Event type: Seminar
Events
Eduardo Garcia-Juarez
(University of Pennsylvania)
Analysis of Moving Interfaces in Incompressible Flows
Show/Hide Abstract
Interfaces that evolve with a fluid flow abound in nature and engineering. They are subject to intense research in many different fields, from meteorology to medical sciences or the petroleum industry. However, the mathematical analysis of these free boundary problems is still emerging, which impedes a deeper understanding necessary for accurate numerical methods. In this talk, we will give an overview of certain fluid-fluid and fluid-structure interfaces problem (inhomogeneous Navier-Stokes, Boussinesq, Muskat and Peskin models). The focus is placed on global-in-time results (versus finite-time singularities), with initial interfaces that are not just small perturbations. The analysis is thus purely nonlinear. Particular attention is given to initial data in critical spaces and the challenging non-local effects of viscosity contrasts.
Daniel Cooney
(Princeton University)
PDE Models of Multilevel Selection
Show/Hide Abstract
Here we consider a game theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on the Prisoners’ Dilemma: a game in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. Comparison principles and an invariant property of the tail of the population distribution are used to characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: if groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.
Daniel Gomez
(University of British Columbia)
Localized Patterns in Bulk-Membrane Coupled Models
Show/Hide Abstract
Turing instabilities in reaction diffusion systems describe potential mechanisms for pattern formation in qualitative models of microbiological processes. A recent direction of research has been to incorporate bulk-membrane coupling (BMC) into these models which introduces a process of attachment and detachment to and from the cell membrane. In these models chemical species can therefore undergo periods of bulk- and membrane-bound diffusion in addition to prescribed kinetics. Linear stability analysis and numerical simulations have revealed that differences between membrane and cytosol diffusivities can trigger Turing-like pattern-forming instabilities in BMC models. We further investigate the role of bulk-membrane coupling by analyzing its effect in a singularly perturbed model where the diffusivity of one membrane-bound species is asymptotically small. In this context, localized solutions are known to exist and can be approximated using asymptotic methods. Additionally, the linear stability and long time dynamics of these localized solutions leads to novel non-local eigenvalue problems and differential-algebraic systems. In this talk we will outline this asymptotic framework and highlight the role of bulk-membrane coupling in the stability properties of localized solutions.
Chuan Xue
(Ohio State University)
Mathematical models in biological pattern formation
Show/Hide Abstract
I will discuss two mathematical problems in biological pattern formation. The first is on modeling concentric ring patterns formed in engineered bacterial colonies. I will first present a hybrid model that incorporates a detailed description of cell movement and cell signaling and explains the underlying mechanism of the ring pattern. I will then present a PDE model derived from the hybrid model that captures the biological phenomena equally well. The second concerns spatial pattern formation in reaction-diffusion systems. I will discuss a computational method that can be used to discover potentially all nonuniform steady states of the PDE system, describing the structure of the spatial patterns. The method uses techniques from numerical algebraic geometry. This talk is based on joint work with Min Tang from Shanghai Jiaotong University and Wenrui Hao from Penn State University.
Erick Matsen
(Fred Hutchinson Cancer Research Center)
Making Bayesian phylogenetics like training a neural network
Show/Hide Abstract
Bayesian posterior distributions on phylogenetic trees remain difficult to sample despite decades of effort. The complex discrete and continuous model structure of trees means that recent inference methods developed for Euclidean space are not easily applicable to the phylogenetic case. Thus, we are left with random-walk Markov Chain Monte Carlo (MCMC) with uninformed tree modification proposals; these traverse tree space slowly because phylogenetic posteriors are concentrated on a small fraction of the very many possible trees. In this talk, I will describe our wild adventure developing efficient alternatives to random-walk MCMC, which has concluded successfully with the development of a variational Bayes formulation of Bayesian phylogenetics. This formulation leverages a “factorization
Gillian Queisser
(Temple University)
Ultrastructural 3D simulations of electrical and calcium dynamics in neurons and networks
Show/Hide Abstract
Neurons make use of their complex cellular and intracellular architecture to process and guide electrical and biochemical signals. To study this structure-function interplay, computational methods are detremental, since many parameters are not directly accessible in an experimental setting. This also means that the detailed three-dimensional morphology of cells and organelles needs to be included in modeling and simulation, which results in complex-domain problems, described by systems of coupled, nonlinear, partial differential equations. We have developed numerical discretization methods and fast solvers to address this general type of biological problem set, with a focus on optimal weak scalability on High Performance Computing infrastructures. We present some of the important biological problems revolving around cellular calcium signaling, coupled to electrical models, and the use of our NeuroBox Toolbox and the multiphysics platform uG4 to solve such ultrastructural 3D neuron models. Selected results show how neurons are capable of using their (intra)cellular architecture to fine-tune their response to exterior/network input.
