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Granger causality: theory and applications to neuroscience data

Mukesh Dhamala, Ph. D.Department of Physics & Astronomy

Neuroscience InstituteGeorgia State University (GSU), Atlanta

GA 30303, USAwww.phy-astr.gsu.edu

Neuroscience Institute

ØGranger Causality (GC): Time and frequency domain versions

ØEstimation approaches: Parametric (P) and Nonparametric (NP)

Validation/demonstration with synthetic data

P and NP comparisons/limitations

Ø Applications to: LFPs, EEG, iEEG and rfMRI

Overview

Granger causality: X Y

X

t

Y

?

Granger causality: X Y

X

t

Y

?

v Are there oscillatory features in X and Y separately? Power

v Are X and Y related in oscillatory behaviors? Coherence

v Is there information flow between X and Y? Granger Causality

v In multiple time series, information flow patterns? Conditional or Multivariate Granger Causality

!"→$ = '("→$ ) *)�

�Oscillatory Activities in Time Series and Spectral Measures

X

tY

ØOscillatory features in X and Y? Power

ØInterdependence between X and Y? Coherence

ØInformation flow between X and Y? Granger Causality

ØIn multiple time series, network flow patterns? Granger Causality (Pairwise & Conditional)

Signal flow

Brain

XY

Time Series, Oscillations and Spectral Measures

Methods Spectral Measures

Parametric(Vector Autoregressive,

State Space Modeling)

Power, Coherence, Granger Causality(Geweke, 1982; 1984; Ding et. al., 2006;

Barnett and Seth, 2014; Solo, 2015)

Nonparametric(Fourier &

Wavelet Methods)

Power, Coherence, Granger causality(Dhamala, et. al., 2008a; 2008b)

Spectral interdependency:

(Geweke, 1982; 1984; Dhamala, 2014)Granger causality: X Y

X

t

Y

?

v Are there oscillatory features in X and Y separately? Power

v Are X and Y related in oscillatory behaviors? Coherence

v Is there information flow between X and Y? Granger Causality

v In multiple time series, information flow patterns? Conditional or Multivariate Granger Causality

!"→$ = '("→$ ) *)�

�Oscillatory Activities in Time Series and Spectral Measures

Estimation of Spectral Measures

Granger causality: a subset of spectral interdependency measures

Spectral interdependency:

(Geweke, 1982; Hosoya, 1991)

C. W. J. Granger

2003 Nobel Laureate

in Economics

(1934 – 2009)

Who is Granger?

ØFor two simultaneously measured time series, the first series is called causal to the second series if the second series can be predicted better by using the knowledge of the first one (Wiener, 1956).

Wiener (1894-1964)

On Causality and the Brain

In the study of brain waves we may be able to obtain electroencephalograms more or less corresponding to electrical activity in different parts of the brain. Here the study of the coefficients of causality running both ways and of their analogues for sets of more than two functions (two processes) may be useful in determining what part of the brain is driving what other part of the brain in its normal activity.

Norbert Wiener, The Theory of Prediction, 1956

Concept of Granger Causality

Granger Causality MeasuresGranger Causality Measures

ØGiven:

ØLinear Prediction Models:

ØGranger Causality (1969):

If var(ex|y) < var(ex), thenY is said to exert a causal influence on X.

Model 1: xn = a1xn-1 + …+amxn-m + …+ex

Model 2: xn = b1xn-1 + …+bkxn-k+…+c1yn-1+…+ckyn-k+…+ex|y

Clive J. Granger2003 Nobel Laureate Economics

Granger Causality: Statistical Definition

Granger causality from Y to X:

= log (var(ex)/var(ex|y))

(time-domain Granger causality (GC))

Granger Causality: Representation

ØGeweke (1982) introduced a spectral representation of time-domain Granger causality:

where is Granger causality spectra.

ØStatistical meaning: spectral decomposition

total power = intrinsic power + causal power

I = log (total power/intrinsic power)

John GewekeUniv of Iowa

Granger Causality: Spectral Version (often referred as Granger-Geweke causality (GGC))

ØGiven:

ØMultivariate Autoregressive Models:

ØSpectral density matrix:

Granger Causality: Parametric Estimation

ØSpectral matrix:

ØPower: diagonal terms

ØCoherence spectra: normalized magnitudeof off-diagonal terms

ØGranger causality:

(H and needed)

Granger Causality: Parametric Estimation (cont’d)

ØFourier and wavelet methods are first used to

estimate the spectral density (Slm(f) = <Xl(f)Xm(f)*)>):

Ø We need H and to calculate Granger causality

Nonparametric Methods

ØSpectral density matrix factorization (Wiener and Masani 1957; Wilson 1972):

where

ØDerivation of H and :

; such that ØGranger causality:

(frequency domain)

(time domain)

(Dhamala, et. al., PRL 2008; NeuroImage, 2008)

Granger Causality: Nonparametric Estimation

Demonstrations with Simulated Time Series

ØModel System:

ØCausality Spectra:

Power and Coherence spectra have peaks at 40 Hz (considering fs = 200 Hz).

