Mathematical Challenges in Population-Based Brain Mapping

Paul Thompson's Talk

Thursday May 24, 2001, 2:00 - 2:30 P.M.

Workshop on Mathematics and Modeling in Brain Mapping, Conference on Imaging in Medicine and Neurosciences

UCLA Institute for Pure and Applied Mathematics, May 21-25, 2001.

These tutorial slides, from Paul's talk, are available on the Web at:

Contact Info: Paul Thompson (, Asst. Prof. of Neurology
UCLA Laboratory of Neuro-Imaging; Tel. (310) 206-2101 ext. 233


As brain imaging databases increase in size and content, it is now possible to develop mathematical algorithms (1) to uncover disease-specific patterns of brain structure and function in human populations, and (2) to reveal the effects of medication, and even genetics, in altering these patterns. We describe our construction of probabilistic atlases that store detailed information on how the brain varies across age and gender, across time, in health and disease, and in large human populations. Specifically, we introduce a mathematical framework based on covariant partial differential equations (PDEs), pull-backs of mappings under harmonic flows, and high-dimensional random tensor fields to encode variations in cortical patterning, asymmetry and tissue distribution in a population-based brain image database (N=200 scans). We use this reference information to detect disease-specific abnormalities in Alzheimer's disease and schizophrenia, including dynamic changes and response to medication over time. We will use illustrative examples to show how patterns of cortical organization, asymmetry, and disease-specific trends can be resolved that are not apparent in individual brain images. Finally, we create four-dimensional (4D) maps that store probabilistic information on the dynamics of brain change in development and disease. Digital atlases that generate these maps show considerable promise in identifying general patterns of structural and functional variation in diseased populations, and revealing how these features depend on demographic, genetic, clinical and therapeutic parameters.

[Title Slide]

Credits/Acknowledgments: Most of the teaching slides in this presentation derive from work by Paul Thompson and his colleagues at the UCLA Laboratory of Neuro Imaging. We would however like to credit several groups and sources for components of some of these teaching slides: Dr. David Rottenberg of the Minneapolis VA Center for his image illustrating the concept of Talairach stereotaxic space; Dr. Michael Mega, for his work on the slides illustrating our Alzheimer's disease brain atlas.

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    Contact Information

  • Mail:

    Paul Thompson, Ph.D.
    Assistant Professor of Neurology
    Laboratory of Neuro Imaging and Division of Brain Mapping
    710 Westwood Plaza, UCLA School of Medicine
    Westwood, Los Angeles CA 90095-1761, USA.

  • E-mail:
  • Tel: (310)206-2101
  • Fax: (310)206-5518