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FMRI Data Analysis II Connectivity Analyses

- Dr. George Andrew James
- Research Associate
- The Wallace H. Coulter Department of Biomedical

Engineering at the Georgia Institute of

Technology Emory University School of Medicine - Tuesday, November 17, 2009

Overview of Connectivity Analyses

- Functional connectivity analyses
- Typically correlational
- Seed analyses
- Component analyses (PCA, ICA, SVD)
- Effective connectivity analyses
- Infer causality
- Structural equation modeling
- Granger causality analysis
- Dynamic causal modeling

Connectivity Analyses

- Advantages
- Require no a priori hypotheses
- Can capture influences not relating to model
- Disadvantages
- Computationally statistically complex
- Less intuitive than model-dependent methods

Correlational Analyses

- Measure correlation between a voxels timecourse

and all other voxels timecourses - Like GLM using a voxels timecourse as the

paradigm

Anterior Cingulate

Orbitofrontal cortex

Thalamus

Subgenual Cingulate

Seed Analysis

- Pick a region of interest (ROI) as seed
- Make SPM where each voxel is color-coded by the

strength of correlation between voxel and ROI

seed

Correlations can be measured in absence of

task! Frequency filtering is essential for

resting correlations!

Peltier 2002

Changing Correlations in Language Network with

Task

He et al, HBM 2003

Changing Correlations in Language Network with

Task

Meaningless pinyins aloud

Meaningful pinyins aloud

Tongue movement

Pinyins chinese words written in English

letters (i.e. xiexie for thank you)

He, HBM 2003

Changes in correlation with time

He, HBM 2003

The Good, The Bad, and the Ugly

Brain activation during naturalistic viewing of

movie stimuli

Fig. 1. Intersubject correlation during free

viewing of an uninterrupted movie segment.( A)

Average percentage of functionally correlated

cortical surface across all pairwise comparisons

between subjects for the entire movie time course

(All), for the regionally specific movie time

course (after the removal of the nonselective

component, Regional) and for the darkness control

experiment (In darkness).( B) 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 theextension of the correlations

beyond visual and auditory cortices.

The Good, The Bad, and the Ugly

Fig. 2. Nonselective activation across regions.(

A) Correlation between the averaged time course

of the VOT cortex in one cortical hemisphere

(correlation seed marked by the red contour) and

the rest of the cortex, shown on unfolded left

and right hemispheres.( B) 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.

Hasson, Science 2004

The Good, The Bad, and the Ugly

Reverse correlation what is subject viewing

during timecourse below?

Hasson, Science 2004

The Good, The Bad, and the Ugly

Reverse correlation what is subject viewing

during timecourse below?

Hasson, Science 2004

The Good, The Bad, and the Ugly

The Good, The Bad, and The Ugly

Intersubject correlation shows us what brain

regions are co-activated across subjects when

watching a movie AND (perhaps more

importantly) brain regions that are not

correlated across subjects!

Component Analyses

- We have a correlation matrix for many ROIs
- How can we simplify or distill this correlation

matrix into a network of regions? - ex visuomotor learning most likely involves

several independent networks that are

simultaneously co-activated. - Well use PCA (principle components analysis) to

extract these networks

Conceptualizing PCA

Conceptualizing PCA

brains total spatial and temporal variance

variance of voxels individual timecourses

Conceptualizing PCA

brains total spatial and temporal variance

variance of voxels individual timecourses

BUT, some voxels better explain the brains

overall variance than others! PCA asks How can

we cluster voxels into components to best

explain the brains variance?

PCA visuomotor example

Visual Stimuli

Subject Response

time

Some brain regions (V1, M1, cerebellum, thalamus,

SMA) should have greater temporal variability

(more variance) than others (Brocas area,

sylvian fissure, amygdala, etc.)

Total Variance

Voxels Variance

How do we do Component Analyses?

- Linear Matrix Algebra
- Eigenvector given a linear transformation,an

eigenvector of that transformation is a nonzero

vector which, when a transformation is applied to

it, may change in length but not direction - Eigenvalue describes manipulation to

eigenvector - 2 same direction, 2x length
- 1 same direction, same length
- -1 opposite direction, same length

How do we do Component Analyses?

Wikipedia, 2008

Principal Components Analysis and Singular Value

Decomposition

- Given square matrix A with order r x r
- A principal components analysis of A yields
- USU' A
- where U containing the eigenvectors is r x r, S

is a diagonal matrix r x r containing the

eigenvalues - A U(1)S(1)U(1)' U(2)S(2)U(2)'

U(r)S(r)U(r)' - The computed principal components, or latent

variables (LV), are mutually uncorrelated - The first LV accounts for the largest part of A

