First-level fmri modeling. UCLA Advanced NeuroImaging Summer School, 2010

Transcription

1 First-level fmri modeling UCLA Advanced NeuroImaging Summer School, 2010

2 Task on Goal in fmri analysis Find voxels with BOLD time series that look like this

3 Delay of BOLD response Voxel with signal Voxel no signal Voxel signal and drift

4 Voxel with signal Voxel no signal Voxel signal and drift

5 Voxel with signal Voxel no signal Voxel signal and drift Starts off high

6 BOLD issues BOLD response is delayed Convolution FIR modeling BOLD time series suffer from low frequency noise Highpass filtering Prewhitening Precoloring Scaling the data Grand mean scaling Intensity normalization

7 Single voxel time series Recall the GLM

8 How to make a good model Explain as much variability in the data as possible If you miss something it will go into the residual error, e Big residuals Big variance Small t stat

9 Understanding the data Time series drifts down in beginning BOLD response is delayed

10 Simplest Model =

11 Simplest Model =

12 Simplest Model

13 Simplest Model

14 Modeling the delay Hemodynamic response function Real data was used to find good models for the hemodynamic response Stimulus HRF (double gamma)

15 Convolution Combine HRF and expected neural response Typically model derivative of convolved HRF to adjust for small differences in onset (<1s)

16 Different HRF s Too symmetric Basic shape okay, but no post stimulus undershoot Includes post stimulus undershoot

17 Different HRF s Too symmetric Basic shape okay, but no post stimulus undershoot Includes post stimulus undershoot

18 Different HRF s Too symmetric Basic shape okay, but no post stimulus undershoot Includes post stimulus undershoot

19 Assumptions of canonical HRF BOLD increases linearly Dale & Buckner, 1997

20 Assumptions of canonical HRF The width, height and delay are correct Lindquist & Wager (2007) From what I ve seen only looks like it would work with 1 task

21 Finite impulse response model FIR Make no assumption about the shape of the HRF

22 Constrained basis set Lower the number of regressors in the model by using a basis set Constrained to shapes that are reasonable for HRF shapes

23 Constrained basis set Basis set HRF possibilities

24 FLOBS fmrib Linear Optimal Basis Sets Generates a set of basis sets to model signal Specify ranges for different portions of the hrf

25 Comparison

26 More thoughts about canonical HRF Advantages: Simpler analysis Easily interpretable outcome Simplifies group analysis Disadvantages Biased if canonical HRF is incorrect

27 Unbiased basis sets Advantages Not biased towards a particular shape Allows testing of hypotheses about specific HRF parameters Disadvantages Less powerful Makes group analysis more difficult Tend to overfit the data (i.e., fit noise)

28 We can make a design matrix! Start with task blocks or delta functions Convolve Estimate the GLM and carry out hypothesis!

29 Convolved Boxcar

30 Convolved Boxcar

31 The Noise White noise All frequencies have similar power Not a problem for OLS

32 Colored noise Has structure OLS needs help! More Noise

33 What about the drift? Sources Head motion Cardiac noise Respiratory noise Scanner noise

34 What the noise looks like 1/f structure Power spectra of noise data (Zarahn, Aguirre, D Esposito, NI, 1997)

35 More noise Average spectrum of principal components (Mitra & Pesaran, Biophysical Journal, 1999)

36 More noise Low frequency noise Average spectrum of principal components (Mitra & Pesaran, Biophysical Journal, 1999)

37 More noise Breathing Average spectrum of principal components (Mitra & Pesaran, Biophysical Journal, 1999)

38 More noise Cardiac Average spectrum of principal components (Mitra & Pesaran, Biophysical Journal, 1999)

39 The 1-2 punch Punch 1: Highpass filtering FSL uses gaussian weighted running line smoother SPM fits a DCT basis set Punch 2: Prewhitening We ll get to that later There s also a thing called lowpass filtering (precoloring), but generally it isn t so great and nobody uses it

40 Highpass filtering Simply hack off the low frequency noise SPM: Adds a discrete cosine transform basis set to design matrix

41 Highpass filtering FSL: Gaussian-weighted running line smoother Step 1: Fit a Gaussian weighted running line

42 Highpass filtering FSL: Gaussian-weighted running line smoother Step 1: Fit a Gaussian weighted running line Fit at time t is a weighted average of data around t

43 Highpass filtering Step 2: Subtract Gaussian weighted running line fit IMPORTANT: Must apply filter to both the data and the design. FSL has apply temporal filter box in design setup. Leave it checked!

44 Highpass filtered design (FSL) If it wasn t filtered, this trend wouldn t be here.

45 Filter below.01 Hz Highpass filtering

46 Filter cutoff High, but not higher than paradigm frequency Look at power spectrum of your design and base cutoff on that Block design: Longer than 1 task cycle usually twice the task cycle Event related design: Larger than 66 s (based on the power spectrum of a canonical HRF of a single response)

47 High-pass Filtering Removes the worst of the low frequency trends High-pass From S. Smith

48 Highpass filtering What does it do to the signal?? Signal Power spectrum Lowpass filter Highpass filter Woolrich et al, NI 2001

49 Highpass filtering What does it do to the signal?? Signal Power spectrum Lowpass filter Highpass filter

50 Highpass filtering What does it do to the signal?? Signal Power spectrum Lowpass filter Highpass filter

51 Filtering conclusions Lowpass filtering Idea is to swamp out high frequency noise Easily removes important signal in ER designs Choose cutoff to remove noise, but avoid your signal Highpass filtering Removes low frequency drift We typically avoid designs with low frequencies, so highpass filtering is always used