Y Z

Fourier Transform-Based Granger Causality

X2 X1

Wavelet Transform-Based Granger Causality

Conditional (Multivariate) Granger causality

and its demonstrations with simulated data

Conditional causality:

(Geweke,1984)Y indirectly influences X

FY®X|Z = FYZ®X – Fz®x= FYZ*®X*

Conditional Causality: direct or indirect?

Y indirectly influences X(Dhamala, et. al., NeuroImage, 2008)

Example 1: Conditional Granger causality

(Wen, et. al., Phil. Trans. R. Soc. A, 2013)

Example 2: Conditional Granger causality

Y X

fk = (0.122, 0.391, 0.342)

(Dhamala, et. al., NeuroImage, 2008)

ØUncertainty in model order selection

ØSharp oscillatory spectral features often not captured

Parametric Approach: difficulty and limitation

ØEffect of data length

short data: lower estimates, but correct directions

Nonparametric Approach: limitation

Granger Causality in neuroscience

(Friston, Moran, Seth, Current Opinion in Neurobiology, 2013)

2014: State-space approach to Granger causality

(Barnett and Seth, 2014)

0 20 40 60

-15

-10

-5

0

5

10

15

Pow

er (1

)

0 20 40 60

0

1

2

3

4G

C: 1

2|3

0 20 40 60-0.5

0

0.5

GC

: 13|

2

0 20 40 60-0.5

0

0.5

GC

: 21|

3

0 20 40 60

-15

-10

-5

0

5

10

15

Pow

er (2

)

0 20 40 60-0.5

0

0.5G

C: 2

3|1

0 20 40 60frequency

-0.5

0

0.5

GC

: 31|

2

0 20 40 60frequency

-0.5

0

0.5

GC

: 32|

1Power and Granger Causality (GC) Spectra (blue: parametric with p = 3, green: nonparametric)

0 20 40 60frequency

-15

-10

-5

0

5

10

15

Pow

er (3

)

GC spectra from VAR and SS methods

0 10 20 30 40 50 60frequency

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

GC

:12

|3

VAR estimates dispersionTrueSS, p=3VAR, p=3

Parametric (VAR) and Nonparametric MethodsParametric: VAR and SS

(Dhamala, et al., NeuroImage, 2018; includes codes)

1 2 3

- PNAS article by Stokes and Purdon, 2017- Commentaries by (i) Barnett et al., (ii) Faes et al., and (iii) reply by Stokes and Purdon, 2017

Excellent agreement between VAR and SS!

Granger-Geweke causality (GGC) problematic? No!

0 10 20 30 40 50 60frequency

0

0.5

1

1.5

2

2.5

3

3.5

GC: 1

2

(A) Granger Causality spectra

1 (50Hz-Transmitter) 2 (10Hz-Receiver)1 (50Hz-Transmitter) 2 (30Hz-Receiver)

0 10 20 30 40 50 60frequency

0

0.5

1

1.5

2

2.5

3

3.5

-log(1

-Coh

(1,2))

(B) Total Interdependence spectra

1 (50Hz-Transmitter) - 2 (10Hz-Receiver)1 (50Hz-Transmitter) - 2 (30Hz-Receiver)

0 10 20 30 40 50 60frequency

-10

-5

0

5

10

15

20

Powe

r

(C) Power spectra

Total Power (50Hz-Transmitter)Intrinsic Power (10Hz-Receiver)Causal Power (10Hz-Receiver)Intrinsic Power (30Hz-Receiver)Causal Power (30Hz-Receiver)

0 10 20 30 40 50 60frequency

0

0.5

1

1.5

2

2.5

3

3.5

GC: 1

2

(D) GC spectra at different coupling and intrinsic noise

1 2, coupling = 0.51 2, noise variance = 0.5

0 0.2 0.4 0.6 0.8 1noise variance

0

1

2

3

4

5

6

Integ

rated

GC:

1 2

(E) Effect of intrinsic noise on GC

-1 -0.5 0 0.5 1coupling strength

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Integ

rated

GC:

1 2

(F) Effect of coupling on GC

Integrated GC

v GGC consistent with other interdependency measures (not problematic at all!)v GGC definition allows for intrinsic and causal power estimationv GGC depends on intrinsic noise and coupling strength

1 2

(Dhamala, et al., NeuroImage, 2018; includes codes)

Y X

fk = (0.122, 0.342, 0.391)

(Dhamala, et. al., NeuroImage, 2008)

v The nonparametric approach works better in recovering complex spectral features!0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

frequency

-10

-8

-6

-4

-2

0

2

log(Gr

anger C

ausalit

y): Y

X

Y(AR + sinusoids at 0.122,0.342,0.391) X (AR)

Nonparametric ApproachParametric Approach - SS, p = 76Parametric Approach - VAR, p = 76

Parametric and Nonparametric Methods on Sinusoidal Driving

Application to Local Field Potentials (Monkeys)

(Bressler, et. al. 1993; Brovelli, et. al. 2004; Dhamala, et ., 2008)

Experiments conducted by Dr. Nakamura at NIMH (Bressler, et.al. 1993; Brovelli, et. al. 2004)

ØSubject depressed a hand lever

ØReleased for Go-stimuli

Experiment: Sensorimotor Task with Go/No-Go

Ø-100 to 20 ms = Network Analysis Segment

during which hand pressure on the lever

was maintained

Network Analysis Segment of LFPs

(Brovelli, et. al. 2004)

Sensorimotor Beta (14 – 30 Hz) Network

Fourier-Based Granger Causality

Wavelet-Based Granger Causality

ØS1àM1 consistent with the known role of S1 for a sustained motor output.

