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fMRI Analysis Fundamentals
With a focus on task-based analysis and SPM12
fMRI Modeling
• Modeling goal: explain as much variability as possible
• Anything that isn’t accounted for will go into “residual error”, e — want to minimize e
• Smaller residuals -> greater significance
Analysis Options/History
• Subtraction: calculate difference of “on” image minus “off”, for example [Ogawa et al. 1992]
• Correlation: test for similarity of time series to stimulus series
• General Linear Model (GLM): a generalization of the above approaches– Regression type framework– Matrix-based formulation of linear models– Review paper: Poline & Brett 2012, “The general linear
model and fMRI: does love last forever?”** http://www.ncbi.nlm.nih.gov/pubmed/22343127
GLM
• GLM encompasses ANOVA, ANCOVA, t-test• GLM in equation form:
Y = XB + ε (after demeaning Y and usually X)Y: voxel data (column/s). X: model (“design matrix”)B: coefficients (slopes) of fit lines (“betas”)ε: residual error
GLM
• SPM’s approach is “mass univariate”: one separate equation to solve for each voxel
• Essentially, we are fitting a multiple regression model at each voxel:
• We know y and all xs, want to determine βs (and error):
Example with single x:
y = β1x1 + β2x2 + … + c
y = βx + c + ε
this line is a 'model' of the data
slope β = 0.23
Intercept c = 54.5
• β: slope of line relating x to y– ‘how much of x is
needed to approximate y?’
• ε = residual error– the best estimate of β
minimises ε: deviations from line
– Assumed to be independently, identically and normally distributed (IID)
y = βx + c + ε
Source: “Idiot's guide to General Linear Model & fMRI”
Regression Example
Source: “Functional MRI data analysis” (C. Pernet)
Our constant term
Source: “Idiot's guide to General Linear Model & fMRI”
x (task)
constant term
covariates (6)
(note: now .nii)
Y
data ve
ctor
(v
oxel ti
me serie
s)
=
= X
design m
atrix
b1
b2
parameters
+
+
error v
ector
GLM Matrix View
Source: Rik Henson, “General Linear Model”
SPM12
• As with preprocessing, uses Batch Editor• Can set up with our previously preprocessed
motor data (mot_sp task)• Or, can use this copy:
/net/ms3T/sample/mot_sp/swr*
SPM Settings
• Directory: specify a new directory for results of each model
• Units for design: seconds is much easier!• Interscan interval (TR): 2• Data & Design: select sw* files (139)• We will manually enter three conditions…
Left, Right, and Rest
mot_sp Conditions
• Left– Onsets: 18 60 116 158 186 242– Duration: 10
• Right– Onsets: 32 74 102 144 200 228– Duration: 10
• Rest– Onsets: 46 88 130 172 214 256– Duration: 10
Data Adjustment: Global Effects
• fMRI BOLD signal values are dimensionless, and vary across subjects/regions
• Hypothetical case:– region 1 has baseline 2000, changed signal of 2050
(+50)– region 2 has baseline 800, changed signal of 840
(+40)• Is it better to compare signal change? (50 >
40) Or is proportional value better? (2.5% < 5%)
Global Effects
• In PET, absolute change is meaningful — and early fMRI work used PET methods
• In fMRI, proportion of change believed to be more relevant
• Note: other possibilities could be considered, e.g. z-transformation
Global Effects
• SPM terms:– Global scaling: adjust each volume (TR) to have
same mean (PET holdover, not recommended)– Grand mean scaling: scale so “grand mean” has a
particular value (100); automatic in SPM– “grand mean” (g): across all voxels and timepoints
Grand mean scaling amounts to multiplying all voxels in a session by 100/g
Why Not Global Scaling?
• Problem: true global is unknown, and volume mean may be unreliable proxy– Large signal changes over some area can confound
global with local changes– Possible consequence: artifactual deactivations
after global scaling• Other ways to account for volume-to-volume
drift (high-pass filtering)
fMRI Noise
• Scanner signal commonly “drifts” slowly over time
• Physiological fluctuations (heartbeat, breathing) add other noise
• Goal: find and remove any structured noise– Convenient to use frequency domain, especially
for periodic changes– “Linear” drift approximated by “1/f” noise (long
period)
Frequency Domain
Source: Handbook of Functional MRI Data Analysis
Source: Human Brain Function ch. “Issues in Functional Magnetic Resonance Imaging”
Removing Noise
• Nyquist Theorem: if sample rate is insufficient, samples can appear to have a lower frequency
Example: are blue dots from blue or red curve?
