Granger Causality Analysis of fmri Data: Techniques, Caveats and Applications

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

fMRI Deriving Connectivity

• Functional connectivity

– “Temporal correlations between remote

neurophysiological events”

neurophysiological events”

• Effective connectivity

– “Influence one neuronal system exerts over

another”

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)=(x

(t),x

(t),... x

(t))

be the data

vector wherein x

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

_{(f)), or the non-normalized directed transfer}

∑

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 M

(f) is the minor of the cross-spectrum

matrix between the time series.

∑

Statistical Testing using Surrogate Data

Original time series

Phase randomized

0 10 20 30 40 NULL SURROGATE DISTRIBUTION

Significant !

5s

12s

fMRI Impulse-Response Function

τ

2s

A Poor Man’s Application of

Granger Analysis: Investigation

of Slow Causal Influences

of 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.

ROIs and Integrated Time Courses

Effective Connectivity during Fatigue

First Window Middle Window Last Window

Deshpande et al., HBM (2009)

• 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

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 Influences

BOLD Signal and LFP

Simulations

• LFP signal X

sampled at 1ms

interval. Y

obtained by shifting it

by d

ms

W = full width at half maximum K = scaling factor

• TR=0.5, 1, 1.5 and 2 seconds

fMRI simulation from LFP

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

Y(blue)_→X(red): 0.0024 LFP convolved with HRF X(red)_→Y(blue): 1.8 Y(blue)_→X(red): 0.7 X(red)→Y(blue): 80 Y(blue)_→X(red): 21

HRF Difference (0.5 s) Opposite the Neuronal Delay

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): 0.85 Y(blue)_→X(red): 1.73 SNR=50 X(red)_→Y(blue): 0.9 ± 0.3 Y(blue)_→X(red): 1.6 ± 0.2 Reverse direction inferred

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

• 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

• Top: Better

• Bottom: Poorer

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

E ff e c ti v e C o n n e c ti v it y M a tr ix

"Better" group "Poor" group

F u n c ti o n a l C o n n e c ti v it y M a tr 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) … x_n(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 lag correlation effects, we introduce the zero-• In order to account for the 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)

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

            = = 1 0.5 0.5 1 Cov and 0 0 0 0 A(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

TR Zero-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

TR Zero-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 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

TR Zero-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 0.29 ±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

TR Zero-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 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

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

DMN HCMN DAN FPCN Frontal Parietal Temporal Cingulate

DMN HCMN DAN FPCN Frontal Parietal Temporal Cingulate

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

Deshpande et al., NeuroImage (2010)

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

• SVC successfully learned patterns of functional connectivity

capable of predicting MDD from HC

capable of predicting MDD from HC

– Uncovered differences not discovered by t-test analysis

• Feature selection substantially improved the prediction

accuracy of SVC

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 DLPFC _DLPFCRight

Prediction of PCE status with functional/effective connectivity

Left amygdala Right amygdala

Medial PFC Left Parietal cortex Right Parietal cortex ACC PCC

Prediction of PCE status with functional/effective connectivity

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

Data Processing

Mean signal as reference

1. ROI definition by regular GLM

V1 and FFA

2. BH modulated cortical voxels

3. Voxel-wise signal latency

ROI signal extraction 4. Voxel-wise latency correction Reference Each voxel

Shift to find the best correlation

FFA original V1 original FFA corrected V1 corrected Shift Latency applied

5. Compare GCA results of uncorrected & corrected data

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 LGN Right LGN Visual cortex Left primary motor Right primary motor SMA

GCA Results

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