Granger Causality Analysis of fMRI Data:...

Granger Causality Analysis of fMRI Data:

Techniques, Caveats and Applications

Xiaoping Hu

Wallace H. Coulter Department of Biomedical Engineering

Emory University and Georgia Tech

Atlanta, GA, USA

[email protected]

fMRI Deriving Connectivity

• Functional connectivity

– “Temporal correlations between remote

neurophysiological events” neurophysiological events”

• Effective connectivity

– “Influence one neuronal system exerts over

another”

Friston et al

Granger Causality Analysis• Granger causality is based

on the concept of temporal precedence information

• If including past values of Y improves the prediction of future values of X, then Y is said to have a causal

Time n

T

I

M

Y is said to have a causal influence on X

• Originally invented by Granger for stock market prediction and awarded Nobel prize in economics in 2003

Time series X Time series Y

Time n-k

M

E

Multivariate Granger Causality and Directed

Transfer Function

• Let X(t)=(x1(t),x2(t),... xk(t)) be the data vector wherein xk is the time series, a multivariate autoregressive model with model parameters A(n) of order p is given by

• Transforming the equation to the frequency domain, one has the transfer matrix, H(f)(=A-

1(f)), or the non-normalized directed transfer

∑=

+−=

p

n

tntnt1

)()()()( EEEEXXXXAAAAXXXX

)()()()()( 1 fEfHfEfAfX ==−

1(f)), or the non-normalized directed transfer function (DTF).

• H(f) is multiplied by partial coherence to emphasize direct connections and summed over all frequencies to obtain direct DTF, where the partial coherence is given by

and Mij(f) is the minor of the cross-spectrum matrix between the time series.

∑=

f

ijijij ffhdDTF )()( η

)()(

)()(

2

fMfM

fMf

jjii

ij

ij =η

Statistical Testing using Surrogate Data

Original DataRetain Power

SpectrumRandomize Phase

Surrogate DataCalculate dDTFon Surrogates

2500 Times

EmpiricalNull

Distribution

Original time series

Phase randomized

0 10 20 30 40

NULL SURROGATE DISTRIBUTION

Significant !

Deshpande et al, HBM, 30: 1361-1373, 2009.

5s

12s

fMRI Impulse-Response Function

τ

2s

Courtesy of Y. Yang, NIDA

A Poor Man’s Application of

Granger Analysis: Investigation

of Slow Causal Influencesof Slow Causal Influences

Motor Fatigue Experiment

• Subjects repeated a hand contraction task guided

by visual feedback (50% maximum force). The

duration of each contraction was 3.5 s, followed

by a 6.5 s rest. by a 6.5 s rest.

• The fatigue task lasted 20 minutes, with a total of

120 contractions performed by each subject.

• fMRI images acquired at every 2 seconds.

Deshpande et al., HBM (2009)

ROIs and Integrated Time Courses

■

▲

*

♥

◄

■ SMA

▲ M1

* S1

♥ P

◄ PM

0 100 200 300 400 500 600-15

-10

-5

0

5

10

15

Time (TRs)0 20 40 60 80 100 120

10

15

20

25

30

35

40

Time


Effective Connectivity during Fatigue

SMA

M1

PM C

P

S1

SMA

M1

PM C

P

S1

SMA

M1

PM C

P

S1

First Window Middle Window Last Window


• In window 1, the strong output from S1 indicates fine tuning of motor activity by sensory feedback.

• In window 2, cerebellum’s role increases indicating more motor control; the shift from window 1 to window 2 likely reflects learning.

• In window 3, the connectivity pattern is similar to that of window 2 but there is a general reduction in connectivity due to fatigue.

Network from raw time series

SMA

M1

P

S1

SMA

M1

P

S1

SMA

M1

P

S1

window 1 window 2 window 3

PM C PM C PM C

• Networks derived from the raw data exhibit more

causal paths that are less significant, with no

apparent driving node(s) and little change with

time.