Leah Edelstein-Keshet
(University of British Columbia)
From Cell polarity to intracellular networks in single and collective cell motility
Show/Hide Abstract
Cell migration plays a central role in embryonic development, wound healing and immune surveillance. In 2008, Yoichiro Mori, Alexandra Jilkine and I published a model for the initial step of cell migration, the front-back chemical polarization that sets a cell’s directionality. (More detailed mathematical properties of this model were described by the same group in 2011.) Since then, progress has been made in investigating how that simple “wave-pinning" mechanism is shaped and tuned by feedback from other proteins, such as actin, from the cell’s environment (extracellular matrix), from interplay with larger signaling networks, and from cell-cell interactions. In this talk I will describe some of this progress, with emphasis on links to experiments on melanoma cell motility. If time permits, I will also briefly describe more recent work on collective cell migration that we are currently undertaking.
Sarah Olson
(Worcester Polytechnic Institute)
Dynamics of movement in complex environments
Show/Hide Abstract
In this talk, we will highlight two different types of movement in viscosity dominated environments
Naomi Leonard
(Princeton University)
Opinion Dynamics with Tunable Sensitivity
Show/Hide Abstract
I will present a general model of continuous-time opinion dynamics for an arbitrary number of agents that sense or communicate over a network and form real-valued opinions about an arbitrary number of options. Drawing from biology, physics, and social psychology, an attention parameter is introduced to modulate social influence and a saturation function to bound inter-agent and intra-agent opinion exchanges. This yields simply parameterized dynamics that exhibit the range of opinion formation behaviors predicted by model-independent bifurcation theory but not exhibited by linear models or existing nonlinear models. Behaviors include rapid and reliable formation of multistable consensus and dissensus states, even in homogeneous networks, as well as ultra-sensitivity to inputs, robustness to uncertainty, flexible transitions between consensus and dissensus, and opinion cascades. Augmenting the opinion dynamics with feedback dynamics for the attention parameter results in tunable thresholds that govern sensitivity and robustness. The model provides new means for systematic study of dynamics on natural and engineered networks, from information spread and political polarization to collective decision making and dynamic task allocation.
This is joint work with Alessio Franci (UNAM, Mexico) and Anastasia Bizyaeva (Princeton).
The talk is based on version 2 of the paper “A General Model of Opinion Dynamics with Tunable Sensitivity”, which will be available on Tuesday October 13, 2020 here: https://arxiv.org/abs/2009.04332v2
Stefano Recanatesi
(University of Washington)
Constraints on the dimensionality of neural representations
Show/Hide Abstract
In the domain of computational/theoretical neuroscience a recently revived question is about the complexity of neural data. This question can be tackled by studying the dimensionality of such data: is neural activity high or low dimensional? How does the geometrical structure of neural activity depend on behavior, learning or the underlying connectivity? In my talk I will show how it is possible to link these three aspects (animal behavior, learning and underlying network connectivity) to the geometrical properties of neural data, with an emphasis on dimensionality phenomena. My results depart from neural recordings and aim at building understanding of neural dynamics by means of theoretical and computational tools. Such tools are mainly borrowed from the domain of neural networks dynamics, using a blend of large scale dynamical systems and statistical physics approaches.
4
0
https://mathbio.sas.upenn.edu/event_listing_type/seminar/page/5/
https://mathbio.sas.upenn.edu/event_listing_type/seminar/page/3/
