Robert M. Miura - US grants

This award furnishes travel support for participants in the Conference on Frontiers in Applied and Computational Mathematics, held in May 2005 at the New Jersey Institute of Technology. Topics of the conference include mathematical biology, fluid dynamics, nonlinear waves, optics, and applied probability and statistics. Significant funding is allocated to support participation of students, postdoctoral fellows, and junior faculty.

The conference will bring together established researchers, junior faculty, postdoctoral fellows, and graduate and undergraduate students to discuss the use of mathematical modeling to treat difficult problems in the biomedical and health sciences, in the physical and social sciences, and in engineering and technology.

This meeting provides an opportunity for interaction between leading experts and junior researchers in important related areas of applied and computational mathematics. The conference will provide a valuable opportunity for postdocs and students to broaden their understanding and to learn about open problems in these areas.

Conference web page:
http://math.njit.edu/Events/FACM05/

This project investigates free boundary problems governed by nonlinear partial differential equations where localized heating controls the dynamics of the formation of singularities in surface-tension-driven flows. Thermal effects can lead to interesting dynamics that are very different from the dynamics in isothermal cases. The project will investigate the role temperature effects on viscosity and surface tension play in shaping viscous threads. The problems under study involve nonuniform fluid cylinders and cylindrical tubes with viscosity and/or surface tension that change rapidly with temperature. To better understand the collapse mechanism of fluid cylinders or tubes, the project studies model problems that describe: 1) pinch off of cylindrical threads; 2) collapse of a cylindrical hole in an infinite medium; 3) collapse of tubes; 4) annealing of tubes, and 5) extensions of these problems with flow.

This project analyzes the effects of temperature on threads of viscous material, which are used in many manufacturing processes. Thin viscous threads (such ink jets in printing) pinch off into droplets due to the effect of surface tension, which tends to minimize the surface area of the liquid. Pinch-off phenomena also occur in other applications, e.g., in thin films that arise in coating flows. Heating can significantly modify surface tension, which leads to thermocapillary effects, in which temperature gradients cause surface tension gradients that drive flows. Manufacturing processes require glass and polymeric materials to be at high temperatures to deform them, which leads to thermoviscous effects due to due to the resultant large changes in viscosity. Examples include the pulling of fiber optic cables and formation of glass microelectrodes used in electrophysiology. In recent years, semiconductor nanoclusters and high-purity nanowires have been produced using various techniques, and heating is normally required in the annealing stage. In all these processes, heating plays a crucial role in controlling the shape of final products. This project will contribute to detailed knowledge of the effects of heating on viscosity and surface tension, potentially leading to improved manufacturing processes for these materials.

Research on restricted ion diffusion in cellular media refers to the alteration of the diffusion flux of ions in tissue because of the intervening geometry of cellular membranes with variable permeability. Modeling, mathematical, and computational tools are developed for a broad class of problems in biological tissues that have complicated geometry and connectivity, specifically the brain. The brain-cell microenvironment is treated as a porous medium using volume averaging that has been used to study the diffusion of substances at the macroscopic level. Mathematical models are derived to study the effects on complex brain phenomena of permeability of cell membranes, the geometrical and topological structure of brain tissue including neurons and glial cells, and the connectivity of glial cells and of the vascular network. These models retain essential mathematical features of restricted diffusion of ionic species in biological tissue containing cells, namely coupled systems of diffusion equations with nonlinear source terms, and analytical and computational techniques are developed for studying these models. Additional mechanisms, believed to be important in certain applications, are studied mathematically and computationally. These studies of the brain include treating the intracellular spaces of neurons and glial cells separately and treating the extracellular space as a separate compartment. In addition, the influence of degenerate sources in the model equations is studied theoretically.

A specific application of this research is to the effects of restricted diffusion of ions due to the complex geometry of cells in the brain on cortical spreading depression (CSD), a nonlinear chemical and electrical (slow) wave phenomenon in the cortices of different brain structures. Although physiologists have studied spreading depression (SD) for more than 65 years, identification of the precise mechanisms involved in the propagation of SD waves has remained elusive. CSD has been implicated in migraine with aura; however, we still do not fully understand how the known and postulated mechanisms that are involved in CSD conspire to produce this enigmatic phenomenon. A prominent neurophysiologist has said, "No matter how many channel proteins we sequence, how many neuromodulators we identify and how many neural networks we construct, if we cannot explain spreading depression, we do not understand how the brain works." Mechanisms involved in CSD include ionic diffusion, membrane ionic currents, osmosis, effect of spatial buffering on fast transport of extracellular potassium ions, and effects of the vascular tree. CSD waves are characterized by large ionic concentration changes in the cortex of many different brain structures in different animals. It is essential to have a good understanding of the mechanisms underlying CSD waves to develop a thorough understanding of how the mammalian brainfunctions. The continuum frameworks developed earlier by researchers studying CSD are extended and generalized in order to carry out more detailed and careful studies of the putative mechanisms that may be involved in experimental CSD wave propagation. Also, as a step towards bridging the continuum and single neuron approaches to modeling CSD, electrodiffusion in a network of neurons is also studied.