curso uam 2013

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Introducción a la técnica y análisis de
resonancia magnética funcional
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Jorge L. Armony
McGill University
Montreal
Canada
Potencial de acción
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Interacciones neuronales
Kandel, Schwartz & Jessel: Principles of Neural Science
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Técnicas de neuroimagen funcional:
Señal electrodinámica
Electrodos de profundidad
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Electroencefalografía (EEG)
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Magnetoencefalografía (MEG)
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Técnicas de neuroimagen funcional:
Señal hemodinámica
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J Physiol. (1890), 11: 85-158
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These facts seem to us to indicate the existence of an automatic
mechanism by which the blood-supply of any part of the
cerebral tissue is varied in accordance with the activity of the
chemical changes which underlie the functional action of that
part.
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Resonancia magnética funcional (IRMf/fMRI)
Iron
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La hemoglobina
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- four globin chains
- each globin chain contains a heme group
- at center of each heme group is an iron atom (Fe)
- each heme group can attach an oxygen atom (O2)
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PNAS (1936), 22: 210-216
Over ninety years ago, on November 8, 1845, Michael Faraday investigated the
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magnetic properties of dried blood and made a note "Must try recent fluid blood."
If he had determined the magnetic susceptibilities of arterial and venous blood, he
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would have found them to differ by a large amount (as much as twenty per cent
for completely oxygenated and completely deoxygenated blood); this discovery
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without doubt would have excited much interest and would have influenced
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appreciably the course of research on blood and hemoglobin.
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Hemoglobin:
Magnetic Properties
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Oxy-Hb (four O2) is diamagnetic → no ΔB effects
Deoxy-Hb is paramagnetic → if [deoxy-Hb] ↓ → local ΔB ↓
B0
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Oxyhemoglobin
Diamagnetic
χ ~ -0.3
oxyHb
deoxyHb
Deoxyhemoglobin
Paramagnetic
χ ~ 1.6
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oxyHb
deoxyHb
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Hemoglobin: Magnetic Properties
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B0
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neuronal activity ↑
tissue energy demand ↑
neurovascular
coupling
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Glucose, O2 consumption ↑
fMRI relevant
physiological
correlates
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blood flow and volume ↑
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local dHb content of blood ↓
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local dHb-induced magnetic field disturbance ↓
BOLD fMRI signal ↑
(Blood Oxygen Level Dependent signal)
Source: B. Pike, MNI
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Relationship between BOLD signal and neural activity
LFP: local field potentials
MUA: multiunit activity
SDF: Spike density function
Logothetis et al. (2001). Nature 412: 150-157
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First Functional Images
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Kwong et al. (1992) PNAS 89: 5675-5679
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Hemodynamic response
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Source: Robert Cox’s web slides
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7
30 slices
VOXEL
(Volumetric Pixel)
Un volumen:
64 x 64 x 30 voxels
4 mm
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4 mm
Resolution in x (e.g. 4mm)
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Matrix in the xy plane (e.g., 64 x 64)
4 mm
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Resolution in y (e.g. 4mm)
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Re
so
(e. lutio
g.
4 m n in z
m)
122880 datos!
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Imagenes funcionales
time
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Experimental Design in Neuroimaging
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Most studies use a categorical, subtractive design
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CATEGORICAL:
Two (or more) levels of a given category (one of them usually serves as control)
Examples: Happy, Fearful and Neutral Faces, Remembered and Forgotten words, Pictures
and Fixation Cross
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SUBTRACTIVE:
Directly compares two conditions (A-B). Uses the notion of cognitive subtraction
(Donders, 1868), which relies on the assumptions of pure insertion and linearity
Cogntitive subtraction = Mathematical Subtraction
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(principle of pure insertion: Donders, 1868)
–
FEAR
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=
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>
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=
+
FEAR
–
=
FEAR
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Preprocessing
• Corrects for non-task-related variability in
experimental data
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• Usually done without consideration of
experimental design; thus, pre-analysis
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• Sometimes called post-processing, in reference
to being done after acquisition
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• Attempts to remove, rather than model, data
variability
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Source: http://www.biac.duke.edu
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• Realignment
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Preprocessing Steps
• Coregistration
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• Slice timing
• Normalization
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• Spatial Smoothing
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Inter-scan movement: Realignment
People move, even if they don’t realize!
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Need for motion correction
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Two steps:
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1. Registration: Determine the 6 parameters that describe the rigid-body
transformation between each image and a reference image (usu. first
in series).
