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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