By the Numb3rs Fall 2014


Attentional modulation of neuronal variability constrains circuit models
Principal Investigator:  Brent Doiron
Sponsor: Simons Foundation Collaboration on the Global Brain

Each time you look at a picture on a wall, even though you see the same thing, the electrical activity of neurons in your brain’s visual areas will be slightly different. These slight differences are known as variability, and typically researchers seek to remove variability from their data by taking the average of neural responses over many trials of whatever task they are studying. Then, they build theoretical models based on the averaged data. However, researchers have run into trouble with this approach: many different models seem to explain the same data, making it difficult to determine which model is correct. We propose a radically different approach in which we embrace the variability, and use it to constrain new theoretical models. We will use visual attention to test our model, in which observers focus on particular parts of a complex scene. We will then record the activity of large groups of neurons, and examine how visual attention modulates the variability in those neurons. The goals of our research are threefold. Can our model account for the existing data on how attention changes variability? Can we conduct experiments to determine how attention affects the extent to which variability is shared among neurons? Can we extend our model to this new data we collect? By focusing on variability, we can overcome many of the shortcomings of previous models, establishing a new approach for studying neural circuits in the visual system and other areas of the brain.

Theoretical Models of Shape Formation: Analysis, Geometry and Energy Scaling Laws

Principal Investigator: Marta Lewicka
Sponsor: NSF

Abstract: Recently there has been sustained interest in growth-induced morphogenesis (i.e., shape formation), particularly of low-dimensional structures such as filaments, laminae, and their assemblies, which arise routinely in biological systems and their artificial mimics. The physical basis for morphogenesis can be presented in terms of a simple principle: differential growth in a body leads to residual strains that generically result in changes of its shape. Eventually, the growth patterns are expected to be, in turn, regulated by these strains, so that this principle might well be the basis for the physical self-organization of biological tissues. Such topics lie at the interface of biology, chemistry and physics, with practical questions of engineering design and others. Residually stressed laminae are present in science and technology in a variety of situations; from atomically thin grapheme (of thickness 1 nm, with a lateral span of a few cm), to the earth's crust (of thickness 10 km, which spans thousands of km laterally). On the everyday scale, there has been much work on trying to understand the mechanics of these laminae when they are actuated, as in a growing leaf, a swelling or shrinking sheet of gel, a plastically strained sheet, etc. Understanding of the laws governing the equilibria and the evolution of such structures has many potential applications. The investigator studies mathematical problems related to the development of the shapes of these low-dimensional structures due to the interplay between growth patterns of the structures and residual strains in the material.

Finite element methods for non-divergence form partial differential equations and the Hamilton-Jacobi-Bellman equation

Principal Investigator: Michael Neilan
Sponsor: NSF

Abstract:  Many models in the sciences and engineering are solved approximately using computational methods, and it is necessary to theoretically justify the reliability of the computed approximations. In addition to providing justification of the numerical methods, the theoretical analysis often provides insight for the development of new methods with improved efficiency, accuracy, and viability. In this project, the investigator and a graduate student will construct, analyze and implement numerical methods for classes of linear and nonlinear partial differential equations arising in stochastic financial models, stochastic differential games, and other applications in finance and engineering. The overall aim of the project is to develop methods that can be implemented using current computational software and to derive explicit estimates of the approximate solutions.

Multiscale domain decomposition methods for flow and mechanics problems

Principal Investigator:  Ivan Yotov
Sponsor: NSF

The primary objective of this work is to develop numerical methods based on variational formulations of systems of partial differential equations coupling free and porous media fluid flows with deformations of the porous solids. These formulations couple through physically meaningful interface conditions free fluid models such as Stokes, Brinkman, or Navier-Stokes equations with single phase or multiphase Darcy flow.  In regions involving deformable porous media the Darcy flow is coupled with elasticity and modeled by the Biot system of poroelasticity. Topics of research include well posedness of the variational formulations, stable and accurate multiscale mortar discretization methods for these multiphysics variational formulations, and efficient parallel non-overlapping domain decomposition algorithms for the solution of the resulting algebraic systems by reducing the coupled global multiscale problem to a coarse scale interface problem.

Multiscale Modeling and Simulation of Multiphase Flow in Porous Media Coupled with Geomechanics
Principal Investigator:  Ivan Yotov
Sponsor: DOE

The main goal of this project is to develop accurate, efficient, and robust physics-based algorithms for modeling and simulation of flow and transport coupled with geomechanics in complex hydrological formations. An advanced modeling and simulation framework for multiphysics systems with multiscale and uncertain input parameters will be developed.  The algorithms will resolve multiple spatial and temporal scales and their interaction across different subsurface multiphysics phenomena. The proposed framework will have the form of a high performance computing simulation software that will allow for addressing a wide range of multiscale applications. The focus will be on applications to multiphase or compositional flows in poroelastic media, including carbon sequestration in saline aquifers, CO2 enhanced oil recovery, production in hydraulically fractured reservoirs, and non-isothermal compositional flows in deformable reservoirs.  These models and methods will be implemented in massively parallel computing environments.