Detecting Sparse Connectivity: MS Lesions, Cortical Thickness, and the ‘Bubbles’ Task in an fMRI Experiment Keith Worsley, Nicholas Chamandy, McGill Jonathan Taylor, Stanford and Université de Montréal Robert Adler, Technion Philippe Schyns, Fraser Smith, Glasgow Frédéric Gosselin, Université de Montréal Arnaud Charil, Alan Evans, Montreal Neurological Institute

Three examples of spatial point data Multiple Sclerosis (MS) lesions Galaxies ‘Bubbles’ I hope to show the ‘connections’ …

Astrophysics 우리의 은하

Sloan Digital Sky Survey, data release 6, Aug. ‘07

What is ‘bubbles’?

Nature (2005)

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

… 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

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

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

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

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

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

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

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

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

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

E.g. Fearful (3000/4=750 trials): 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)

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

Very similar to the proportion of correct bubbles: 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

Happy Sad Fearful Neutral 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

Euler Characteristic Heuristic Euler characteristic (EC) = #blobs - #holes (in 2D) Excursion set Xt = {s: Z(s) ≥ t}, e.g. for neutral face: EC = 0 0 -7 -11 13 14 9 1 0 30 Heuristic: At high thresholds t, the holes disappear, EC ~ 1 or 0, E(EC) ~ P(max Z ≥ t). Observed Expected 20 10 EC(Xt) -10 Exact expression for E(EC) for all thresholds, E(EC) ~ P(max Z ≥ t) is extremely accurate. -20 -4 -3 -2 -1 1 2 3 4 Threshold, t

EC densities of Z above t The result I f Z ( s ) » N ; 1 i a n o t r p c G u d m ¯ e l , 2 < w h V ¡ @ ¢ = ¸ £ P µ x S ¶ ¼ E C \ : g z + A 3 v F W H M 4 ½ ( Z ¸ t ) 1 2 L ( S ) 1 2 Lipschitz-Killing curvatures of S (=Resels(S)×c) EC densities of Z above t Z(s) white noise filter = * FWHM

Results, corrected for search Random field theory threshold: Z=3.92 (P=0.05) 3.82 3.80 3.81 3.80 Saddle-point approx (2007): Z=↑ (P=0.05) Bonferroni: Z=4.87 (P=0.05) – nothing Z~N(0,1) statistic Average face Happy Sad Fearful Neutral

Scale space: smooth Z(s) with range of filter widths w = continuous wavelet transform adds an extra dimension to the random field: Z(s,w) Scale space, no signal 34 8 22.7 6 15.2 4 2 10.2 -2 6.8 -60 -40 -20 20 40 60 w = FWHM (mm, on log scale) One 15mm signal 34 8 22.7 6 15.2 4 2 10.2 -2 6.8 -60 -40 -20 20 40 60 Z(s,w) s (mm) 15mm signal is best detected with a 15mm smoothing filter

Matched Filter Theorem (= Gauss-Markov Theorem): “to best detect signal + white noise, filter should match signal” 10mm and 23mm signals 34 8 22.7 6 4 15.2 2 10.2 -2 6.8 -60 -40 -20 20 40 60 w = FWHM (mm, on log scale) Two 10mm signals 20mm apart 34 8 22.7 6 15.2 4 2 10.2 -2 6.8 -60 -40 -20 20 40 60 Z(s,w) s (mm) But if the signals are too close together they are detected as a single signal half way between them

Scale space can even separate two signals at the same location! 8mm and 150mm signals at the same location 10 5 -60 -40 -20 20 40 60 170 20 76 15 w = FWHM (mm, on log scale) 34 10 15.2 5 6.8 -60 -40 -20 20 40 60 Z(s,w) s (mm)

EC densities of Z above t The result I f Z ( s ) » N ; 1 i a n o t r p c G u d m ¯ e l , 2 < w h V ¡ @ ¢ = ¸ £ P µ x S ¶ ¼ E C \ : g z + A 3 v F W H M 4 ½ ( Z ¸ t ) 1 2 L ( S ) 1 2 Lipschitz-Killing curvatures of S (=Resels(S)×c) EC densities of Z above t Z(s) white noise filter = * FWHM