Nonparametric Granger Causality Spectra

ØNeural Substrate of Motor Control:

(from Gazzaniga, et. al.(2002))

Sensorimotor Granger Causality Network Graph

ØConsistent with the anatomical connections.

Anatomical Connections (Felleman and Van Essen, 1991)

S1à7a is not direct, but mediated by 7b

Application to Epicranial EEG² M. F. Pagnotta, M. Dhamala, G. Plomp, “Benchmarking nonparametric Granger causality:

Robustness against downsampling and influence of spectral decomposition parameters”, NeuroImage 183, 478-494 (2018).

² M. F. Pagnotta, M. Dhamala, G. Plomp, “Assessing the performance of Granger-Gewekecausality: Benchmark dataset and stimulation framework”, Data in Brief 21, 833-851 (2018). (Data and Matlab codes included)

Mattia Pagnota(Univ of Fribourg)

Gijs Plomp(Univ of Fribourg)

(Plomp, et. al., 2014a; 2014b)

(Pagnotta, et al., NeuroImage 183, 478-494 (2018))

Stimulation, Electrode Locations, SEPs, Power and GGC

Application to iEEG of human epilepsy patients² B. Adhikari, C. Epstein, M. Dhamala, “Localizing epileptic seizures with Granger

causality”, Physical Review E 88, 030701 (Rapid Communications) (2013).² C. Epstein, B. Adhikari, R. Gross, J. Willie, M. Dhamala, “High-frequency Granger

causality in analysis of intracranial EEG and in surgical decision-making”, Epilepsia 55, 2038 (2014).

Bhim Adhikari(Physics,Georgia State Univ)

Charles Epstein(Neurophysiologist,Emory Univ)

Robert Gross(Neurosurgeon,Emory Univ)

Jon Willie(Neurosurgeon,Emory Univ)

Where and when does it start?Can GGC help to localize the onset?

iEEG data and Seizure Origin

High-frequency network activity (up to 250 Hz) with GGC can assist in the localization of epileptic seizures.

High-Frequency Network Activity in One Patient

Application to fMRI

² S. Bajaj, B. M. Adhikari, K. J. Friston, M. Dhamala, “Briding the gap: dynamic causal

modling and Granger causality analysis of resting state functional magnetic resonance

imaging”. Brain Connectivity (2016).

² K. Dhakal, M. Norgaard, B. M. Adhikari, K. S. Yun, M. Dhamala, “Higher Node Activity

with Less Functional Connectivity During Musical Improvisation”. Brain Connectivity

(2019).

Gran

ger c

ausa

lity

DCM

(Bajaj et al., 2016 ) In-degree (GC) from all RIOs to R1, R2, R3, and R4

0.1 0.2 0.3 0.4 0.5

Conne

ctivity

streng

th (DC

M) fro

m all R

OIs to

R 1, R2,R 3,an

d R4

0.2

0.3

0.4

0.5

0.6

r = 0.470p = 0.0002

GC and DCM to rfMRI

GC to task-based fMRIfMRI activation Analysis

N = 20, p < 0.0005, k > 20

Directional connectivityGranger Causality

Anatomical basis: fiber tracts enhanced for expert

musicians (N=20) compared to controls (N = 20)

(Dhakal, et. al., Brain Connectivity (2019); Dhakal, et al., in preparation)

Kiran Dhakal

ØGranger causality techniques are useful to test hypotheses about information flow or to explore information flow from time series data.

ØThere are two estimation approaches: parametric (modeling based) and nonparametric (Fourier- and wavelet-transforms based), which can be complementary to each other.

ØGranger causality methods are applicable to a variety of neuroscience data: LFP, iEEG, fMRI, fNIR, EEG, MEG.

Summary

Students and Postdoc Postdoc: Bhim Adhikari (now at Maryland)Graduate Student: Sahil Bajaj (now at Arizona), Kiran Dhakal, Kristy Yun

CollaboratorsDrexel: Hualou LiangEmory: Charles Epstein, Robert Gross, Jon WillieFAU: Steven BresslerFribourg, Switzerland: Matia Pagnotta, Gijs PlompGSU: Martin NorgaardIIS, India: Govindan RangarajanUF: Mingzhou DingUCL, London: Karl J. Friston

FundingGSU: B&B Faculty Grants (08, 011, 013, 016-17)GSU/GaTech: Center for Advanced Brain Imaging Grant (2010)US VA (2008 – 2011)US NSF Career Award (2010 – 2016)

Acknowledgements: Team member and Funding