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2. Transformation: Resampling each image according to the determined
transformation parameters.
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Inter-scan movement: Realignment
Same location in the brain
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Same location in the grid
3 translation parameters
3 rotation parameters
z
roll
pitch
x
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Rigid body movement:
y
yaw
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Realignment: Registration
Small movements are corrected well
TRASLATION
x
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mm
y
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z
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ROTATION
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yaw (y)
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roll (z)
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rad
pitch (x)
Sudden movements are more problematic (especially if correlated with
experimental paradigm)
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Difference between first and last image in one session
AFTER REALIGNMENT
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BEFORE REALIGNMENT
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Optimisation
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* Optimisation involves finding some “best”
parameters according to an “objective function”,
which is either minimised or maximised
* The “objective function” is often related to a
probability based on some model
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Objective
function
Source: John Ashburner
Most probable solution
(global optimum)
Local optimum
Value of parameter
Local optimum
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* Intra-modal
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Objective Functions
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* Mean squared difference (minimise)
* Normalised cross correlation (maximise)
* Entropy of difference (minimise)
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Mutual information (maximise)
Normalised mutual information (maximise)
Entropy correlation coefficient (maximise)
AIR cost function (minimise)
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*
*
*
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* Inter-modal (or intra-modal)
Source: John Ashburner
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Residual Errors from aligned fMRI
* Re-sampling can introduce interpolation errors
* especially tri-linear interpolation
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* Gaps between slices can cause aliasing artefacts
* Slices are not acquired simultaneously
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* rapid movements not accounted for by rigid body model
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* image distortion
* image dropout
* Nyquist ghost
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* Image artefacts may not move according to a rigid body model
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* Functions of the estimated motion parameters can be modelled as
confounds in subsequent analyses
Source: John Ashburner
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Unwarping
Estimate reference from
mean of all scans.
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Estimate new distortion
fields for each image:
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Unwarp time
series.
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Estimate
movement
parameters.
• estimate rate of change of
field with respect to the
current estimate of
movement parameters in
pitch and roll.
Source: John Ashburner
Δϕ
∂B0 ∂ϕ
+Δ
θ
∂B0 ∂θ
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Andersson et al, 2001
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Realignment: Transformation
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Source: John Ashburner
Inter-subject brain differences: Normalization
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People’s brains are different!!
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Normalization
Also useful for reporting coordinates in a standard space (e.g., Talairach and
Tournoux)
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Inter-subject brain differences: After Normalization
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Spatial Normalisation - Procedure
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* Minimise mean squared difference from template
image(s)
Affine registration
Source: John Ashburner
Non-linear registration 33
*
*
*
*
3 translations
3 rotations
3 zooms
3 shears
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* Fits overall shape and size
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* The first part is a 12 parameter affine
transform
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Spatial Normalisation - Affine
* Algorithm simultaneously minimises
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* Mean-squared difference between template and source image
* Squared distance between parameters and their expected values
(regularisation)
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Spatial Normalisation - Non-linear
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Deformations consist of a linear
combination of smooth basis
functions
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These are the lowest frequencies
of a 3D discrete cosine
transform (DCT)
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Algorithm simultaneously minimises
* Mean squared difference between template and
source image
* Squared distance between parameters and their
known expectation
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Non-linear
registration
without
regularisation.
(χ2 = 287.3)
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registration
using
regularisation.
(χ2 = 302.7)
Affine registration.
(χ2 = 472.1)
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Without
regularisation,
the non-linear
Template
spatial
image
normalisation
can introduce
unnecessary
Non-linear
warps.
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Spatial Normalisation - Overfitting
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Why smooth?
