Dynamical Systems Seminars

Fall 2023

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in CCDS 365 . There will be a brief tea in CCDS 365 at 3:45PM.

• September 4th No Seminar (Labor Day) (Affiliation TBD)
  Title:
  Abstract:

• September 11th Stanislav Mintchev (The Cooper Union)
  Title: Excitation propagation in neural media: simple models, as well as some pathologies associated with axonal dilation
  Abstract: Much attention has been given in recent decades to feedforward networks of identical dynamical systems. This is in part because, despite possibly large size, these models often lend themselves to mathematical analysis techniques for low-dimensional (even 1-dimensional, e.g., phase oscillators) phase spaces; also, they present a natural setting for a space-preferred direction of communication, and as such, they are considered relevant paradigms in, e.g., mathematical neuroscience. A specific line of work we have followed is the interpretation of such a network (a chain, to use a simpler term) as an excited or excitable neural medium. In this talk we will review some basic results in this setting and discuss possibilities for how this tableau of phenomenology may afford some plausible mechanisms / explanations for neural conduction pathologies resulting from concussive trauma, which has been associated with neuronal deformations known as focal axonal swellings.

• September 18th Lior Alon (MIT)
  Title: Gaps of Fourier Quasicrystals and Lee-Yang Polynomials
  Abstract: The concept of "quasi-periodic" sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for discrete periodic sets, saying that the Fourier transform of the counting measure of a discrete periodic set is also a counting measure of a discrete periodic set. Fourier Quasicrystals (FQ) generalize this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition. Recently Kurasov and Sarnak provided a method for construction of one-dimensional counting measures which are FQ (motivated by quantum graphs) using the torus zero sets of multivariate Lee-Yang polynomials. In this talk, I will show that the Kurasov-Sarnak construction generates all FQ counting measures in 1D. A discrete set on the real line is fully described by the gaps between consecutive points. A discrete periodic set has finitely many gaps. We show that a non-periodic FQ has uncountably many gaps, with a well-defined gap distribution. This distribution is given explicitly in terms of a an ergodic dynamical system induced from irrational flow on the torus. The talk is aimed at a broad audience, no prior knowledge in the field is assumed. Based on joint works with Alex Cohen and Cynthia Vinzant.

• September 25th Timothy Roberts (Brown U.)
  Title: Snaking of Contact Defects (and why you should beware the infinite-dimensional snakes in the grass)
  Abstract: The Brusselator is one of the oldest systems studied in spatial dynamics, first conceived as a result of Turing’s landmark work on the formation of stripe patterns in the 1960's. Despite the decades and myriad studies since then, it remains a system of interest due to its ability to display a zoo of different complex behaviors. In this work we look at a newly discovered behavior, snaking of contact defects. Numerical studies by Tzou et al. (2013), found that an instability between two distinct types of stable oscillations allows for the production of contact defects: a temporally constant core region sitting in a temporally oscillating background. Their results suggest that these patterns form through a process called snaking which has been deeply explored in the context of spatially localised patterns. However, we will see that in the setting of contact defects, the process is significantly more complex. In this talk I aim to use intuition from both the PDE and the dynamical systems viewpoints to motivate the process of snaking and give you a picture of why it is so different in the infinite-dimensional case.

• October 2nd Olivia Cannon (University Of Minnesota)
  Title: Two case studies in coherent structures: When Fredholm fails
  Abstract: We explore two examples of coherent structure persistence where lack of Fredholm linearization causes classical techniques to fail.
  1) Social dynamics The bounded confidence model is well-known for its dynamics of party formation. We study shifts in these opinion clusters induced by adding a bias term. Using novel, nonlocal center manifold methods, we are able to establish existence for subcritical, strong bias, making use of algebraically simple Taylor expansions of manifolds in function space.
  2) Weakly quenched fronts We study the effect of symmetry-breaking on an interface near a boundary. Starting with the Allen-Cahn equation, breaking the symmetry causes movement of the front interface. We are interested in how weak symmetry-breaking selects an angle between the interface and the boundary, using far-field core techniques analytically. This is work in progress.

• October 9th No Seminar (Indig. People's Day)
  Title:
  Abstract:

• October 16th Peter Topalov (Northeastern U.)
  Title: Spatially quasi-periodic solutions of the Euler equation
  Abstract: Abstract: We prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on $\mathbb{R}^n$. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved. This is a joint work with Xu Sun.

• October 23rd Katie Morrison (U. of Northern Colorado)
  Title: Predicting neural network dynamics from graph structure
  Abstract: Neural networks often exhibit complex patterns of activity that are shaped by the intrinsic structure of the network. For example, spontaneous sequences of neural activity have been observed in cortex and hippocampus, and patterned motor activity arises in central pattern generators for locomotion. In this talk, we will focus on a simplified neural network model known as Combinatorial Threshold-Linear Networks (CTLNs) in order to understand how the pattern of neural connectivity, as encoded by a directed graph, shapes the emergent nonlinear dynamics of the corresponding network. We will see that important aspects of these dynamics are controlled by the stable and unstable fixed points of the network, and show how these fixed points can be determined via graph-based rules. These rules provide a direct link between the structure and function of these networks and give insight into network motifs that produce dynamic attractors.