52 Bandpass filtering High and lowpass filtering Common in functional connectivity analysis since it allows you to focus on a specific frequency Not typically used in standard fmri analyses

53 Model with HP filter Parameter of interest =

54 Model with HP filter Parameter of interest = Use contrast c=( )

55 Convolution & HP filter

56 Punch 2: Prewhitening Highpass filtering Analogous to using a roller to paint a wall you can t get the edges very neatly Prewhitening More precise estimate of correlation like using a brush for the edges

57 Prewhitening Remember Gauss Markov? If our errors are distributed with mean 0, constant variance and not temporally autocorrelated then our estimates are unbiased and have the smallest variance of all unbiased estimators. Uh oh, (even after highpass filtering)

58 Prewhitening Find K such that Premultiply GLM by K

59 Prewhitening Find K such that Premultiply GLM by K Awesome! G-M holds for our *new* model

60 Whitening OLS can be used on whitened model

61 Prewhitening Step 1: Fit the linear regression ignoring temporal autocorrelation Step 2: Use residuals from first regression to estimate temporal autocorrelation to obtain K Step 3: Create prewhitened model and estimate in usual way

62 Estimating V We don t know V, so we estimate it There s a bias problem. SPM uses a global covariance estimate to help with this FSL uses a local estimate, but smoothes it

63 fmri noise Tends to follow 1/f trend Autoregressive (AR) models fit it well

64 Whitening FSL Estimates V locally Step 1: Estimate raw autocorrelations

65 Whitening FSL Estimates V locally Step 1: Estimate raw autocorrelations Lag 1 [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] Take products and average

66 Whitening FSL Estimates V locally Step 1: Estimate raw autocorrelations Lag 2 [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] Take products and average

67 Whitening FSL Estimates V locally Step 1: Estimate raw autocorrelations Lag 7 [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] [e 1 e 2 e 3 e 4 e 5 e 6 e 7 ] Take products and average

68 Whitening FSL Estimates V locally Step 1: Estimate raw autocorrelations Step 2: Smooth spatially Step 3: Within voxel, smooth correlation estimates (Tukey taper) Correlation estimates at high lags aren t estimated well, so they are down-weighted

69 Whitening FSL Time Domain Spectral Domain Raw estimate Woolrich et al., 2001, NI

70 Whitening FSL Time Domain Spectral Domain Tukey Taper Woolrich et al., 2001, NI

71 Whitening SPM Globally estimates correlation Correlation of time series averaged over voxels Structured correlation estimate Scaled AR(1) with correlation 0.2 plus white noise

72 Whitening SPM Only 2 parameters are estimated

73 Comparisons Bandpass (solid), Prewhitening without bias correction (dot-dash), highpass (dashed) PW HP BP BP PW HP Friston, NI 2000

74 Summary of filters Although bandpass (and lowpass) has the best looking bias, it is less efficient Highpass filtering doesn t remove all of the structured noise Prewhitening with bias correction with highpass filtering is the traditional combination

75 Convolution, HP filter, Whitening

76

77 Scaling Grand Mean Scaling Removes intersession variance in global signal due to changes in gain of scanner amplification Allows us to combine data across subjects Whole 4D data set is scaled by a single number Automatically done in software packages Doesn t change variability between time points

78 Scaling Proportional scaling Forces each volume of 4D dataset to have the same mean Also done by modeling the global signal Idea is to remove background activity Problems can occur if true activation is wide spread Negative activations may result

79 Intensity Normalizaton without with Junghofer et al, 2004, NI Signal is lost and negative activation artifacts

80 Other modeling considerations Adding the derivative of the HRF Adding motion parameters to the model

81 Model HRF & Derivative

82 Shifted HRF 1.2 Blue line is sum of HRF and its derivative

83 Temporal derivative We model the derivative, but don t study inferences of it Linquist, et al (NI, 2008) suggest this is a bad idea may lead to bias Generally I d say we don t worry about the canonical HRF too much

84 Collinearity When designing your study, you want your tasks to be uncorrelated Correlation between regressors lowers the efficiency of the parameter estimation Parameter estimates are highly variable Can even flip signs

85 Why is it a problem?

86 Why is it a problem? There are an infinite # of solutions for and etc

87 Collinearity illustration X 1 Cor= -0.2 X 2 X 1 Cor=0.96 X 2

88 Intercept X 1 X 2 Highly variable over experiments Correlated Regressors Inflated for correlated case (green) T statistic Bias can go in either direction

89 Intercept X 1 X 2 Highly variable over experiments Correlated Regressors Inflated for correlated case (green) T statistic Bias can go in either direction

90 Intercept X 1 X 2 Highly variable over experiments Correlated Regressors Inflated for correlated case (green) T statistic Bias can go in either direction

91 Residuals don t change The designs explain the same amount of variability

92 Collinearity You can t fix it after the data have been collected You can t tell from the t statistic if you had collinearity FSL has some diagnostics Absolute value of correlation Bad=white off diagonal Eigenvalues from SVD Bad=near 0

93 You are now first level modeling experts! You know why we convolve, highpass filter and prewhiten If you don t trust the canonical HRF there are other options using basis functions Precoloring (lowpass filtering) typically isn t used because it removes signal

94 Other things you learned Prewhitening was orginally viewed as too biased, but methods have been developed to alleviate this problem Grand mean scaling is necessary in group analyses Proportional scaling is generally frowned upon (although popular in resting state functional connectivity studies)

95 Last things you learned Add derivative as well as motion parameters (unconvolved) to soak up extra variance Check for collinearity! You should think about it before you collect your data.