• Remove residual inter-subject brain differences
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• Allow for the use of Gaussian random field theory (later…)
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Crazy little thing called BOLD
Image acquisition by Sarael Alcauter, National Institute of Psychiatry “Ramón de la Fuente”, Mexico City, Mexico
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=
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Mean (Queen) = 10.77
t(118) = 14.17
Mean (Silence) = 10.26
p = 2 10-27
0.000000000000000000000000002
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=
T-map
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t-values
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t0
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t-values
t0
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TQA: Temporal Queen Area
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Respuestas cerebrales a estímulos visuales
Trejo & Armony (2008)
HGM, México
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Kanwisher et al. (1997)
MIT, USA
Activación en la corteza fusiforme (FFA): Área involucrada en
el procesamiento de rostros
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>
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Respuestas cerebrales a estímulos visuales
Activación en la corteza parahipocámpica: Área involucrada en
el procesamiento de información espacial
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Segregación de la respuesta en corteza temporal ventral a distintos
tipos de estímulos visuales
–
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–
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9
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=
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Casas – Rostros
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Error estándar de la media
Mapa t
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2
1
0
-1
8
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=
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4
2
t > 5.2
(p<0.05 FWE)
0
-2
Mapa t con umbral
-4
Casas – Rostros
Mapa t con umbral
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6
4
=
2
t > 5.2
(p<0.05 FWE)
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-2
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Mapa t
-4
Imagen T1
(estructural)
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Error estándar de la media
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Mapa de activaciones
(SPM)
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NUEVA
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RECORDADA
RECORDADA
Efecto subsecuente de memoria
Subsequently Remembered > Subsequently Forgotten
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NEUTRAL
OLVIDADA
Sergerie, Lepage & Armony (2005). Neuroimage
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Efecto del valor emocional en la memoria
EMOTIONAL
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NEUTRAL
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L
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L DLPFC (-34 32 12)
R DLPFC (48 22 16)
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0.12
0.09
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0.03
Fear
0.05
0.03
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0.06
0
L
0.01
Fear
Happy
Neutral
Happy
Neutral
-0.01
Sergerie, Lepage & Armony (2005). Neuroimage
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What else can we do?
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• Cleaner data (e.g., high-pass filtering, scaling)
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• More sophisticated averaging (modeling)
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• Choose a good threshold
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Modeling the expected response
p = 2 10-13
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Original:
ON-OFF = 13.02, t(80) = 8.6,
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Shifted blocks:
ON-OFF = 12.45, t(80) = 17.7,
p = 10-29
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Modeling the expected response
ASSUMPTIONS:
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Experimental Paradigm
• The response is solely determined by two conditions: the ON and OFF
blocks
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Brain Physiology
• There is a delay of about 6 sec between onset/offset of the block and the
observed signal
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Brain-Design Interactions
• The response is the same within a given block
• The response is the same for all blocks belonging to the same condition
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Modeling the expected response
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10
5
-5
-10
0
10
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30
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60
70
80
90
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60
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90
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-15
AM
0
25
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5
0
C
-5
-10
-15
0
10
20
30
40
A better model for the HRF
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0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
10
20
30
40
50
60
70
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-0.8
80
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Convolve with a model of hrf
0.8
0.6
0.4
0.2
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0
-0.2
-0.6
0
10
20
30
40
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-0.8
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-0.4
50
60
25
20
15
70
80
10
5
0
-5
-10
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0
10
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60
70
80
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β
+
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=
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Modeling the data: The General Linear Model
y
x
ε
Who’s who in the GLM
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DURING
Given by
the
scanner
yi = xi β + εi
BEFORE
data model
You
build it
error
parameter
Part of the data
not accounted
by the model
NEVER
The
weight of
the model
AFTER
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Fitting the model to the data
In search of a criterion
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0
-5
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yi
Mi
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0
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-15
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-10
50
60
70
Try to minimize:
∑(yi – Mi)2
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Height
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Sum of squares
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0
-5
-10
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AM
10
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80
90
Height
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Sum of squares
-20
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Choosing a height (parameter) of 8 minimizes the “distance” between the data and
the model.
ORDINARY LEAST SQUARES (OLS) SOLUTION
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+ β2
+ β3
+
+ β3 x3
+ ε
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= β1
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A bigger (and better) model
y
=
β1 x1
+ β 2 x2
System of Linear Equations
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Now we are left with a SLE of N independent equations and p unknowns
Three possibilities:
1. N = p
Only one solution
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X = A-1b
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2. N < p
Infinite number of solutions (underdetermined system)
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3. N > p
No solution (overdetermined system)
y = Xβ + ε
We try to find
β̂
N
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2
ˆ
ε
∑ t is minimal (as small as possible)
t =1
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such that
(an estimate of the true parameters β )
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Given
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Ordinary Least Squares Estimator
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εˆ = y − Xβˆ (residuals)
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In matrix form…
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β1
β2
+
β3
y
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=
=
X
β
+
ε
General Linear Model (GLM)
p
1
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X
=
+
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N
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y
p
1
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β
N:
N:number
numberofofscans
scans
p:p:number
numberofofregressors
regressors(explanatory
(explanatoryvariables)
variables)
y = Xβ + ε
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1
N
ε
Given
y = Xβ + ε ,
we look for
y = Xβ̂
y
ˆ
β = = X −1 y = inv( X ) y
X
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If X was a number, then
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Looking for a solution…
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But X is a matrix, and it typically has no inverse (it is not square)
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y1 = x11β1+x12 β 2+…+x1p βp
y2 = x21β1+x22 β 2+…+x2p βp
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…
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yN = xN1β1+xN2 β 2+…+xNp βp
More equations (scans) than unknowns (conditions)
Overdetermined System
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Settling for a pseudo-solution…
(
pinv ( X ) = X X
T
)
−1
XT
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So, we use the pseudo-inverse (we choose the Moore-Penrose version)
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βˆ = pinv( X ) y = ( X T X ) −1 X T y
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But… this pseudo-solution is the OLS solution!