Effect of Slow Sampling and

Hemodynamic Response on Fast

Causal InfluencesCausal Influences

BOLD Signal and LFP

Logothetis, Nature, 2001

Simulations

• LFP signal X sampled at 1ms interval. Y obtained by shifting it by d ms

• Hemodynamic impulse response modeled by modeled by Gamma functions

A = time to peak

W = full width at half maximum

K = scaling factor

• TR=0.5, 1, 1.5 and 2 seconds

Desphande & Hu, NeuroImage, 52: 884-96, 2010.

fMRI simulation from LFP

Original LFP time series X(red)→Y(blue): 0.3Y(blue)→X(red): 0.0024

LFP convolved with HRF

X(red)→Y(blue): 1.8Y(blue)→X(red): 0.7

X(red)→Y(blue): 80Y(blue)→X(red): 21

HRF Difference (0.5 s) Opposite the Neuronal Delay

X(red)→Y(blue): 44Y(blue)→X(red): 12

No Noise

Noise (SNR=50) added to simulated

fMRI

X(red)→Y(blue): 29±5Y(blue)→X(red): 15±4

HRF Difference (0.5 s) Opposite the Neuronal Delay

preserving HRF shape

HRFs shifted in time; shape preserving (results below)

HRFs with different rise time; shape altering (results above)

Physiologically feasible model

Aguirre et al, NeuroImage 1998

preserving (results below) shape altering (results above)

No Noise

X(red)→Y(blue): 0.85Y(blue)→X(red): 1.73

SNR=50

X(red)→Y(blue): 0.9 ± 0.3Y(blue)→X(red): 1.6 ± 0.2

Reverse directioninferred

Desphande, Sathian & Hu. Effect of Hemodynamic Variability on

Granger Causality Analysis of fMRI. NeuroImage 52: 884-96, 2010.

•In the absence of HRF variability, even tens of milliseconds of

neuronal delay can be inferred from GC analysis of fMRI.

•In the presence of HRF delays which oppose neuronal delays, the

minimum detectable neuronal delay may be hundreds of milliseconds.

•In the more realistic scenario of unknown neuronal and hemodynamic

delays within their normal physiological range, the accuracy of

detecting the correct multivariate network from fMRI is well above

chance and up to 90% with faster sampling.

•Under all conditions, faster sampling and low measurement noise

improve the sensitivity of GC analysis of fMRI data.

• Aim: to investigate the neural circuitry underlying tactile spatial acuity at the human finger pad

• Spatial task: linear, 3-dot arrays, applied to the immobilized right index finger pad using a computer-

Tactile Spatial Acuity Experiment

finger pad using a computer-controlled, MRI-compatible, pneumatic stimulator

• Control task: Temporal offset stimulus instead of spatial offset

Stilla et al., J Neurosci 07

• Activity specific for

spatial processing

revealed activity in a

distributed fronto-

parietal cortical

network

• Levels of activity in

right posterior right posterior

intraparietal sulcus

(pIPS) significantly

predicted individual

acuity thresholds


135• Multivariate Granger

causality

relationships among

selected ROIs

• Top: Better

2

66 • Top: Better

• Bottom: Poorer

Stilla et al., J Neurosci, 27, 11091, 2007

What determines acuity ?

• Regression shows that in the better group, the paths predicting acuity converged from the left postcentral sulcus and right frontal eye field onto converged from the left postcentral sulcus and right frontal eye field onto the right pIPS.

• These connections were selective for the spatial task

• Their weights predicted the level of right pIPS activity

• Conclusion: The optimal strategy for fine tactile spatial discrimination involves interaction in the pIPS of a top-down control signal, possibly attentional, with somatosensory cortical inputs, reflecting either visualization of the spatial configurations of tactile stimuli or engagement of modality independent circuits specialized for fine spatial processing


Comparing functional connectivity and

Granger-based effective connectivity

Eff

ective C

onnectivity

Matr

ix

"Better" group "Poor" group

Functional C

onnectivity

Matr

ix

R=-0.08, p=0.45 R=-0.02, p=0.85

Correlation-purged Granger Causality

• Given n time series X(t) = [x1(t) x2(t) … xn(t)], the traditional VAR model of order p is given below

