Neurodegeneration will undoubtedly become a major challenge in medicine and public health because of demographic changes worldwide. More than 45 million people are living with dementia today and this number is expected to triple by 2050. Recent studies have reinforced the hypothesis that the prion paradigm--the templated growth and spreading of misfolded proteins--could help explain the progression of a variety of neurodegenerative disorders. However, our current understanding of prion-like growth and spreading is rather empirical. Here we show that a physics-based reaction-diffusion model can explain the growth and spreading of misfolded protein in a variety of neurodegenerative disorders. We combine the classical Fisher-Kolmogorov equation for population dynamics with anisotropic diffusion and simulate misfolding across a representative section of the brain and across the brain as a whole. Our model correctly predicts amyloid-beta deposits and tau inclusions in Alzheimer's disease, alpha-synuclein inclusions in Parkinson's disease, and TDP-43 inclusions in amyotrophic lateral sclerosis. Our results suggest that misfolded proteins in various neurodegenerative disorders grow and spread according to a universal law that follows the basic physical principles of nonlinear reaction and anisotropic diffusion. A more quantitative understanding of the timeline of neurodegeneration could have important clinical implications, ranging from estimating the socioeconomic burden of neurodegeneration to designing clinical trials and pharmacological intervention.
Ellen Kuhl is a Professor of Mechanical Engineering at Stanford University. She received her PhD from the University of Stuttgart in 2000 and her Habilitation from the University of Kaiserslautern in 2004. Her area of expertise is Living Matter Physics, the design of theoretical and computational models to predict the acute and chronic behavior of living structures. Ellen has published more than 200 peer-reviewed journal articles and edited two books; she is an active reviewer for more than 20 journals at the interface of engineering and medicine and an editorial board member of seven international journals in her field. Ellen is currently the Chair of the US National Committee on Biomechanics, an Executive Member of the US Association for Computational Mechanics, and the Chair of the Biomechanical Engineering Group at Stanford. She is a Fellow of the American Institute for Mechanical and Biological Engineering and a founding member of the Living Heart Project, a translational research initiative to revolutionize cardiovascular science through realistic simulation. Ellen received the National Science Foundation Career Award in 2010, was selected as Midwest Mechanics Seminar Speaker in 2014, and received the Humboldt Research Award in 2016.
We discuss geometry-based statistical learning techniques for learning approximations to certain classes of high-dimensional dynamical systems.
In the first scenario, we consider systems that are well-approximated by a stochastic process of diffusion type on a low-dimensional manifold. Neither the process nor the manifold are known: we assume we only have access to a (typically expensive) simulator that can return short paths of the stochastic system, given an initial condition. We introduce a statistical learning framework for estimating local approximations to the system, for stitching these pieces together and form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system not only at small time scales, but also at very large time scales (under suitable assumptions on the dynamics). We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems high-dimensions, that are well-approximated by stochastic diffusion-like equations.
In the second scenario we consider a system of interacting agents: given only observed trajectories of the system, we are interested in estimating the interaction laws between the agents. We consider both the mean-field limit (i.e. the number of agents going to infinity) and the case of a finite number of agents, with an increasing number of observations. We show that at least in particular cases, where the interaction is governed by an (unknown) function of distances, the high-dimensionality of the state space of the system does not affect the learning rates. We prove that in these case in fact we can achieve an optimal learning rate for the interaction kernel, equal to that of a one-dimensional regression problem. We exhibit efficient algorithms for constructing our estimator for the interaction kernels, with statistical guarantees, and demonstrate them on various simple examples.
Dr. Mauro Maggioni is a Bloomberg Distinguished Professor of Mathematics, and Applied Mathematics and Statistics at Johns Hopkins University. He works at the intersection of harmonic analysis, approximation theory, high-dimensional probability, statistical and machine learning, model reduction, stochastic dynamical systems, spectral graph theory, and statistical signal processing. He received his B.Sc. in Mathematics summa cum laude at the Universita degli Studi in Milano in 1999, and the Ph.D. in Mathematics from the Washington University, St. Louis, in 2002. He was a Gibbs Assistant Professor in Mathematics at Yale University till 2006, when he moved to Duke University, becoming Professor in Mathematics, Electrical and Computer Engineering, and Computer Science in 2013. He received the Popov Prize in Approximation Theory in 2007, an NSF CAREER award and Sloan Fellowship in 2008, and was selected as a Fellow of the American Mathematical Society in 2013.
Speaker Affiliation: Distinguished Professor, Department of Mathematics, The University of Tennessee Distinguished Scientist & Group Leader, Computational & Applied Mathematics, Oak Ridge National Laboratory
This tutorial will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest to the UQ community is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements. These include novel weighted $\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods.
