#
Course Webpage

Special Topics:

Introduction to Low Dimensional Modeling

ME225BC, Winter Quarter 2004

### Meets: MWF 1:00-1:50, starting January 5

Bldg 387, Room 101

Course Description:

Many biological and physical systems of interest have fine details which
may not be necessary for a basic understanding of the system's behavior.
Through appropriate simplifications, low dimensional mathematical models
may be constructed which capture qualitative, and perhaps also quantitative,
aspects of the system's dynamics. This course will cover the development
and analysis of such models for problems arising in neuroscience and,
as time permits, fluid dynamics. More broadly, it will teach the student
a variety of mathematical modeling and analysis techniques which can be
applied to problems from engineering, physics, chemistry, and biology.

Specific topics to be covered include:

Neuroscience Modeling:

basics of neuroscience
conductance-based neuron models, such as the Hodgkin-Huxley equations
simple reductions of conductance-based neuron models
nullcline analysis of neuron models
integrate-and-fire neuron models
phase response curves and isochrons for periodically firing neurons
dynamics of coupled neurons
reduction of coupled neuron systems to phase models
analysis of phase-locked solutions for coupled neuron systems
analysis of bursting neuron models
Fluid Dynamics Modeling, as time permits:

basics of fluid dynamics
derivation of low-dimensional models using Galerkin projection
analysis of the Lorenz equations for fluid convection, a prototypical
chaotic system
analysis of a model for shear flow turbulence derived using Galerkin
projection
basics of center manifold reduction
derivation of low-dimensional models using symmetry methods
analysis of a model for binary fluid convection derived using symmetry
methods
derivation of low-dimensional models using the proper orthogonal
decomposition
analysis of a model for shear flow turbulence derived using the proper
orthogonal decomposition
pseudospectra analysis of linear models for shear flow turbulence

Questions? Email
Jeff Moehlis
at
moehlis@engineering.ucsb.edu
Course Syllabus

Homework - problem sets and solutions