson's disease, a vital yet not well understood treatment for patients whose symptoms are otherwise intractable. We conclude the minisymposium with a talk (by Pranay Goel from the Indian Institute for Science Education and Research) regarding the management of glucose for patients with diabetes including new modelling strategies that may allow for better management of this disease.

Individual cells such as neurons and cardiomyocytes are the building blocks of detailed biological models. The dynamical equations of these cells are usually complicated, but state reduction techniques can make them more analytically tractable. One such technique, represented here, is isostable reduction. The colormap on the left gives the isostables of an excitable system along with three example trajectories. The right panel shows V as a function of time for each trajectory. Trajectories that start on the same isostable approach the fixed point together, in a well-defined sense. (D. Wilson and J. Moehlis. Extending phase reduction to excitable media: theory and applications. To appear in SIAM Review.)


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