Many a computational software scientist starts as a “domain scientist” who discovers that computation can accelerate scientific discovery and ends up contributing to the software infrastructure for scientific exploration. As a domain scientist, he/she is expected to understand the mathematics that underlies the domain (physics, chemistry, etc.). Yet once they become software scientists, few master the fundamental mathematics that underlies programming: the so-called Hoare Calculus that underlies goal-oriented programming. Did you know that you can prove a program correct? That the Principle of Mathematical Induction is fundamental to understanding loops? That you can derive programs hand-in-hand with their proofs of correctness? That the derivation process yields families of algorithms from which the highest performing can be chosen? In this talk, we illustrate how the science of programming matrix operations has allowed us to develop open source software libraries that are exceptionally robust and high performing.
Robert van de Geijn is professor of computer science and member of the Institute for Computational Engineering and Sciences. He received his Ph.D. in Applied Mathematics from the University of Maryland, College Park.
His interests are in linear algebra, high-performance computing, parallel computing, and formal derivation of algorithms. He heads the FLAME project, a collaboration between UT Austin, Universidad Jaume I (Spain), and RWTH Aachen University (Germany). This project pursues foundational research in the field of linear algebra libraries and has led to the development of the libflame library, a modern, high-performance dense linear algebra library that targets both sequential and parallel architectures. One of the benefits of this library lies with its impact on the teaching of numerical linear algebra, for which van de Geijn received the UT President’s Associates Teaching Excellence Award. He has published several books and more than 100 refereed publications.
Abnormal neural oscillations are implicated in certain disease states, for example repetitive firing of injured axons evoking painful paresthesia, and rhythmic discharges of cortical neurons in patients with epilepsy. In other clinical conditions, the pathological state manifests as a vulnerability of an oscillator to switch off, for example prolonged pauses in automatic breathing commonly observed in preterm infants. I will present theory and experimental observations on the initiation and termination of neural rhythms at the cellular, tissue and organism levels. The findings suggest how small appropriately tuned noisy inputs could silence a neural oscillator or, conversely, could promote rhythmic activity. Noise-sensitive neurons have intrinsic properties that yield interesting physiological properties on the edge of a bifurcation, affording remarkable adaptive capacities to circuits that require rapid and efficient on-off switching; between multiple modes of activity (e.g., quiescence, repetitive firing, bursting) and between multiple functions (e.g., breathing, swallowing, coughing, and vocalization). I will illustrate the therapeutic potential of stochastic stimulation for promoting stability of breathing and preventing central apnea in preterm infants.
David Paydarfar is Professor and inaugural Chair of the Department of Neurology at the Dell Medical School at The University of Texas at Austin. He previously served as Professor and Executive Vice Chair of the Department of Neurology at the University of Massachusetts Medical School, and as Associate Faculty of the Wyss Institute for Biologically Inspired Engineering at Harvard University. Paydarfar received his B.S. in Physics (summa cum laude) from Duke University and M.D. from the University of North Carolina at Chapel Hill, and completed his residency training in neurology at the Massachusetts General Hospital and Harvard Medical School. He is a Fellow of the American Neurological Association and an Investigator of the Clayton Foundation for Research.
Biological membranes have a complex composition with hundreds of different lipids and a high protein concentration. The nature of the lateral structure of membranes is hotly debated as experiments reach increasingly higher spatial and temporal resolution and simulations increasingly larger time and length scales. Coarse-grained simulations with the Martini model have enabled a significant jump in time and length scale for detailed simulations, and currently can reach of the order of 100 microseconds on systems of ca. 100 x 100 nm size on relatively available computers. We are particularly interested the interactions between lipids and membrane proteins. The local environment around membrane proteins is uniquely shaped by the protein surface, resulting in a local composition and membrane properties that differ significantly from the average properties of the lipids that make up the membrane model. This may play an important role in shaping the lateral structure of biological membranes. This type of simulation also enables detailed studies on more specific interactions. I will illustrate this with simulations of lipid interactions with P-glycoprotein, a human ABC transporter involved in multidrug resistance.
In the framework of steady-state diffusion problems, we show how the ideas of static condensation and hybridization lead to the introduction of the hybridizable discontinuous Galerkin methods.
Professor Cockburn received his Ph.D from University of Chicago in 1986 under the direction of Jim Douglas, Jr. He has spent all his academic career at University of Minnesota where he is now Distinguished McKnight University Professor. His research interests include the development of Discontinuous Galerkin methods for nonlinear conservation laws, second-order elliptic problems, electro-magnetism, wave propagation and elasticity.