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Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological Institute, McGill

Detecting connectivity between images: MS lesions, cortical thickness, and the 'bubbles' task in an fMRI experiment. Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological Institute, McGill Philippe Schyns, Fraser Smith, Psychology, Glasgow Jonathan Taylor ,

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Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological Institute, McGill

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  1. Detecting connectivity between images: MS lesions, cortical thickness, and the 'bubbles' task in an fMRI experiment Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological Institute,McGill Philippe Schyns, Fraser Smith, Psychology,Glasgow Jonathan Taylor, Stanford and Université de Montréal

  2. What is ‘bubbles’?

  3. Nature (2005)

  4. Subject is shown one of 40 faces chosen at random … Happy Sad Fearful Neutral

  5. … but face is only revealed through random ‘bubbles’ • First trial: “Sad” expression • Subject is asked the expression: “Neutral” • Response: Incorrect 75 random bubble centres Smoothed by a Gaussian ‘bubble’ What the subject sees Sad

  6. Your turn … • Trial 2 Subject response: “Fearful” CORRECT

  7. Your turn … • Trial 3 Subject response: “Happy” INCORRECT (Fearful)

  8. Your turn … • Trial 4 Subject response: “Happy” CORRECT

  9. Your turn … • Trial 5 Subject response: “Fearful” CORRECT

  10. Your turn … • Trial 6 Subject response: “Sad” CORRECT

  11. Your turn … • Trial 7 Subject response: “Happy” CORRECT

  12. Your turn … • Trial 8 Subject response: “Neutral” CORRECT

  13. Your turn … • Trial 9 Subject response: “Happy” CORRECT

  14. Your turn … • Trial 3000 Subject response: “Happy” INCORRECT (Fearful)

  15. Bubbles analysis • E.g. Fearful (3000/4=750 trials): Trial 1 + 2 + 3 + 4 + 5 + 6 + 7 + … + 750 = Sum Correct trials Thresholded at proportion of correct trials=0.68, scaled to [0,1] Use this as a bubble mask Proportion of correct bubbles =(sum correct bubbles) /(sum all bubbles)

  16. Results • Mask average face • But are these features real or just noise? • Need statistics … Happy Sad Fearful Neutral

  17. Statistical analysis • Correlate bubbles with response (correct = 1, incorrect = 0), separately for each expression • Equivalent to 2-sample Z-statistic for correct vs. incorrect bubbles, e.g. Fearful: • Very similar to the proportion of correct bubbles: Z~N(0,1) statistic Trial 1 2 3 4 5 6 7 … 750 Response 0 1 1 0 1 1 1 … 1

  18. Comparison • Both depend on average correct bubbles, rest is ~ constant • Z=(Average correct bubbles • average incorrect bubbles) • / pooled sd Proportion correct bubbles = Average correct bubbles / (average all bubbles * 4)

  19. Results • Thresholded at Z=1.64 (P=0.05) • Multiple comparisons correction? • Need random field theory … Z~N(0,1) statistic Average face Happy Sad Fearful Neutral

  20. Euler Characteristic = #blobs - #holes Excursion set {Z > threshold} for neutral face EC = 0 0 -7 -11 13 14 9 1 0 Heuristic: At high thresholds t, the holes disappear, EC ~ 1 or 0, E(EC) ~ P(max Z > t). • Exact expression for E(EC) for all thresholds, • E(EC) ~ P(max Z > t) is extremely accurate.

  21. The details …

  22. 2 Tube(S,r) r S

  23. 3

  24. B A

  25. 6 Λ is big TubeΛ(S,r) S r Λ is small

  26. 2 ν U(s1) s1 S Tube S Tube s2 s3 U(s3) U(s2)

  27. Z2 R r Tube(R,r) Z1 N2(0,I)

  28. Tube(R,r) R z t-r t z1 Tube(R,r) r R R z2 z3

  29. Summary

  30. Random field theory results • For searching in D (=2) dimensions, P-value of max Z is (Adler, 1981; W, 1995): • P(max Z > z) • ~ E( Euler characteristic of thresholded set ) • = Resels× Euler characteristic density (+ boundary) • Resels (=Lipschitz-Killing curvature/c) is • Image area / (bubble FWHM)2 = 146.2 • Euler characteristic density(×c) is • (4 log(2))D/2zD-1 exp(-z2/2) / (2π)(D+1)/2 • See forthcoming book Adler, Taylor (2007)

  31. Results, corrected for search • Thresholded at Z=3.92 (P=0.05) Z~N(0,1) statistic Average face Happy Sad Fearful Neutral

  32. Bubbles task in fMRI scanner • Correlate bubbles with BOLD at every voxel: • Calculate Z for each pair (bubble pixel, fMRI voxel) – a 5D “image” of Z statistics … Trial 1 2 3 4 5 6 7 … 3000 fMRI