Without a higher sample rate, red is undersampled: we attribute to blue what might be from red
aliasing
Noise Sources
• What about physiological noise sources?– 1 Hertz = 1 cycle per second– fMRI sampling rate ≈ 0.5 Hz (~0.3-0.6 = TR 3.33-
1.67) – Nyquist limit: frequencies > ½ sample rate are
aliased (appear partially at other frequencies)
• Heart rate ≈ 1 Hz: no way • Breathing rate ≈ 0.1-0.3 Hz: no?
Filtering Noise
• Just zap all frequencies below Nyquist limit?– No: the task is a legitimate source of periodic
variation, too…– 30s alternating blocks = 1/60 Hz frequency
• “High-pass filter”: pass (leave intact) signal at frequency greater than some x (and remove slower variations)
• x = 1/128 Hz in SPM (based on typical data)• You can test your own data!
Power Spectral Density
• Using ART tool
• Note: only task, not signal, here
HPF (@ 1/128)
Implementation of HPF
• High-pass filtering options:– Directly filter the data (fit a model, subtract low
frequency trends): FSL– Use covariates for various frequencies: SPM
Source: Mumford, “First-level Statistics”, UCLA NITP 2008
Collinearity
• In regression, correlated regressors can’t be uniquely solved for, so interpretability suffers– Individually, regressors may have no significant
impact– Overall, a model may nonetheless have low error
• Not a big deal if “nuisance covariates” (such as for HPF) are correlated
• A problem if you want to assess individual covariates
Example: Task-correlated Motion
• Incidental: e.g., subject nods when responding “yes”
• Design related: e.g., if task is “press a button to get a reward when you spot a target”– When looking for “reward processing” areas, you
will get motor areas as well– Need a more careful design to distinguish motor
and reward activations here
Noise & Modeling
• “white noise”– AKA, in SPM-speak, “sphericity”– all frequencies equally represented– No problem for least-squared estimation
• “colored noise”– AKA “non-sphericity”– has structure; problems for least-squares estimation– highpass filtering helps (for low frequency noise)– “whitening”: alter the covariance matrix toward white
noise
Autocorrelation and Whitening
• Autocorrelation: in general, cross-correlation of a signal with itself (under various lags)
• In fMRI, successive timepoints are correlated• Can whiten using an autoregressive (AR) model
– AR(1): previous timepoint (+ noise) contributes to value of current timepoint
– AR(2): previous 2 timepoints (etc.)• SPM99: AR(1) with a fixed weight (correlation) of 0.2• Later SPMs: AR(1), correlation estimated in first pass
– SPM lingo: "hyperparameter estimation”– This is why appearance of design matrix in SPM changes after model
estimation
Modeling HRF (Redux)
Lindquist et al. 2009
• Recall: signal comes from the BOLD effect, and is assumed to track neural activity
• Knowing/assuming an HRF shape, we can predict BOLD response to stimuli
HRF Options
• In SPM, multiple basis functions:– canonical HRF– canonical HRF with derivatives– Finite Impulse Response (FIR)– Fourier– Gamma
• The default choice is canonical HRF, and we’ll focus on that
SPM Canonical HRF
• Difference of Gammas
Gamma Distribution (Wikipedia)
% returns a hemodynamic response function % FORMAT [hrf,p] = spm_hrf(RT,[p]); % RT - scan repeat time % p - parameters of the response function (two gamma functions) % % defaults % (seconds) % p(1) - delay of response (relative to onset) 6 % p(2) - delay of undershoot (relative to onset) 16 % p(3) - dispersion of response 1 % p(4) - dispersion of undershoot 1 % p(5) - ratio of response to undershoot 6 % p(6) - onset (seconds) 0 % p(7) - length of kernel (seconds) 32 % % hrf - hemodynamic response function
SPM’s HRF Code (spm_hrf.m)
% Copyright (C) 1996-2014 Wellcome Trust Centre for Neuroimaging
% Karl Friston% $Id: spm_hrf.m 6108 2014-07-16 15:24:06Z guillaume $
%-Parameters of the response function%--------------------------------------------------------------------------p = [6 16 1 1 6 0 32];if nargin > 1 p(1:length(P)) = P;end
%-Microtime resolution%--------------------------------------------------------------------------if nargin > 2 fMRI_T = T;else fMRI_T = spm_get_defaults('stats.fmri.t');end
%-Modelled hemodynamic response function - {mixture of Gammas}%--------------------------------------------------------------------------dt = RT/fMRI_T;u = [0:ceil(p(7)/dt)] - p(6)/dt;hrf = spm_Gpdf(u,p(1)/p(3),dt/p(3)) - spm_Gpdf(u,p(2)/p(4),dt/p(4))/p(5);hrf = hrf([0:floor(p(7)/RT)]*fMRI_T + 1);hrf = hrf'/sum(hrf);
That’s it!