(largest variance), and the next LV accounts for

the second largest variance not related to the

first LV

PLS Workshop 2008 University of Toronto

Principal Components Analysis and Singular Value

Decomposition

- Eigenvalues
- Indicate the proportion of total variance in the

matrix that is captured by a each LV - If ?i is the ith eigenvalue from a PCA on a

matrix

PLS Workshop 2008 University of Toronto

Principal Components Analysis and Singular Value

Decomposition

- Conceptual - Regression Analogy
- Step 1 Derive a latent variable (LV) that

accounts for as much of matrix A as possible - LV1 S1 u1X1 u2X2 u2X3
- where S1 is a constant scaling factor

(eigenvalue) and uj is the weight for the Xj in

LV1 - Step 2 Regress LV1 out of matrix A and repeat

step. Note that because we have remove LV1 from

the data, LV2 is necessarily orthogonal to LV1 - Note this will not work in practice, it's only

an analogy

PLS Workshop 2008 University of Toronto

Principal Components Analysis and Singular Value

Decomposition

1.0000 0.5685 0.2558 0.5685 1.0000 0.2424 0.2558 0

.2424 1.0000

Correlation Matrix (A)

PLS Workshop 2008 University of Toronto

In other Words

Helpful hint in Matlab, just use command

U,Seig(A)

A correlation matrix A U S U

1.0000 0.5685 0.2558 0.5685 1.0000 0.2424 0.2558 0

.2424 1.0000

0.6402 -0.2898 -0.7115 0.6357 -0.3202

0.7024 0.4313 0.9019 0.0207

1.7369 0 0 0 0.8318

0 0 0 0.4313

0.6402 0.6357 0.4313 -0.2698 -0.3202 0.6357 -0.711

5 0.7024 0.0207

LV1 U1 S1 U1

0.7118 0.7068 0.4796 0.7068 0.7019 0.4763 0.4796 0

.4763 0.3232

0.6402 0.6357 0.4313

1.7369

0.6402 0.6357 0.4313

Principal Components Analysis and Singular Value

Decomposition

LVi UiSiUi'

0.7118 0.7068 0.4796 0.7068 0.7019 0.4763 0.4796 0

.4763 0.3232

LV1

0.0699 0.0772 -0.2174 0.0772 0.0853 -0.2402 -0.217

4 -0.2402 0.6766

LV2

1.0000 0.5685 0.2558 0.5685 1.0000 0.2424 0.2558 0

.2424 1.0000

0.2183 -0.2156 -0.0064 -0.2156 0.2128 0.0063 -0.0

064 0.0063 0.0002

LV3

PLS Workshop 2008 University of Toronto

Principal Components Analysis and Singular Value

Decomposition

- For non-square matrices, we use singular value

decomposition (SVD) rather than principal

components analysis - Given matrix B, that is r x c, an SVD of B yield
- USV' B, where U is r x r, S is a diagonal

matrix r x r, and V is c x r

PCA limitations

- If task-related fMRI changes are only a small

part of total signal variance capturing the

greatest variance in the data may reveal little

information about task-related activations.

(McKeown, 1998) - ex V1 and images of disgust vs. horror
- Components must be orthogonal, making

components difficult to conceptualize and less

significant as their order increases.

Independent Component Analysis

- Related to PCA, ICA deconvolves a mixture of

signals into sources. - Generally accepted as more powerful and sensitive

than PCA. - GIFT, Matlabs FastICA

McKeown (1998)

Another ICA illustration

(McKeown, 1998)

Conceptualizing ICA

Axis 1

Axis 2 (PCA)

Axis 2 (ICA)

ICA Comparisons

3 participants performed the Stroop test. ICA

yielded multiple components including one whose

timecourse closely matched the paradigm (shown

right)

(McKeown, 1998)

(McKeown 1988)

Additional comments A voxel can contribute to

multiple components. ICA reveals non-task

specific components. ICA could be valuable for

masking unwanted voxels (i.e. slowly-varying

activity)

Regional Homogeneity

- Regional homogeneity estimates how correlated a

voxel is with its immediate neighbors i.e. a

regions homogeneity

Regional Homogeneity and Anesthesia

Peltier, Kerssens, Hamann, Sebel, Byas-Smith

Hu. (2005). NeuroReport, 16, 285-288.

Regional homogeneity describes how strongly a

brain region communicates with its immediate

neighbors. This analysis provides insight into

aberrant connectivity patterns within a neural

region. We have demonstrated progressive

reduction in the local coherence of frontal and

sensorimotor cortices with increasing anesthesia.

Regional Homogeneity and Epilepsy

James Drane. (unpublished)

Preliminary findings The hippocampus in the

epileptogenic hemisphere shows less regional

homogeneity than its counterpart.

Effective Connectivity

- Unlike correlational methods (aka functional

connectivity), effective connectivity attempts to

find causal relationships - Simultaneous influences among variables

(structural equation modeling, dynamic causal

modeling) - Temporal influences among variables (Granger

causality analysis)

Structural Equation Modeling

- SEM is a statistical technique to assess both the

strength and directionality of interactions

between variables - SEM aka path analysis or causal modeling
- SEM is traditionally confirmatory
- SEM assess how well a model fits a given dataset

i.e. SEM tests the model, not the data!