Random field theory for scale-space g b t w L z K u v ( ) ? ; 1 2 ¸ d x m y = + ¡ 4 ¼ 3

Rotation space: Try all rotated elliptical filters Unsmoothed fMRI data: T stat for visual stimulus Threshold Z=5.25 (P=0.05) Maximum filter

EC densities of Z above t The result I f Z ( s ) » N ; 1 i a n o t r p c G u d m ¯ e l , 2 < w h ¸ £ = V ¡ @ ¢ P µ x S ¶ ¼ E C \ : g z + A 3 v F W H M 4 ½ ( Z ¸ t ) 1 2 L ( S ) 1 2 Lipschitz-Killing curvatures of S (=Resels(S)×c) EC densities of Z above t Z(s) white noise filter = * FWHM

Theorem (1981, 1995) L e t T ( s ) , 2 S ½ < b a m o h i r p c n d ¯ l . X = f : ¸ g x u E C \ N w Z - y G 1 ; » V @ I £ R z j Example: the chi-bar random field, a special case of a random field of test statistics for the magnitude of an fMRI response in the presence of unknown delay of the hemodynamic response function.

Example: chi-bar random field ¹ Â = m a x · µ ¼ 2 Z 1 c o s + i n Z1~N(0,1) Z2~N(0,1) s2 s1 Excursion sets, Rejection regions, X t = f s : ¹ Â ¸ g R t = f Z : ¹ Â ¸ g Threshold t Z2 Search Region, S Z1

Beautiful symmetry: L i p s c h t z - K l n g u r v a e ( S ) E C d e \ X t ) = D d L ½ R Adler & Taylor (2007), Ann. Math, (submitted) Beautiful symmetry: L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ½ ( R ) Steiner-Weyl Tube Formula (1930) Taylor Gaussian Tube Formula (2003) Put a tube of radius r about the search region λS and rejection region Rt: ¸ = S d µ @ Z s ¶ Z2~N(0,1) Rt r Tube(λS,r) Tube(Rt,r) r λS Z1~N(0,1) t-r t Find volume or probability, expand as a power series in r, pull off coefficients: j T u b e ( ¸ S ; r ) = D X d ¼ ¡ 2 + 1 L P ( T u b e R t ; r ) = 1 X d 2 ¼ ! ½

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

Thresholding? Cross correlation random field Correlation between 2 fields at 2 different locations, searched over all pairs of locations, one in S, one in T: Bubbles data: P=0.05, n=3000, c=0.113, T=6.22 P µ m a x s 2 S ; t T C ( ) ¸ c ¶ ¼ E f : g = d i X j L ½ n ¡ h 1 ! + b k l Cao & Worsley, Annals of Applied Probability (1999)

MS lesions and cortical thickness Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex Data: n = 425 mild MS patients 5.5 5 4.5 4 Average cortical thickness (mm) 3.5 3 2.5 Correlation = -0.568, T = -14.20 (423 df) 2 Charil et al, NeuroImage (2007) 1.5 10 20 30 40 50 60 70 80 Total lesion volume (cc)

MS lesions and cortical thickness at all pairs of points Dominated by total lesions and average cortical thickness, so remove these effects as follows: CT = cortical thickness, smoothed 20mm ACT = average cortical thickness LD = lesion density, smoothed 10mm TLV = total lesion volume Find partial correlation(LD, CT-ACT) removing TLV via linear model: CT-ACT ~ 1 + TLV + LD test for LD Repeat for all voxels in 3D, nodes in 2D ~1 billion correlations, so thresholding essential! Look for high negative correlations … Threshold: P=0.05, c=0.300, T=6.48

Cluster extent rather than peak height (Friston, 1994) Choose a lower level, e.g. t=3.11 (P=0.001) Find clusters i.e. connected components of excursion set Measure cluster extent by resels Distribution: fit a quadratic to the peak: Distribution of maximum cluster extent: Find distribution for a single cluster Bonferroni on N = #clusters ~ E(EC). Z D=1 extent t Peak height s Cao and Worsley, Advances in Applied Probability (1999)

Three examples of spatial point data Multiple Sclerosis (MS) lesions Galaxies ‘Bubbles’ I hope I have shown the connections ... 너를 감사하십시요 !