Basics of fMRI Time-Series Analysis
Douglas N. Greve
fMRI Analysis Overview
Subject 1
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
Subject 2
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
C
First Level
GLM Analysis
X
C
Higher Level GLM
Subject 3
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
X
C
C
Subject 4
Preprocessing
MC, STC, B0
Smoothing
Normalization
Raw Data
First Level
GLM Analysis
X
C
2
Overview
•
•
•
•
•
Neuroanatomy 101
fMRI Contrast Mechanism
Hemodynamic Response
“Univariate” GLM Analysis
Hypothesis Testing
3
Neuroantomy
• Gray matter
• White matter
• Cerebrospinal Fluid
4
Functional Anatomy/Brain Mapping
5
Visual Activation Paradigm
Flickering
Checkerboard
Visual, Auditory, Motor, Tactile, Pain, Perceptual,
Recognition, Memory, Emotion, Reward/Punishment,
Olfactory, Taste, Gastral, Gambling, Economic, Acupuncture,
Meditation, The Pepsi Challenge, …
• Scientific
• Clinical
• Pharmaceutical
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MRI Scanner
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Magnetic Resonance Imaging
T1-weighted
Contrast
BOLD-weighted
Contrast
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Blood Oxygen Level Dependence (BOLD)
Oxygenated
Hemoglobin
(DiaMagnetic)
Neurons
Deoxygenated
Hemoglobin
(ParaMagnetic)
Lungs
Contrast Agent
Oxygen
CO2
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Functional MRI (fMRI)
Stimulus
Localized
Neural
Firing
Localized
Increased
Blood Flow
Localized
BOLD
Changes
Sample BOLD response in 4D
Space (3D) – voxels (64x64x35, 3x3x5mm^3, ~50,000)
Time (1D) – time points (100, 2 sec) – Movie
Time 1
Time 2
Time 3 …
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4D Volume
64x64x35
85x1
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Visual/Auditory/Motor Activation Paradigm
15 sec ‘ON’, 15 sec ‘OFF’
• Flickering Checkerboard
• Auditory Tone
• Finger Tapping
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Block Design: 15s Off, 15s On
Voxel 1
Voxel 2
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Contrasts and Inference

OFF
2
OFF
ON
2
 ON
Voxel 1
t
 ON   OFF
2
2
( N ON  1) ON
 ( N OFF  1) OFF
( N ON  N OFF  2) 2
Voxel 2
Contrast ON  OFF
2
2
( NON  1) ON
 ( NOFF  1) OFF
Var(Contrast) 
( NON  NOFF  2) 2
2
 ON ,  ON
, N ON  Mean,Var, N in ON
2
 OFF ,  OFF
, N OFF  Mean,Var, N in OFF
p = 10-11, sig=-log10(p) =11
p = .10, sig=-log10(p) =1
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Statistical Parametric Map (SPM)
+3%
0%
-3%
Contrast Amplitude
ON-OFF
“Massive Univariate Analysis”
-- Analyze each voxel separately
Contrast
Amplitude
Variance
(Error Bars)
Significance
t-Map (p,z,F)
(Thresholded
p<.01)
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Statistical Parametric Map (SPM)
Signficance Map
sig=-log10(p)
Signed by contrast
“Massive Univariate Analysis”
-- Analyze each voxel separately
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Hemodynamics
• Delay
• Dispersion
• Grouping by simple time point inaccurate
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Hemodynamic Response Function (HRF)
Time-to-Peak (~6sec)
Dispersion
TR (~2sec)
Equilibrium
(~16-32sec)
Undershoot
Delay (~1-2sec)
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Convolution with HRF
• Shifts, rolls off; more accurate
• Loose ability to simply group time points
• More complicated analysis
• General Linear Model (GLM)
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GLM
Data fom
one voxel
= Task
Task
Baseline Offset
(Nuisance)
+ base
base=off
Task=onoff
•Implicit Contrast
•HRF Amplitude
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Matrix Model
y
=
X
*

Observations
Task
base
=
Vector of
Regression
Coefficients
(“Betas”)
Data from
one voxel
Design Matrix
Regressors
Design Matrix
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Two Task Conditions
Observations
y
=
X
*

Odd
Even
base
=
Design Matrix
Data from
one voxel
Design Matrix
Regressors
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Working Memory Task (fBIRN)
“Scrambled”
16s
Encode
Distractor
Probe
Stick Figs
Images
Stick Figs
16s
16s
16s
0. “Scrambled” – low-level baseline, no response
1. Encode – series of passively viewed stick figures
Distractor – respond if there is a face
2. Emotional
3. Neutral
Probe – series of two stick figures (forced choice)
4. Following Emotional Distractor
5. Following Neutral Distractor
fBIRN: Functional Biomedical Research Network (www.nbirn.net)
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Five Task Conditions
y
=
=
X
*