X(t) = A(1)X(t-1) + A(2)X(t-2) + ... + A(p)X(t-p) + E(t)

where A(1) … A(p) are the coefficients of the model and E(t) is the model error

• In order to account for the zero-lag correlation effects, we introduce the zero-• In order to account for the zero-lag correlation effects, we introduce the zero-lag term

X(t) = A' (0)X(t) + A' (1)X(t-1) + A' (2)X(t-2) + ... + A' (p)X(t-p) + E' (t)

• The inclusion of the zero-lag term affects the value of other coefficients and hence A'(1) … A' (p) ≠ A(1) … A(p)

• GC obtained from A'(1) … A' (p) are linearly independent of zero-lag correlation, which we call correlation-purged GC (CPGC)

Deshpande & Hu, TBME, 57: 1446-1456, 2010

Simulation

• CASE 1: Consider two time series x(n) and y(n) modeled as a first order VAR process such that the causal influence between them is zero but the instantaneous correlation is nonzero

==

10.5

0.51Covand

00

00A(1)

• Assuming x(n) and y(n) represent LFPs sampled at 1ms, they were convolved with HRF and downsampled 1000/2000 times to simulate convolved with HRF and downsampled 1000/2000 times to simulate fMRI series with TRs of 1 s and 2 s

• CASE 2: Subsequently, in order to demonstrate the efficacy of CPGC for recovering neuronal causal influences from fMRI, we generated x(n)and y(n) such that a unidirectional causal influence exists from x(n) to y(n) with no correlation between them. The corresponding fMRI time series, x' (n) and y' (n), were derived and zero-lag correlation, GC and CPGC were calculated from them


TRZero-lag correlation Granger causality Correlation-purged Granger causality

x' (n) ↔ y' (n) x' (n) → y' (n) y' (n) → x' (n) x' (n) → y' (n) y' (n) → x' (n)

1 s 0.49 ± 0.05 0.47 ± 0.01 0.47 ± 0.01 0.01 ± 0.02 0.01 ± 0.02

2 s 0.49 ± 0.06 0.40 ± 0.07 0.40 ± 0.07 0.00 ± 0.09 0.00 ± 0.09



1 ms 0.5 ± 0.09 0.00 ± 0.01 0.00 ± 0.01 0.00 ± 0.01 0.00 ± 0.01

Simulation 1: Only correlation and no causality in LFP data



1 s 0.29 ± 0.09 0.47 ± 0.1 0.27 ± 0.09 0.21 ± 0.02 0.01 ± 0.02

2 s 0.29 ± 0.09 0.40 ± 0.09 0.20 ± 0.07 0.14 ± 0.02 0.00 ± 0.02



1 ms 0.0 ± 0.09 0.5 ± 0.01 0.0 ± 0.01 0.5 ± 0.01 0.0 ± 0.01

Simulation 2: Only causality and no correlation in LFP data


Functional connectivity

• Temporal correlations of low frequency

fluctuations exist in the brain, even at “rest”– Biswal et al. Magn Reson Med 34:537 (1995)

• Connectivity of functionally related areas

– Examples: Motor, visual, language, “default mode”

networks

• Can characterize changes in brain state

Resting State Networks (RSNs)

• Internally directed cognitive processing (specifically, self

referential and mental simulation) by Default Mode

Network (DMN)

• Obtained using posterior cingulate (PCC) seed

• Internally directed cognitive processing (specifically, memory encoding and retrieval ) by Hippocampal Cortical

Memory Network (HCMN)

• Obtained using hippocampus (HC) seed

• Externally directed cognitive processing by Dorsal

Attention Network (DAN)

• Obtained using middle temporal (MT) seed

• Executive control of anti-correlated DMN/HCMN and

DAN by Fronto-parietal Control Network (FPCN)

• Obtained usinganterior prefrontal (aPFC) seed

Deshpande et al., NeuroImage (2010)

DMN

HCMN

DAN

FPCN

Frontal

Parietal

Temporal

Cingulate

PCC and pIPL: The Transit Hubs

• Central location on layout ideal for this role

• High resting state metabolism

• PCC seed-based correlation analysis will only reveal DMN and HCMN ROIs

• Drives the characterization of different groups of ROIs in different networks

• Functional segregation of different networks is rather a soft boundary


Memory Encoding

• Predominant inputs to HF from:

