Optimal decisions: From neural spikes, through stochastic differential equations, to behavior.

Speaker: 

Professor Philip Holmes

Institution: 

Princeton Univ.

Time: 

Thursday, January 13, 2005 - 4:00pm

Location: 

McDonnell Douglas Auditorium

There is increasing evidence from in vivo recordings in monkeys
trained to respond to stimuli by making left- or rightward eye
movements, that firing rates in certain groups of `visual' neurons
mimic drift-diffusion processes, rising to a (fixed) threshold prior
to movement initiation. This supplements earlier observations of
psychologists, that human reaction time and error rate data can be
fitted by random walk and diffusion models, and has renewed interest
in optimal decision-making ideas from information theory and
statistical decision theory as a clue to neural mechanisms.

I will review some results from decision theory and stochastic
ordinary differential equations, and show how they may be extended and
applied to derive explicit parameter dependencies in optimal
performance that may be tested on human and animal subjects. I will
then describe a biophysically-based model of a pool of neurons in a
brainstem organ - locus coeruleus - that is implicated in widespread
norepinephrine release. This neurotransmitter can effect transient
gain and response threshold changes in cortical circuits of the type that
the abstract drift-diffusion analysis requires. I will argue that, in
spite of many gaps and leaps of faith, a rational account of how neural
spikes give rise to simple behaviors is beginning to emerge.

This work is in collaboration with Eric Brown, Rafal Bogacz, Jeff
Moehlis and Jonathan Cohen (Princeton University), and Ed Clayton,
Janusz Rajkowski and Gary Aston-Jones (University of Pennsylvania).
It is supported by the National Institutes of Mental Health.

Transport in celestial and molecular systems

Speaker: 

Professor Jerry Marsden

Institution: 

Caltech

Time: 

Thursday, January 27, 2005 - 4:00pm

Location: 

MSTB 254

This topic is concerned with the application of computational
dynamics tools to the problem of transport in celestial mechanics and
molecular systems. The celestial problems are typified by computing the
probability that a member of the Hilda group of asteroids (which are
just inside the orbit of Jupiter) cross the orbit of Mars in a certain
interval of time. In the molecular context, the computation of the
ionization rate of the Rydberg atom will be used to illustrate the
techniques.

Geometric Properties and Non-Blowup of 3D Incompressible Euler Flow

Speaker: 

Professor Tom Hou

Institution: 

Caltech

Time: 

Thursday, March 10, 2005 - 4:00pm

Location: 

MSTB 254

Whether the 3D incompressible Euler equation can develop a finite time
singualrity from smooth initial data has been an outstanding open problem.
It has been believed that a finite singularity of the 3D Euler equation
could be the onset of turbulence. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equation.
Further, we show that there is a sharp relationship between the geometric
properties of the vortex filament and the maximum vortex stretching. By
exploring this geometric property of the vorticity field, we have obtained
a global existence of the 3D incompressible Euler equation provided that
the normalized unit vorticity vector has certain mild regularity property
in a very localized region containing the maximum vorticity. Our assumption
on the local geometric regularity of the vorticity field is consistent
with recent numerical experiments. Further, we discuss how viscosity may
help preventing singularity formation for the 3D Navier-Stokes equation,
and present a new result on the global existence of the viscous Boussinesq
equation.

Attractors for equations of mathematical physics

Speaker: 

Prof, Mark Vishik

Time: 

Thursday, February 5, 2004 - 4:00pm

Location: 

MSTB 254

The lecture will cover the following topics:
1. Global attractor for an autonomous evolution equation. Examples. between the attractor and the family of complete solutions.
2. Fractal dimension of a global attractor. Examples.
3. Nonautonomous evolution equations and corresponding processes. Uniform global attractor of a process.
4. Global attractor of the nonautonomous 2D Navier Stokes system. Translation-compact forcing term. Relation between the uniform attractor and the family of complete solutions. Nonautonomous 2D Navier-Stokes system with a simple attractor.
5. Kolmogorov epsilon-entropy of the global attractor of a nonautonomous equation. Estimates of the epsilon-entropy. Examples.
6. Some open problems.

Geometric Motion in Plasmas

Speaker: 

Prof. Marshall Slemrod

Institution: 

University of Wisconsin

Time: 

Thursday, January 29, 2004 - 4:00pm

Location: 

MSTB 254

This talk outlines recent work by Feldman, Ha, and Slemrod on the dynamics of the sheath boundary layer which occurs in a plasma consisting of ions and electrons. The equations for the motion are derived from the classical Euler- Poisson equations. Of particular interest is that the boundary layer interface moves via motion by mean curvature where the acceleration of the front (not the velocity) is proportional to the mean curvature of the front.

Twisted K-theory and moduli spaces

Speaker: 

Prof. C. Teleman

Institution: 

Cambridge University

Time: 

Thursday, December 11, 2003 - 11:00am

Location: 

MSTB 254

The notion of topological field theory has stymied topologists partly because it assigns to spaces quantities that are multiplicative under disjoint union; traditional homological or homotopical constructions are additive. In this talk I will survey how the use of an old "multiplicative" object in topology ("the spectrum of
units" in the class of vector spaces) leads to a successful formulation of a simple (but non-trivial) 2-dimensional field theory (the "Verlinde ring" and its deformations) and to new topological results about the moduli space of vector bundles on a Riemann surface. This is based on joint work with Freed-Hopkins and with Woodward.

Regularization of differential equations

Speaker: 

Prof. David Nualart

Institution: 

Spain Academician, visiting Kansas University

Time: 

Thursday, November 6, 2003 - 4:00pm

Location: 

MSTB 254

In this talk, he will discuss the regularization effect of the noise in ordinary and partial differential equations. The main results are the existence and uniqueness of strong solutions for nonlinear equations when the drift coefficient is not Lipschitz. The proofs of these results are based on the Girsanov transformation of measure. Some recent regularization results by fractional noise will be also presented.

The semiclassical focusing nonlinear Schroedinger equation

Speaker: 

Prof. S. Venakides

Institution: 

Duke

Time: 

Thursday, October 16, 2003 - 4:00pm

Location: 

MSTB 254

The NLS equation describes solitonic transmission in
fiber optic communication and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures.

The IVP for the NLS equation is solvable by the method of inverse scattering. The initial spectral data of the Zakharov Shabhat (ZS) operator, a particular linear operator having the solution to NLS as the potential, are calculated from the initial data of the NLS; they evolve in a simple way as a result of the integrability of the problem, and produce the solution through the inverse spectral transformation.

In collaboration with A. Tovbis, we have developed
a one parameter family of initial data for which the derivation of the spectral data is explicit. Then, in collaboration with A. Tovbis and X. Zhou, we have obtained the following results:

1) We prove the existence and basic properties of the
first breaking curve (curve in space-time above which the character
of the solution changes by the emergence of a new
oscillatory phase) and show that for pure radiation
no further breaks occur.

2) We construct the solution beyond the first break-time.

3) We derive a rigorous estimate of the error.

4) We derive rigorous asymptotics for the large
time behavior of the system in the pure radiation case.

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