>> hrf = spm_hrf(2)
hrf =
0 0.0866 0.3749 0.3849 0.2161 0.0769 0.0016 -0.0306 -0.0373 -0.0308 -0.0205 -0.0116 -0.0058 -0.0026 -0.0011 -0.0004 -0.0001
>> plot(hrf)
SPM Canonical HRF
Note: TR units! (TR = 2)
Derivatives
• Basis functions can be added together to explain fMRI time series
• SPM offers to expand the canonical HRF basis set with two additions:– Time derivative– Dispersion derivative(you can choose “none”, “time only”, or “time + dispersion”)
SPM Time Derivative
Idea: shift the HRF earlier or later
This is implemented by a “+” bulge before the HRF peak and a “–” bulge after
(this shifts the HRF earlier; to shift later, can assign a negative weight)
stimulus
SPM Dispersion Derivative
Idea: make the HRF wider or narrower
Implemented using two “–” bulges around the peak (and “+” in the center to compensate)
(this narrows the HRF; to widen, can assign a negative weight)
SPM Derivatives
• The time derivative counters misalignment of HRF onset (subject/region has faster or slower HRF, or slice timing effects)
• The dispersion derivative counters shorter/longer HRFs
• However, note that “counter” here really means “model small deviations as a nuisance covariate”…
Time Derivative Advantages
• Lindquist et al. 2009:– Even minor misspecification of the HRF can
increase Type I error (bias and loss of power)– Derivative very accurate for small shifts (< 1 s),
progressively worse as shift increases
Calhoun et al. 2004:– TD reduces error variance in first level models
• Pernet 2014:– Using TD improved model R2
Time Derivative Concerns
Della-Maggiore et al. 2002, Calhoun et al. 2004:– Not so helpful for second level (group) analysis:
random effects models ignore first level variance– “Amplitude bias”: if HRF delay varies, different
voxels experience different “adjusted” HRF amplitudes (for >1s shifts especially)
• Pernet 2014:– Presence of derivative changes parameter estimates
for canonical HRF terms, sometimes drastically
Phew…
• Now we can estimate the model we set up
• This will generate certain output files, including images for each beta
Contrasts
• Once we have solved a GLM model, we have “betas” for variables (e.g., conditions)– SPM uses marginal (Type IV) sum-of-squares: each
term is estimated after accounting for all others– Or, put another way, condition order doesn’t matter
• Motor task model:– Left, right, rest conditions – Can test for individual effects (e.g. Left > 0)– Can test for differences (E.g. Left > Right)
Task Design Considerations
• How the task is organized has many implications for signal (and modeling)
• Ultimately, everything should be guided by the experimental question (see Task Design talk)
• Main options:– Blocked design: compare blocks of time– Event-related design: aggregate responses to
individual stimuli
Blocked vs. Event-Related Designs
• Because of HRF summation, blocks have high signal but low separability
• Can potentially separate stimulus responses (event related design)
D’Esposito 2000, Seminars in Neurology
Simple, “Slow” ER Design
Source:Andysbrainblog
“Rapid” Event-Related Design
Source:Andysbrainblog
Fixed interval!
Randomized/Jittered Design
Source:Andysbrainblog
Variable interval
Power Spectrum View
• Periodic stimuli: power mainly at one freq.• Random stimuli: power spread out• Can think of the HRF “filtering” frequencies too;
only “slow” events end up having powerSource: The Clever Machine blog