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DE Seminars: Fall 2004

Monday September 13

Title Homoclinic and heteroclinic orbits in nonlocal, synaptically coupled, excitatory neuronal networks
Speaker Linghai Zhang
Lehigh University

Abstract Homoclinic and heteroclinic orbits play significant roles in the propagation of activation of cells in neuronal networks, their existence, conductance velocity and exponential stability is worth of a rigorous mathematical investigation. For a large class of nonlinear singularly perturbed systems of nonlocal evolution equations arising from the study of nonlocal, synaptically coupled excitatory neuronal networks,
u_t + f (u; w) = (alpha – Gamma u){K*[H(u -theta)]},
w_t + g(u, w) = 0,
under different conditions on various parameters, the author proves the existence of both homoclinic and heteroclinic orbits. He also investigates the exponential stability and instability of the orbits. He applies Banach contraction mapping principle, Leray-Schauder’s fixed point principle and implicit function theorem to study the existence, and he uses linearized stability criterion, complex analytic functions and asymptotic analysis to establish the stability. This work extends known results to the case Gamma> 0 and to general nonlinearities. Key words and phrases: excitatory neuronal networks, integral-differential equations, homoclinic and heteroclinic orbits, existence and uniqueness, fixed point principle, wave-speed, exponential stability, linear differential operators, eigenvalue problems, eigenvalue functions

Monday September 20

Title Regularization and DEs with rough right hand sides
Speaker Thomas I. Seidman

We consider an autonomous DE, nominally in the form: $x’=f(x)$, and seek a weak notion of solution (involving a regularization of $f$) under which existence can be assured, even for very rough $f$. [The notion of `regularization’ here is suggested by Fillipov’s treatment for a class of such problems and notions of set-valued derivative used in optimization theory.]

Monday October 4

Title Homogenization in partial differential equations
Speaker Rouben Rostamian

“Homogenization” in the context of differential equations refers to a limiting process whereby the frequency of one or more highly oscillatory coefficients tends to infinity. The central question in the theory of homogenization is the manner in which the solutions of such differential equations behave in the limit. I will present case studies for three differential equations arising from simple physics. The equations are those of: (1) heat conduction, (2) fluid flow, and (3) elasticity. I will demonstrate the process of homogenization in these cases and point out certain common features. In particular, I will focus on the question of how geometrical symmetries in the unit cell affect the mechanical symmetries in the corresponding homogenized materials.

Monday October 11

Title Distributed Coordination of Mobile Agents: From bird flocking to synchronization of coupled oscillators
Speaker Ali Jadbabaie
University of Pennsylvania

In this talk we provide a rigorous analysis of several distributed coordination and flocking algorithms which have appeared in various disciplines such as statistical physics, biology, computer graphics over the past 2 decades as a mechanism for demonstrating emergence of collective behaviors (such as social aggregation, schooling, flocking and synchronization) using only local interactions. Simulation studies of these algorithms have appeared in the computer graphics literature as a landmark result in `artificial life’ , in the evolutionary biology literature as a possible mechanism for fish schooling and bird flocking and in the physics literature as an example of `emergence of collective behavior’, and have been featured as cover stories in the journals Nature and Scientific American. We will show that the above phenomenon can be analytically explained, using tools from systems and control theory and graph theory. It is shown that when a suitably defined proximity graph induced by the neighboring relation is `connected in time’, collective behavior such as flocking will occur. We will also show that the synchronization phenomena in a network of coupled nonlinear oscillators (a well studied problem in physics and dynamical systems) can also be explained using the same approach. Utilizing these results, we provide a biologically plausible mechanism for flocking and synchronization.

Monday October 25

Title Analyzing interchannel crosstalk in a multichannel optical fiber analog tranmsission system
Speaker Brian Marks

Analog transmission through optical fibers has multiple applications, from radio and cable television to radar and signal buffering. Optical fibers are appealing because of their compactness, light weight, low loss, and insensitivity to electromagnetic interference. However, in an analog setting, optical fibers can degrade fidelity of signals due to their inherent dispersion and nonlinearity. In particular, crosstalk between channels due to the nonlinear response of the fiber is a major penalty. Although light in optical fibers is typically modeled by variants of the nonlinear Schroedinger equation, the dynamics of the crosstalk can be reduced to a much simpler set of differential equations. I will discuss the mechanisms that give rise to crosstalk in these analog systems and discuss ways of mitigating it.

Monday November 1

Title A fish thalamic dendrite: an example for ‘geometric’ cable theory?
Speaker Jonothan Bell

I will introduce people to one of the strangest shaped dendrites in the animal kingdom, but one with wonderful geometry to apply a variable diameter cable theory to. I will then apply Green’s function techniques to solve the problem.

Monday November 8

Title Dynamical Attractors which are Nonchaotic and Strange
Speaker Ram Ramaswamy
Institute for Advanced Study, Princeton

Attractors in quasiperiodically forced dynamical systems can be strange (geometrically fractal) but nonchaotic. As a result, the motion is aperiodic, but nevertheless has no sensitivity to initial conditions. This allows for new strategies in control and synchronization.

Monday November 15

Title A numerical model for simulating cell adhesion in hydrodynamic flows
Speaker Charles Eggleton

Cell adhesion in the presence of hemodynamic flow plays a fundamental role in many of the physiological processes occurring in the circulatory system. These adhesive interactions are influenced by processes taking place at drastically different length scales: (1) Fluid forces which tend to deform the cells are on the length scale of the cell diameter (~10 mm) whereas (2) the range of receptor-ligand bonds mediating cell adhesion is ~ 100 nm. The coupling of forces acting at these disparate length scales as well as their effects on cell deformation makes the simulation of cell adhesion computationally intensive. To investigate the role of cell deformability on cellular adhesion, we developed a three-dimensional computational model based on the immersed boundary method which simulates rolling of a deformable cell on a selectin-coated surface under shear flow with a stochastic description of receptor-ligand bond interaction. The effect of other suspended cells in the plasma is neglect in these simulations such that the rolling cell experiences hydrodynamic forces. We observed that rolling velocity increases with increasing membrane stiffness and this effect is larger at high shear rates. The bond lifetime, number of receptor-ligand bonds and the contact area between cell and substrate decreased with increasing membrane stiffness. This study shows that cellular properties along with the kinetics of selectin-ligand interactions affect leukocyte rolling on selectin-coated surfaces. Ongoing simulations of cell-cell adhesion in bulk shear flows will be discussed.

Monday December 6

Title Dynamics and plasticity
Speaker Evelyn Sander
George Mason University

This lecture concentrates on applying dynamical systems to computational neuroscience. We concentrate on a model for short term synaptic changes within populations of different types of neurons. Such changes are known as short term plasticity. We model the behavior observed in experiments studying epileptic seizures. No previous knowledge of computational neuroscience is assumed.