But correlation doesnt imply causality! (so how

does SEM work?)

Interpreting Factor Analysis / SEM

Observed covariance matrix

Test1 Test2 Test3 Test4

Test1 1 .6 .2 .3

Test2 .6 1 .1 .3

Test3 .2 .1 1 .7

Test4 .3 .3 .7 1

Interpreting Factor Analysis

Find values of a, b, etc. so that predicted

covariance best matches observed covariance.

Factor 1

d

b

a

c

Test 1

Test 2

Test 3

Test 4

u

v

w

x

Predicted covariance matrix

Observed covariance matrix

Test1 Test2 Test3 Test4

Test1 a2u ab ac ad

Test2 ab b2v bc bd

Test3 ac bc c2w cd

Test4 ad bd cd d2x

Test1 Test2 Test3 Test4

Test1 1 .6 .2 .3

Test2 .6 1 .1 .3

Test3 .2 .1 1 .7

Test4 .3 .3 .7 1

Interpreting Factor Analysis

r

Factor 1

Factor 2

d

b

a

c

Test 1

Test 2

Test 3

Test 4

u

v

w

x

Predicted covariance matrix

Observed covariance matrix

Test1 Test2 Test3 Test4

Test1 1 .6 .2 .3

Test2 .6 1 .1 .3

Test3 .2 .1 1 .7

Test4 .3 .3 .7 1

Test1 Test2 Test3 Test4

Test1 a2u ab arc ard

Test2 ab b2v brc brd

Test3 arc brc c2w cd

Test4 ard brd cd d2x

Interpreting Factor Analysis

r 0

Factor 1

Factor 2

d

b

a

c

Test 1

Test 2

Test 3

Test 4

u

v

w

x

Predicted covariance matrix

Observed covariance matrix

Test1 Test2 Test3 Test4

Test1 1 .6 .2 .3

Test2 .6 1 .1 .3

Test3 .2 .1 1 .7

Test4 .3 .3 .7 1

Test1 Test2 Test3 Test4

Test1 a2u ab 0 0

Test2 ab b2v 0 0

Test3 0 0 c2w cd

Test4 0 0 cd d2x

Examples of SEM from neuroimaging

Path loading of PP?ITp significantly increases as

subjects learn object/spatial associations.

(Büchel 1999)

Structural Equation Modeling

SMA supplementary motor area PM premotor

(R/L) M1 primary motor (R/L) Zhang et al., 2005

- Path weighting expresses strength of connection.

(Analagous to correlation... but directional!)

Granger Causality Analysis

- Directly measures temporal associations
- Given two ROI timecourses
- X(t)x1 x2 x3 xN and Y(t)y1 y2 y3 yN
- Build autoregressive model so that past values of

X(t) can predict future values. - If including past values of Y improves the

ability for past values of X to predict current

value of X, then we say Y granger causes X

Granger Causality Analysis

Experiment Subjects engage in motor fatigue

task sinusoidal contraction of a hand weight

Activation of many regions Motor, premotor, SMA,

cerebellum, S1, parietal

Granger Causality Analysis

M1 SMA PM S1 C P

Window-1 Cin 19 15 7 9 16 11

Window-1 Cout 8 13 7 25 10 13

Window-2 Cin 15 21 15 9 16 8

Window-2 Cout 8 14 11 23 18 10

Window-3 Cin 11 13 9 9 15 8

Window-3 Cout 7 8 9 18 14 9

References

- Huettel SA, Song AW McCarthy G. (2004).

Functional Magnetic Resonance Imaging. Sinauer

Associates Inc Sunderland, Massachusetts USA. - Peterson SE, Fox PT, Snyder AZ Raichle ME.

(1990). Activation of the extrastriate and

frontal cortical areas by visual words and

word-like stimuli. Science, 249(4972),

1041-1044. - SPM http//www.fil.ion.ucl.ac.uk/spm/
- AFNI http//afni.nimh.nih.gov/afni

References

- McKeown MJ, Makeig S, Brown GG, Jung T-P,

Kindermann SS, Bell AJ Sejnowski TJ. (1998).

Analysis of fMRI data by blind separation into

independent spatial components. Human Brain

Mapping, 6, 160-188. - Zhuang, J. C., LaConte, S. M., Peltier, S.,

Zhang, K., Hu, X. (2005). Connectivity

exploration with structural equation modeling an

fMRI study of bimanual motor coordination.

NeuroImage, 25, 462-470.

Thanks!

BOLD-fMRI

Vasomotor Tone Adapting...? Fast or slow

Uncoupling

Neural Activation

Stim.

rCBO

rCBF

Vo2

( K, H, NO, PO2

Adenosine...? )

empirical

- deoxyHb ---gt rate of MR signal decay (1/T2)
- rf pulse sequence sensitive to the decay effect

A(t) A0 e - TE /T2(t)

Analysis of Variance (ANOVA)

Interaction effects is the whole greater than

the sum of the parts? ex thalamic response to

simultaneous visual and auditory stimuli