Navigating the Brain
Mark P. Wachowiak, Ph.D.
Department of Computer Science and Mathematics
Nipissing University
April 27, 2007
Outline
• Basic brain anatomy
• Brain imaging
– Magnetic resonance imaging (MRI)
– Computed tomography (CT)
– Functional imaging
• Brain navigation
• Future directions
Mathematics Awareness Month
Interdisciplinary Nature of Brain
Research
•
•
•
•
•
•
•
Neuroscientists
Physicians
Psychologists
Mathematicians
Biologists
Engineers
Computer scientists
Neurons
• Electrically excitable cells
in the nervous system.
• Transmit and process
information.
• Dendrites
– Conduct electrical impulses
from other neurons or cells
towards the cell body.
• Axons
– Conduct impulses away
from the cell body to other
neurons.
http://faculty.uca.edu/~benw/biol1400/pictures/neuron.jpg
Hodgkin-Huxley Model of Neurons
• First mathematical model of
neurons (1952).
• Models electrical
characteristics of the cells
• Based on systems of
nonlinear ordinary
differential equations.
• Starting point for modern,
advanced neuron models.
www.nobel.org
Alan Lloyd
Hodgkin
Andrew
Fielding Huxley
Cerebrum
• Largest part of the
brain.
• Higher brain
functions:
– Thought
– Action
– Vision
– Memory
http://serendip.brynmawr.edu/bb/kinser/Structure1.html#cerebrum
Cerebellum
• Associated with
the regulation and
coordination of
movement,
posture, and
balance.
Medulla Oblongata
• Relays nerve
signals between
the brain and the
spinal cord.
• Involuntary
functions:
– Breathing
– Blood pressure
– Heart rate
– Reflexes
Sulci and Gyri
• Sing. sulcus, gyrus
• Sulcus
Gyrus
– Fissure in the brain
tissue.
– Interhemispheric fissure
– divides the brain into
left and right
hemispheres.
• Gyrus
– Elevated “hill” areas
between sulci.
Atamai
Sulcus
White Matter
• Found in the brain
and spinal cord.
• Consists of
insulated
(myelinated) nerve
fibers (axons).
• Responsible
transmitting and
conducting
information.
http://www.brainexplorer.org/brain-images/white_matter.jpg
Grey Matter
• Consists of the
bodies of neurons.
• Responsible for
information
processing.
• Generates responses
to stimuli.
http://www.brainexplorer.org/brain-images/white_matter.jpg
Neuroimaging
Types of Neuroimaging
• Structural
– Magnetic resonance imaging
– Computed tomography
– Ultrasound
• Functional
– Functional MRI
– Positron emission tomography
– Single photon emission computed
tomography
Magnetic Resonance Imaging
• Excellent for clearly visualizing structures
in soft tissues, such as the brain.
• Very commonly used in:
– Diagnosis
– Image-guided surgery and therapy
• By adjusting scanning settings, specific
features can be detected.
• MRI images are 2D slices through the
body at a specific location.
MRI Scanner
http://psyphz.psych.wisc.edu/
Proton Precession
Hydrogen protons precess about an axis, like a
“wobbling” spinning top.
Proton Precession in Tissue
Randomly-oriented hydrogen protons precess.
Application of Magnetic Field
Magnetic field
A strong magnetic field is applied in a specified direction.
The protons align with the magnetic field.
Application of RF Pulse
A strong, sudden RF
(radiofrequency) pulse
is applied in a direction
orthogonal to the
magnetic field.
Magnetic field
Protons are briefly
placed into a highenergy state.
RF pulse
RF Pulse is Turned Off
Magnetic field
Energy is released
as the protons
return to their lowenergy orietation
within the
magnetic field.
MRI Image Formation
• When the RF pulse is turned off, the
hydrogen protons return to their natural
alignment within the magnetic field.
• Energy is released.
• The coil detects this signal and sends it to
a computer for processing.
• The signal consists of complex values
which have real and imaginary
components.
Complex Numbers
i  1
c  a  bi
c  a b
2
Imaginary number
Complex number
2
Magnitude
Fourier Transform
• Determine the frequency
components of a signal.
• From a complex
frequency
representation, recover
the original signal.
• Involves calculus and
integration of complexvalued functions.
ocw.mit.edu
Jean Baptiste Joseph Fourier
(1768-1830)
Obtaining Frequency Information
Fourier Transform
+
Fast Fourier Transform
• A very efficient method to
compute the Fourier
transform of a signal.
• Developed in 1965 by J.W.
Cooley and John Tukey
(AT&T Labs).
• One of the “top ten”
algorithms of the 20th
century.
www.ieee.org, www.math.brown.edu
James W. Cooley
John W. Tukey
MRI Image Formation
Fourier
Transform
Magnitude information from signal
Phase information from signal
MRI Visualization
• A series of 2D
MRI images can
be combined
together to form a
3D volume.
