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Lefschetz Center for Dynamical Systems | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Organization
Index Activities |
LCDS Events and Seminars (2009/2010)
Wednesday, September 23, 2009
Abstract: Kinetic theory provides a coarse-grained alternative to the integrate-and-fire neuronal network description. In the limit of infinitely short conductance responses, a Boltzmann-type differential-difference equation can be derived for the probability density function of the neuronal voltage. A Fokker-Planck and a mean-field equation can be derived in the limit of small and vanishing conductance fluctuations, respectively. The talk will present detailed solutions to these equations, describing both the steady asynchronous and synchronously-oscillating states of the network, and will also discuss the effects of the network architecture. The mean-field provides exact solutions for the steady asynchronous state. For scale-free neuronal networks, it can be used to argue that the distributions of the firing rates and neuronal activity correlations are also scale free. The steady asynchronous state is also described by asymptotic solutions of the Fokker-Planck equation, using the size of the neuronal conductance fluctuations as the small parameter. in addition, the Fokker-Planck equation can also be used to describe the likelihood and temporal period of synchronous network oscillations, in which all the neurons fire in unison. The likelihood of synchrony is computed combinatorially using the network oscillation period and the voltage probability distribution. The oscillation period is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analyticaly via an eigenfunction expansion. The voltage probability distribution is found using a Central-Limit-Theorem-type argument via a calculation of the voltage cumulants. Differences between oscillations in all-to-all coupled and scale-free networks will also be discussed.
Wednesday, September 23, 2009
Monday, September 28, 2009
Abstract: We will begin by considering the second initial boundary problem in narrow domains of width є≪ 1 for linear second order differential equations with nonlinear boundary conditions. The solution of such a problem converges as є ↓ 0 to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in smooth narrow domains. However, if the domain is asymptotically non-smooth then the situation is a lot more involved and one is led to study the large deviations principle (LDP) for continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process Xt in R that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator DvDu, where v and u are two strictly increasing functions, v is right continuous and u is continuous. Here, we shall consider the LDP for Markov processes whose infinitesimal generator is є DvDu where 0<є≪ 1. This result generalizes the classical LDP results for the class of one dimensional "classical" stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator DvDu. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.
Wednesday, October 21, 2009
Abstract: I will talk about generalized traveling waves for scalar reaction diffusion equations in an inhomogeneous environment. As the name suggests, these solutions generalize the traditional notion of a traveling wave, although the wave profile is not fixed and the wave speed may not be well-defined. These solutions are stable attractors. If the environment has a certain statistical structure, then the asymptotic wave speed is well-defined, and the interface moves like a Brownian motion with positive drift.
Wednesday, October 21, 2009
Abstract: Burgers vortices are explicit solutions of the three-dimensional Navier-Stokes equations which are often used to model the vortex filaments observed in turbulent flows. Despite obvious limitations, this model describes in a correct way the fundamental mechanisms which are responsible for the persistence of coherent structures in three-dimensional turbulence. In this perspective, an important problem is to determine the stability of Burgers vortices with respect to perturbations in the largest possible class. This question has been open for almost three decades, and rigorous answers have been obtained so far for small Reynolds numbers only, or in the particular case of two-dimensional perturbations. In this talk, I shall show how a detailed analysis of the linearized operator allows to prove the stability of Burgers vortices with respect to three-dimensional perturbations, for any given value of the circulation number. This is a joint work with Yasunori Maekawa (Kobe University).
Wednesday, October 28, 2009
Abstract: The speaker will discuss recent work with A.Its and I.Krasovsky on the asymptotics of Toeplitz determinants with Fisher hartwig singularities.
Monday, November 2, 2009
Monday, November 9, 2009
Abstract: The model of a periodic potential with a defect is a common one in many areas of physics and applied mathematics. We consider a model with a dislocation type defect, where there is an asymptotic phase shift in the periodic potential between plus and minus infinity. Using Evan's function techniques and the Hamiltonian structure of the problem we prove an index theorem that counts the number of point eigenvalues created in the gaps in the continuous spectrum, and show that in the large energy limit the number of defect eigenvalues in a particular gap depends on the solvability of a certain Diophantine approximation problem.
Monday, November 16, 2009
Abstract: We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunov's equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role.
Wednesday, November 18, 2009
Abstract: Our ability to perceive and respond to transient external stimuli arises from the brain's ability to generate sustained, spatially localized activity in response to transient inputs. Mathematical models designed to represent this process in networks of neurons must provide a mechanism for recruitment of neurons into an active group as well as a mechanism to limit the spread of activity and hence maintain its localization. Analysis of such models often focuses on how different localization features, such as long-range inhibition, contribute to the existence and stability of sustained, localized activity patterns. In this talk, I will discuss work on the recruitment of neuronal oscillators, in contexts where desynchronization of inputs or competition complicate outcomes. I will consider discrete models, most of which are not specific to neurons, as well as a continuum limit. This work involves a geometric perspective, and the talk assumes no background knowledge of neuroscience.
Wednesday, November 18, 2009
Abstract: KdV equation is a standard model of weakly nonlinear long waves on the surface of shallow water. It will be shown that in KdV with periodic boundary conditions, high frequency solutions evolve almost as the linear ones for large time. For KdV (or some other dispersive equations) on the real line such behavior could be expected due to the dispersive decay. While on the circle (i.e. periodic boundary conditions) such dispersive decay is not possible, the dispersion manifests itself in averaging out nonlinearity over high frequency solutions. This result is obtained by the application of normal form transformations in the appropriate spaces. The integrability properties of KdV are not used, so similar results could be obtained for other KdV like equations. The interaction of these high frequency solutions with a cnoidal wave will be discussed, too. This work has been motivated by an attempt to explain some phenomena in nonlinear optics and fluid dynamics. This is a joint work with M.B. Erdogan and N. Tzirakis (also University of Illinois).
Monday, November 30, 2009
Wednesday, December 2, 2009
ArchivesListings of past seminars presented by LCDS are archived by year for your viewing. |
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| Last change: Nov. 10, 2009 |
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