• October 30th Zachary Kilpatrick (U. of Colorado-Boulder)
  Title:Stochastic dynamics of wandering bumps in neural fields: Mechanisms for stabilizing parametric working memory
  Abstract: To navigate the world, organisms temporarily store and retrieve information in a temporal and capacity limited way. Such working memory computations are often accomplished by spatially structured persistent activity in cortical networks, well modeled by integrodifferential equations known as neural fields. Their associated integral kernels correspond to the weight and polarity of synaptic connectivity in neuronal networks, producing and predicting specific spatiotemporal patterns of activity. Standing pulse solutions to these equations are often referred to as "bumps" whose positions encode the remembered location of a continuum stimulus feature like color, orientation, or position. Effects of network heterogeneities and noise upon the evolving position of bumps can be accounted for perturbatively, using linear or weakly nonlinear analysis to describe the asymptotic dynamics. We explain the mathematical underpinnings of these approaches and key results relating how network architecture shapes the stochastic wandering of bumps, providing a mechanistic theory of how short term estimates can be stabilized when considering (a) spatial heterogeneities in architecture; (b) separate excitatory/inhibitory populations; (c) spatial noise correlations; (d) interareal connections; and (e) short term plasticity.

• November 6th Thomas Fai (Brandeis U.)
  Title: Mathematical models of organelle size control and scaling
  Abstract: Why do organelles have their particular sizes, and how does the cell maintain them despite the constant turnover of proteins and biomolecules? To address this fundamental biological question, we formulate and study mathematical models of organelle size control rooted in the physicochemical principles of transport, chemical kinetics, and force balance. By studying the mathematical symmetries of competing models, we arrive at a hypothesis describing general principles of organelle size control. In particular, we consider flagellar length control in the unicellular green algae Chlamydomonas reinhardtii, and develop a minimal model in which diffusion gives rise to a length-dependent concentration of depolymerase at the flagellar tip. We show how noise may be used to fit model parameters and explain how similar principles may be applied to other examples of organelle size and scaling such as the ratio of nucleus to cell volume.

• November 13th Susanna Haziot (Brown U.)
  Title: Desingularization and long-term dynamics of solutions to fluid models
  Abstract: The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. The Peskin problem describes the flow of a Stokes fluid through the heart valves. In this talk, we investigate the small data critical regularity theory for these two models, and in particular, the desingularization of interfaces with small corners. The first part is joint work with E. Garcia-Juarez, J. Gomez-Serrano and B. Pausader. The second part is joint work with E. Garcia-Juarez.

• November 20th No Seminar
  Title:
  Abstract:

• November 27th Peter Thomas (Case Western University)
  Title: A Universal Description of Stochastic Oscillators
  Abstract: Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. In joint work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we introduce a nonlinear transformation of stochastic oscillators to a new complex-valued function $Q^*_1(\mbx)$ that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function $Q^*_1(\mbx)$ is the eigenfunction of the Kolmogorov backward operator with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled stochastic oscillators.
  Joint work with:
  Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona),
  Boris Gutkin (Ecole Normale Supérieure, Paris),
  Benjamin Lindner (Humboldt University, Berlin)

• December 4th Daniele Avitabile (Vrije Universiteit Amsterdam)
  Title: Bump Attractors and Waves in Networks of Integrate-and-Fire Neurons
  Abstract: Bump attractors are localised patterns observed in in vivo experiments of neurobiological networks. They are important for the brain's navigational system and specific memory tasks. A bump attractor is characterised by a core in which neurons fire frequently, while those away from the core do not fire. We uncover a relationship between bump attractors and travelling waves in a classical network of excitable, leaky integrate-and-fire neurons. This relationship bears strong similarities to the one between complex spatiotemporal patterns and waves at the onset of pipe turbulence. We define and study analytical properties of the voltage mapping, an operator transforming a solution's firing set into its spatiotemporal profile. This operator allows us to construct localised travelling waves with an arbitrary number of spikes at the core, and to study their linear stability. A homogeneous "laminar" state exists in the network, and it is linearly stable for all values of the principal control parameter. We show that one can construct waves with a seemingly arbitrary number of spikes at the core; the higher the number of spikes, the slower the wave, and the more its profile resembles a stationary bump. As in the fluid-dynamical analogy, such waves coexist with the homogeneous state, are unstable, and the solution branches to which they belong are disconnected from the laminar state. We provide evidence that the dynamics of the bump attractor displays echoes of the unstable waves

• December 11th Paul Holst (Universität Bremen)
  Title: Well-posedness and long-term behaviour for geophysical fluid models with a subgrid parametrization
  Abstract: Numerical simulations of large scale geophysical flows typically require unphysically strong dissipation for numerical stability. Towards energetic balance various schemes have been devised to re-inject this energy, in particular by horizontal kinetic energy backscatter. In order to gain insight into the impact of such schemes from a mathematical viewpoint, we view these as a modification of the fluid momentum equations on the continuum level, i.e., as partial differential equations. In this talk, we examine two geophysical fluid models, namely the 2D Euler equations and the 3D primitive equations with bottom drag, equipped with a backscatter parameterization. We discuss the global well-posedness of both models and, in particular, their long-term behavior. We prove global well-posedness exploiting that the vector fields are divergence free and using anisotropic Sobolev spaces. For the long-term behavior of these models, essentially two outcomes emerge dependent on the choice of parameters in the backscatter parameterization: Either the dynamics become trivial, i.e., they asymptotically converge towards the zero attractor, or the so-called 'grow-up' phenomenon occurs, i.e., there exist solutions that grow unboundedly and exponentially over time. In the case of the 2D Euler equations, we prove stability of unbounded growth.

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