• This volume can
then be used to
generate realistic
visualizations and
models.
MS Lesions
http://www.med.harvard.edu/AANLIB/cases/case5/mr2/035.html
Computed Tomography (CT)
• Tomography
– Imaging in sections, or slices.
• Computed
– Geometric processing used to reconstruct an
image.
– Computerized algorithms
Computed Tomography (2)
• Uses X-rays
– Dense tissue, like bone, blocks x-rays.
– Gray matter weakens (attenuates) the x-rays.
– Fluid attenuates even less.
• A computerized algorithm (filtered
backprojection) reconstructs an image of
each slice.
CT Image Formation
X-ray tube
X-ray
X-ray detector
Computed Tomography
http://fitsweb.uchc.edu/student/selectives/TimHerbst/intro.htm
CT Image Formation
Backprojection
CT Image Reconstruction – 6
Slices
CT Image Reconstruction – 12
Slices
CT Image Reconstruction – Final
Image
fMRI
• Functional MRI – used to investigate brain
function.
• Enables watching brain activity in vivo.
• Measures haemodynamic response.
– Changes in oxygen content of the blood occur
as the result of neuronal activity.
Interdisciplinary Nature of fMRI
• Physics
– Hardware tools
• Electrophysiology
– Neuronal behaviour
• Psychology
– Cognitive psychology
• Statistics
– Making sense of observations
• Neuroanatomy
Blood Oxygen Level Dependent
fMRI (BOLD)
Signal
increase
Signal
decrease
http://en.wikipedia.org/wiki/Neuroimaging
fMRI
Active areas while subjects remembered information presented visually
Active areas while subjects remembered information presented aurally
Active areas for both types
http://mednews.stanford.edu/stanmed/2005fall/brain-main.html
Complementary Imaging
Techniques
MRI
CT
http://www.med.harvard.edu/AANLIB/hms1.html
fMRI
Brain Navigation
Mathematical Challenges in
Neuroimaging
• Segmentation
– Identifying structures or abnormalities from 2D
or 3D brain images.
– Development of models to help plan surgery
and therapy.
– Concepts from computer graphics, geometry,
topology, probability theory.
Mathematical Challenges in Brain
Imaging
• Registration
– Aligning and combining images from the
same or different type of image.
– Useful in simulation, modeling, and in
planning surgical procedures.
– Employs concepts from probability theory,
information theory, geometry, topology,
optimization, parallel computing, and many
other areas.
MRI Visualization and
Segmentation
Atamai
Segmentation – Differential Geometry
Automatically computed network of 3D curves
lying deep in the cortex (sulcal fundi), colorcoded according to the curvature.
G. Sapiro, SIAM News, Volume 40, Number 2, March 2007
Registration and Fusion
MRI
Histology cryosection
MRI
Ultrasound
PET
MRI + Ultrasound
CT-to-MRI Registration
Brain Warping
• Nonlinear registration.
• Used to match features in
structurally different brains.
• Uses:
– Geometry
– Topology
– Probability
– Calculus
https://www.rad.upenn.edu/sbia/dgshen/HAMMER/brainWarping.htm
Segmentation and Registration
Segmentation of
the brain surface
from MRI scans
Registration of
fMRI onto
segmented brain
surface to display
activation areas
Virtual Reality Planning System for
Neurosurgery
Atamai
Neurosurgery Planning
3D models generated from MRI and CT images.
Atamai
Surgical Planning with MRI and fMRI
MRI and fMRI registration, and the 3D
reconstruction of a tumour.
Tumour segmentation is carried out prior to
the surgery.
Neurosurgeons now have complex
information available to decide the best
strategy for removing the tumour.
Future Areas
• Functional imaging for to relieve acute and
chronic pain.
• Modeling to develop better therapies for:
– Alzheimer’s disease
– Multiple sclerosis
– Brain tumours
– Strokes
– Psychiatric disorders
– Other neurological and brain diseases
Other Areas of Cross-fertilization
• Electroencephalography
(EEG)
http://www-sop.inria.fr/odyssee/research/benar-clerc-etal:06/oddball-orig-erpimage.png
Other Areas of Cross-fertilization
• Electroencephalography
(EEG)
• Artificial neural networks
http://www.math.ntnu.no/~elenac/diplomoppgaver/neurons.jpg
Other Areas of Cross-fertilization
• Electroencephalography
(EEG)
• Artificial neural networks
• Dynamical systems
http://www.nd.edu/~malber/images/classes/lorenz3d.gif
Other Areas of Cross-fertilization
• Electroencephalography
(EEG)
• Artificial neural networks
• Dynamical systems
• Modeling of brain
processes
Moving Through the Visual Cortex
http://people.scs.fsu.edu/~burkardt/fun/misc/brain.html
Thank You.
http://www.nipissingu.ca/numeric