Encode
EmotDist
NeutDist
EmotProbe
NeutProbe
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y=X*
=
GLM Solution
Encode
EmotDist
NeutDist
EmotProbe
NeutProbe
• Set of simultaneous equations
• Each row of X is an equation
• Each column of X is an unknown
• s are unknown
• 142 Time Points (Equations)
• 5 unknowns
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Estimation of s
y  X  n, y  s  n, n ~ N (0,  n2 )
Unknowns:  and  n2
ˆ  ( X T X ) 1 X T y P aramet erEst imat es
sˆ  Xˆ
nˆ  y  sˆ
T
ˆ
n
nˆ
2
ˆ n 
DOF
Unbiased :
Signal Est imat e
Residual (Noise Est imat e)
Residual Variance
E ( ˆ )  True , E (ˆ n2 )   n2
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Estimates of the HRF Amplitude
=
Encode
EmotDist
NeutDist
EmotProbe
NeutProbe
y  X  n, ˆ  ( X T X ) 1 X T y
ˆEncode  Hemodynamic amplit udein responset o Encode
ˆ
 Hemodynami
c amplit udein responset o Emot ionalDist ract or
EmotDist
ˆNeutDist  Hemodynamic amplit udein responset o Neut ralDist ract or
ˆEmotProbe  Hemodynamic amplit udein responset o P robefollowingEmot ionalDist ract or
ˆNeuttProbe  Hemodynamic amplit udein responset o P robefollowingNeut ralDist ract or
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Hypotheses and Contrasts
Which voxels respond more/less/differently to the Emotional
Distractor than to the Neutral Distractor?
ˆEDist  ˆNDist
ˆEDist
ˆEDist   NDist
ˆNDist
ˆEDist  ˆNDist
  ˆEDist  ˆNDist  0,  0,  0
Contrast: Assign Weights to each Beta
  c Encode ˆEncode  c EDist ˆEDist  c NDist ˆNDist  c EProbe ˆEProbe  c NProbe ˆNProbe
c Encode  0
c EDist  1
c NDist  1
c EProbe  0
c NProbe  0
C  0
1
1
0
0 Cont rastMat irx
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Hypotheses
• Which voxels respond more to the Emotional Distractor
than to the Neutral Distractor?
• Which voxels respond to Encode (relative to baseline)?
• Which voxels respond to the Emotional Distractor?
• Which voxels respond to either Distractor?
• Which voxels respond more to the Probe following the
Emotional Distractor than to the Probe following the
Neutral Distractor?
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1. Encode
2. EDist
3. NDist
4. EProbe
5. NProbe
Which voxels respond more to the Emotional Distractor than to
the Neutral Distractor?
• Only interested in Emotional and Neutral Distractors
• No statement about other conditions
Condition: 1 2 3 4 5
Weight
0 +1 -1 0 0
Contrast Matrix
C = [0 +1 -1 0 0]
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1. Encode
2. EDist
3. NDist
4. EProbe
5. NProbe
Which voxels respond to Encode (relative to baseline)?
Only interested in Encode
• No statement about other conditions
Condition: 1 2 3 4 5
Weight
+1 0 0 0 0
Contrast Matrix C = [+1 0 0 0 0]
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1. Encode
2. EDist
3. NDist
4. EProbe
5. NProbe
Which voxels respond to the Emotional Distractor (wrt baseline)?
• Only interested in Emotional Distractor
• No statement about other conditions
Condition: 1 2 3 4 5
Weight
0 +1 0 0 0
Contrast Matrix
C = [0 +1 0 0 0]
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1. Encode
2. EDist
3. NDist
4. EProbe
5. NProbe
Which voxels respond to either Distractor (wrt baseline)?
• Only interested in Distractors
• Average of two Distractors
• No statement about other conditions
Condition: 1
Weight
0
2 3 4 5
½ ½ 0 0
Contrast Matrix
C = [0 0.5 0.5 0 0]
• Don’t have so sum to 1, but usual
• Could have used an F-test instead of average
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1. Encode
2. EDist
3. NDist
4. EProbe
5. NProbe
Which voxels respond more to the Probe following the Emotional
Distractor than to the Probe following the Neutral Distractor?
• Only interested in Probes
• No statement about other conditions
Condition: 1 2 3 4 5
Weight
0 0 0 +1 -1
Contrast Matrix
C = [0 0 0 +1 -1]
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Contrasts and the Full Model
y  X  n, y  s  n, n ~ N (0,  n2 )
ˆ  ( X T X ) 1 X T y P aramet erEst imat es
T
ˆ
n
nˆ
2
ˆ n 
DOF
ˆ  Cˆ
Residual Variance,
nˆ  y  Xˆ
Cont rast


1
ˆ
ˆ      C ( X T X ) 1 C T ˆ n2 Cont rastVariance Est imat e
J
J  Rows in C
ˆ
Cˆ
t DOF 

t - T est (univariate)
ˆ 
C ( X T X ) 1 C T ˆ n2
2

ˆ 1ˆ
FDOF, J  ˆ T 


F - T est (mult ivariat e)
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fMRI Analysis Overview
Subject 1
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
Subject 2
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
C
First Level
GLM Analysis
X
C
Higher Level GLM
Subject 3
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
X
C
C
Subject 4
Preprocessing
MC, STC, B0
Smoothing
Normalization
Raw Data
First Level
GLM Analysis
X
C
36
Time Series Analysis Summary
• Correlational
• Design Matrix (HRF shape)
• Estimate HRF amplitude (Parameters)
• Contrasts to test hypotheses
• Results at each voxel:
• Contrast Value
• Contrast Value Variance
• p-value (Volume of Activation)
• Pass Contrast Value and Variance up to higher
level analyses
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