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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 Announcement

    Course Syllabus

    Homework - problem sets and solutions