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A Geometric Perspective
OLS estimates
Projection matrix P
x2
yˆ = Xβ̂
x1
Design space defined by X
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e = Ry
R= I −P
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Residual forming
matrix R
e
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yˆ = Py
P = X ( X T X )−1 X T
y
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T
−1 T
ˆ
β = (X X ) X y
Slide from Klaas Stephan
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Basic Model
Parametric Modeling
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Linear Modulation
(habituation)
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Categorical Modeling
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Other Modulation
(e.g., performance, physio)
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Independent Blocks
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Event-related Designs
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Event-related Designs
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GLM: The L stands for Linear
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It assumes that the superposition principle holds
If
A
C
B
D
then
A+C
B+D
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GLM: And the M stands for Model
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Introducing the Temporal Derivative
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hrf alone
hrf + d(hrf)/dt
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Basis Functions
Gamma functions
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HRF, temporal derivative and dispersion
Fourier set (sines and cosines)
Basis Functions: Choosing the Right Model
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Few, model-based functions (e.g., synthetic HRF)
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All models are wrong, but some are useful
– George Box
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Good: Easy to analyze and interpret in terms of hemodynamic
activity
Bad: May fail to capture real responses that do not fit the
assumed behaviour
Many, general basis functions (e.g., Fourier set)
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Good: No a priori assumptions about the shape of the response. Can
capture unexpected responses (e.g., longer delay/duration)
Bad: Difficult to interpret physiologically. They may capture nonhemodynamic responses (noise, artifacts)
Statistical Inference
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–Why don't you say at once “it’s a miracle”?
–Because it may be only chance.
F. Dostoevsky, Crime and Punishment
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Where in the brain is my experimental parameter (β ) significantly bigger than zero?
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10
8
6
=
4
2
0
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std (βˆ )
=
C
t=
βˆ
14
-2
-4
t-map
df = N − p
p-values
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Statistical Inference
Where in the brain is one experimental parameter (β1 ) significantly bigger than
another one (β2 ) ?
β1 > β2
β1 - β2 > 0
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Hypothesis:
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1∗β1 + (-1)∗β2 = [1 -1]
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Contrast: Linear combination of parameters
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c = [1 -1]
c = [1 -1 0 0] if we included mean and linear trend in the model
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cβˆ
t=
std (cβˆ )
β1
β2
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My p-value is smaller than yours…
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If P is between .1 and .9 there is certainly no reason to suspect the hypothesis
tested. If it is below .02 it is strongly indicated that the hypothesis fails to account
for the whole of the facts. We shall not often be astray if we draw a conventional
line at .05.
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R. Fisher, Statistical Methods for Research Workers (1925)
Multiple comparisons across the brain
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There are ~100,000 voxels in the brain!!
No correction (p < 0.05 uncorrected)
Good: Easy, minimize Type II errors
Bad: Too many false positives (Type I errors) Æ 5% of 100,000 = 5,000 voxels!
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Bonferroni correction (p < 0.05/100,000 = 0.0000005)
Good: Easy, minimize Type I errors
Bad: Too stringent (worst-case scenario). Too many Type II errors
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Gaussian random field theory (spatial smoothing)
Good: Works well for spatially-correlated data. Reasonable results
Bad: Still fairly stringent. Removes some specificity (due to smoothing)
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False discovery rate (FDR)
Good: Fewer Type II errors while still controlling for Type I errors
Bad: Significance of a voxel depends on significance of other voxels
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“Pseudo-Bonferroni” correction (p < 0.001)
Good: Somewhere between 0.05 uncorrected and 0.05 corrected
Bad: Completely arbitrary
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•