• Parietal ROIs provide perceptual content

• DAN ROIs may provide the context, i.e. those perceptual

contents which are being attended to



Integration and Control

• aPFC is at the apex of the control hierarchy

• Implicated in integrating the outcomes of multiple cognitive operations

• Integration of internal and external representations from the anti-correlated

DMN/HCMN and DAN systems:

• The input to R aPFC from PCC brings the internal representations from memory

• Inputs from bilateral MT in the DAN bring the information about the external environment

• Our results support Vincent’s hypothesis: aPFC seeded FPCN to be a control network

• SVC successfully learned patterns of functional connectivity

capable of predicting MDD from HCcapable of predicting MDD from HC

– Uncovered differences not discovered by t-test analysis

• Feature selection substantially improved the prediction

accuracy of SVC

– Methods that incorporate reliability performed the best

PCE may affect behavior and functioning by increasing baseline arousal and altering the

excitatory/inhibitory balancing mechanisms involved in cognitive resource allocation.

PCE is associated with activation changes in different regions, so can connectivity changes

across these regions be used to predict PCE?

Medial PFC

Left DLPFCRight

DLPFC

Prediction of PCE status with functional/effective connectivity

Left amygdala Right amygdala

Medial PFC

Left Parietal

cortex

Right Parietal

cortex

ACC

PCC

Deshpande et al., 2010 PLoS ONE 5 (12):e14277

Prediction of PCE status with functional/effective connectivity

Deshpande et al., 2010 PLoS ONE 5 (12):e14277

Correction of HRF latency with breath holding

Breath Holding

3.8 sec 3.8 sec 3.8 sec 3.0 sec 11.4 sec

inhale inhale inhale inhale

get ready

hold release

… 16 repetitions

Face Perception

90 faces randomly presented

ISI = 3.92sec (mean) ± 2.1sec (SD)

2 fMRI scan runs, ~6min

TR = 1sec or 2sec

Chang, Neuroimage 2008

Data Processing

Mean signal

as reference

1. ROI definition by regular GLMV1 and FFA

2. BH modulated cortical voxels

3. Voxel-wise signal latency

ROI signal

extraction4. Voxel-wise

latency correction

Reference

Each voxel

Shift to find the best correlation

FFA

original

V1 original

FFA

corrected

V1 corrected

ShiftLatency

applied

5. Compare GCA results of

uncorrected & corrected dataUncorrected Corrected

Feed into GCA

Results

Effect of temporal resolution on GCA

Block (30sec) design, visual (flashing checker board) motor (finger tapping) task

Parallel imaging with TR=2.4sec, 0.6sec, and 0.3sec, 4 subjects

Six ROIs defined according to fMRI activation

Left LGNRight LGN

Visual

cortex

Left primary

motor

Right primary

motor

SMA

GCA Results

TR=2.4sec LGCN ->

V

TR=0.6sec

TR=0.3sec

LGC -> V

LGN -> Motor

LGC -> V

LGN -> Motor

SMA -> Primary Motor

Time-frequency dynamics of default

mode effective connectivity

Summary

• Granger causality analysis can infer causal influences

(neuronal delays) between different brain regions from

fMRI data although there are limitations.

• Granger causality analysis can also be applied to resting-

state fMRI data to infer instantaneous correlation and

causal influences.

• Connectivity measures, particularly the combination of

functional and effective connectivities, can be used to

improve prediction.

Acknowledgements

• Funding: NIH, Georgia Research Alliance

• Students and Postdocs: Cameron Craddick, Priya

Sanatham, Gopi Deshpande, Stephen LaConte, Zhihao Li, Sanatham, Gopi Deshpande, Stephen LaConte, Zhihao Li,

John Sexton, Andy James, and Scott Peltier

• Collaborators: Claire Coles, Helen Mayberg, Paul

Holtzheimer, Clint Kilts, Krish Sathian, Stephan Hamann,

and Mary Ellen Lynch

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