  33. Discussion: thresholding • Thresholding in advance is vital, since we cannot store all the ~1 billion 5D Z values • Resels=(image resels = 146.2) × (fMRI resels = 1057.2) • for P=0.05, threshold is Z = 6.22 (approx) • The threshold based on Gaussian RFT can be improved using new non-Gaussian RFT based on saddle-point approximations (Chamandy et al., 2006) • Model the bubbles as a smoothed Poisson point process • The improved thresholds are slightly lower, so more activation is detected • Only keep 5D local maxima • Z(pixel, voxel) > Z(pixel, 6 neighbours of voxel) > Z(4 neighbours of pixel, voxel)

  34. Discussion: modeling • The random response is Y=1 (correct) or 0 (incorrect), or Y=fMRI • The regressors are Xj=bubble mask at pixel j, j=1 … 240x380=91200 (!) • Logistic regression or ordinary regression: • logit(E(Y)) or E(Y) = b0+X1b1+…+X91200b91200 • But there are only n=3000 observations (trials) … • Instead, since regressors are independent, fit them one at a time: • logit(E(Y)) or E(Y) = b0+Xjbj • However the regressors (bubbles) are random with a simple known distribution, so turn the problem around and condition on Y: • E(Xj) = c0+Ycj • Equivalent to conditional logistic regression (Cox, 1962) which gives exact inference for b1 conditional on sufficient statistics for b0 • Cox also suggested using saddle-point approximations to improve accuracy of inference … • Interactions? logit(E(Y)) or E(Y)=b0+X1b1+…+X91200b91200+X1X2b1,2+ …

  35. MS lesions and cortical thickness • Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex • Data: n = 425 mild MS patients • Lesion density, smoothed 10mm • Cortical thickness, smoothed 20mm • Find connectivity i.e. find voxels in 3D, nodes in 2D with high • correlation(lesion density, cortical thickness) • Look for high negative correlations …

  36. 5.5 5 4.5 4 3.5 3 2.5 2 1.5 0 10 20 30 40 50 60 70 80 n=425 subjects, correlation = -0.568 Average cortical thickness Average lesion volume

  37. Thresholding? Cross correlation random field • Correlation between 2 fields at 2 different locations, searched over all pairs of locations • one in R (D dimensions), one in S (E dimensions) • sample size n • MS lesion data: P=0.05, c=0.325 Cao & Worsley, Annals of Applied Probability (1999)

  38. Normalization • LD=lesion density, CT=cortical thickness • Simple correlation: • Cor( LD, CT ) • Subtracting global mean thickness: • Cor( LD, CT – avsurf(CT) ) • And removing overall lesion effect: • Cor( LD – avWM(LD), CT – avsurf(CT) )

  39. Histogram 5 5 Same hemisphere Different hemisphere x 10 x 10 0.1 0.1 2.5 2.5 0 0 2 2 -0.1 -0.1 1.5 1.5 correlation -0.2 -0.2 correlation 1 1 -0.3 -0.3 0.5 0.5 -0.4 -0.4 -0.5 0 -0.5 0 0 50 100 150 0 50 100 150 0.1 1 0.1 1 0 0 0.8 0.8 -0.1 -0.1 0.6 0.6 correlation correlation -0.2 -0.2 0.4 0.4 -0.3 -0.3 0.2 0.2 -0.4 -0.4 -0.5 0 -0.5 0 0 50 100 150 0 50 100 150 distance (mm) distance (mm) threshold threshold ‘Conditional’ histogram: scaled to same max at each distance threshold threshold

  40. Science (2004)

  41. fMRI activation detected by correlation between subjects at the same voxel The average nonselective time course across all activated regions obtained during the first 10 min of the movie for all five subjects. Red line represents the across subject average time course. There is a striking degree of synchronization among different individuals watching the same movie. Voxel-by-voxel intersubject correlation between the source subject (ZO) and the target subject (SN). Correlation maps are shown on unfolded left and right hemispheres (LH and RH, respectively). Color indicates the significance level of the intersubject correlation in each voxel. Black dotted lines denote borders of retinotopic visual areas V1, V2, V3, VP, V3A, V4/V8, and estimated border of auditory cortex (A1).The face-, object-, and building-related borders (red, blue, and green rings, respectively) are also superimposed on the map. Note the substantial extent of intersubject correlations and the extension of the correlations beyond visual and auditory cortices.

  42. What are the subjects watching during high activation? Faces …

  43. … buildings …

  44. … hands

  45. Thresholding? Homologous correlation random field • Correlation between 2 equally smooth fields at the same location, searched over all locations in R (in D dimensions) • P-values are larger than for the usual correlation field (correlation between a field and a scalar) • E.g. resels=1000, df=100, threshold=5, usual P=0.051, homologous P=0.139 Cao & Worsley, Annals of Applied Probability (1999)

  46. Detecting Connectivity between Images: the 'Bubbles' Task in fMRI Keith Worsley, McGill Phillipe Schyns, Fraser Smith, Glasgow

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