Analytical Techniques Hypothesis Driven Data Driven Principal Component Analysis (PCA) Independent Component Analysis (ICA) Fuzzy Clustering Others Structural equation modeling

Matrix Notation of fMRI Data 1 voxel BOLD signal t=1 t=2 t=3 t=4 . . . Voxels X time Data Matrix Slice 1

Calculating level of Significance significance: ~ t statistic i/i  X G +  = fMRI Data + = Variability explained by the model total variability noise

SPM Nomenclature for Design Matrix G (interesting) H (non-interesting) Covariate Indicator variable G1 Gc Hc H1 Global activity Linear trends E.g. dose of drug subject Design matrix G

Some General Linear Model (GLM) Assumptions: Design matrix known without error the design matrix is the same everywhere in the brain the ’s follow a Gaussian distribution the residuals are well modeled by Gaussian noise the voxels are temporally aligned each time point is independent of the others (time courses of voxels are white) each voxel is independent of the others

Inclusion of Global Signal in Regression Hypothesis Regression Coefficients  < 0!!! Global signal Hypothesis Test voxel < 5 degrees difference between Global Signal & Hypothesis !

“Reference Function, R” Inclusion of Global Covariate in Regression: Effect of non orthogonality 1 2 X1 X’1 db1 db2 b = (GTG)-1GTX 2 db2 X1 X’1 1 “Reference Function, R”

Consider an fMRI experiment with only 3 time points

Consider an fMRI experiment with only 3 time points

Analysis of Brain Systems Correlation viewed as a projection reference function R1 R2 R2 Corr(R2, ref) ref Corr(R1, ref) Although R1 and R2 both somewhat correlated with the reference function, they are uncorrelated with each other R1

Principal Component Analysis (PCA) Voxel 1 Voxel 2 Voxel 1 Voxel 2 Voxel 3 PC1 t Voxel 3 Eigenimage + time course

Independent Component Analysis (ICA) Without knowing position of microphones or what any person is saying, can you isolate each of the voices?

Independent Component Analysis (ICA) Assumption: each sound from speaker unrelated to others (independent)

Some ICA assumptions g(C) : Position of microphones and speakers is constant (mixing matrix constant) Sources Ergodic The propagation of the signal from the source to the microphone is instantaneous Sources sum linearly Number of microphones equals the number of speakers In Bell-Sejnowski algorithm, the non-linearity approximates the cdf of the sources g(C) :

Independent Component Analysis (ICA) Independent Sources (individuals’ speech) time Mixing matrix = Data S ?M X Goal of ICA: given Data (X), can we recover the sources (S), without knowing M? Independent Components time = Data X W Weight matrix C g(C) : ‘InfoMax’ algorithm: Iteratively estimate W, so that: Key point: maximizing H(y) implies that rows of C are maximally independent Goal of ICA: Find W, so that Kullback-Leibler divergence between f1(C) and f2(S) is minimized ?

Independent Component Analysis (ICA) Non task-related activations (e.g. Arousal) Task Measured Signal Machine Noise Pulsations Assumption: spatial pattern from sources of variability unrelated (independent)

time M Mixing #1 t = 1 #2 t = 2 n t = n COMPONENT MAPS MEASURED The fMRI data at each time point is considered a mixture of activations from each component map Mixing COMPONENT MAPS MEASURED fMRI SIGNAL #1 S t = 1 time #2 ‘mixing matrix’, M S t = 2 S n S t = n

Selected Components: Consistently task-related Transiently Abrupt head movement Quasi-periodic Slowly-varying Slow head movement Activated Suppressed

Increasing spatial independence between components Comparison of Three Linear Models PCA (2nd order) 4th order ICA (all orders) r = 0.46 r = 0.85 r = 0.92 Increasing spatial independence between components

Statistically Independent Are Two Maps Independent? 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ... 0.1, 1.2, 1.3, -1.9, ... -0.1, 4.2 ... ? A B Statistically Independent = å i q p B A ICA (all orders) Identical 2nd-order statistics Higher- order statistics Comon’s 4th order = å i B A PCA (2nd order) Decorrelated

Derived Independent Components ICA Component Histogram of voxel values for component map z > 1 A component map specified by voxel values 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ... Component map after thresholding associated time course

Unexpected Frontal-cerebellar activation detected with ICA Self-paced movement Rest Movie

A Transiently task-related (TTR) component (active during first two trials) Martin J. McKeown, CNL, Salk Institute, martin@salk.edu

Single trial fMRI Trial 1 ICA component time course Aligned ICA component spatial distribution (a) (b)

Single trial fMRI (c) (d) (e) 19-sec All p < 10-20

Assessing Statistical Models PRESS Statistic: ^ -i +  G Eliminate 1 time point = Data How well does G-i match data? Gives some idea of the influence of the ith time point

Hybrid Techniques Hypothesis Driven Data Driven Exp Con

HYBICA: L arm pronation/supination hypothesis Hybrid activation

Use of HYBICA for Memory Load Hypothesis testing

Use of HYBICA for Memory Load Hypothesis testing Maintenance

Use of HYBICA for Memory Load Hypothesis testing