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. Slice 1 Data Matrix X Voxels

Calculating level of Significance

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

Covariate Indicator variable

SPM Nomenclature for Design Matrix

G 1 G (interesting) G c H 1 H (non-interesting) H c Global activity Linear trends E.g. dose of drug Design matrix G subject

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

Global Signal Hypothesis Test voxel Regression Coefficients 4 3 -1 -2 2 1 0

Global signal

1

Hypothesis

2  < 0!!!

< 5 degrees difference between Global Signal & Hypothesis !

db 2  2

Inclusion of Global Covariate in Regression: Effect of non orthogonality

X 1 X’ 1 db db 1 2 “ Reference Function, R”  1  2 X 1 X’ 1 b = (G T G) -1 G T X  1

Consider an fMRI experiment with only 3 time points

Consider an fMRI experiment with only 3 time points

Analysis of Brain Systems

reference function R 1 Correlation viewed as a projection

R 2

R 2 Although R1 and R2 both somewhat correlated with the reference function, they are uncorrelated with each other

Corr(R 2, ref) Corr(R 1, ref) ref R 1

Principal Component Analysis (PCA)

Voxel 1 Voxel 2

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

•

Position of microphones and speakers is constant (mixing matrix constant)

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Sources Ergodic

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The propagation of the signal from the source to the microphone is instantaneous

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Sources sum linearly

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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)

?M

S = X Mixing matrix Independent Sources (individuals’ speech) = Data time Goal of ICA : given Data (X), can we recover the sources (S), without knowing M?

W X = C Weight matrix Data = Independent Components time ‘InfoMax’ algorithm: Iteratively estimate W, so that: Goal of ICA: Find W, so that Kullback-Leibler divergence between f 

W

0 , 1 y (C) and f g(C) 2 (S) is minimized ?

g(C) : Key point :

maximizing H(y) implies that rows of C are maximally independent

Independent Component Analysis (ICA)

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

The fMRI data at each time point is considered a mixture of activations from each component map

COMPONENT MAPS

#1

Mixing

#2 ‘mixing matrix’, M n S S S

MEASURED fMRI SIGNAL

t = 1 S t = 2 t = n

Selected Components:

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

Comparison of Three Linear Models PCA (2nd order) 4th order ICA (all orders) r = 0.46

r = 0.85

Increasing spatial independence between components r = 0.92

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 ...

?

Identical 2nd-order statistics A Statistically Independent

i



A i p B i q

B  0 ICA (all orders) Higher order statistics Decorrelated 

i A i B i

 0

Comon’s 4th order

PCA (2nd order)

Derived Independent Components

ICA Component A component map specified by voxel values 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...

Histogram of voxel values for component map z > 1 0 Component map after thresholding associated time course

Unexpected Frontal-cerebellar activation detected with ICA Self-paced movement Rest Movie 0 10 20 30 40 50 60

A Transiently task-related (TTR) component (active during first two trials) Martin J. McKeown, CNL, Salk Institute, [email protected]

ICA component time course

(a)

Aligned

Single trial fMRI

Trial 1

ICA component spatial distribution

(b)

(c) (d) (e)

19-sec All p < 10 -20

Single trial fMRI

Assessing Statistical Models

Voxel # fMRI (X) Data Reference function G = 

PRESS Statistic:

Eliminate 1 time point G  -i = Data +  How well does G  -i match data?

• Gives some idea of the influence of the i th time point + 

Hybrid Techniques

Data Driven Hypothesis Driven

Con Exp Con Exp Con Exp Con Exp

HYBICA:

L arm pronation/supination hypothesis Hybrid activation 0 10 20 30 40 50 60

Use of HYBICA for Memory Load Hypothesis testing

S1

Use of HYBICA for Memory Load Hypothesis testing

Maintenance

Use of HYBICA for Memory Load Hypothesis testing

S2