Physiophysiological interaction (PPI), Granger causality (GCA), and dynamic causal moding (DCM) in resting-state fMRI. These slides are for a pre-conference educational workshop for the biennial conference on resting-state and brain connectivity.

1. Xin Di, PhD
New Jersey Institute of Technology

2.  Definition by Friston (1994):
“temporal correlations between spatially remote
neurophysiological events”
 Regular methods:
Correlation, coherence, PCA/ICA…
 A simple linear model
y  a  a  x  0 1
Connectivity is stable over time
No causality information

3.  Modulation of connectivity by a third region
Physiophysiological interaction (PPI) (Friston et al.,
1997)
 Causal influence (effective causality)
Granger causality analysis (GCA) (Goebel et al., 2003)
Dynamic causal modeling (DCM) (Friston et al., 2003)

4. X1
Y
X2
+ or x ?

5.  Linear relationship
      0 1 1 2 2 y a a x a x
 Model interaction between the two seeds
         0 1 1 2 2 3 1 2 y a a x a x a x x
        0 1 1 2 3 1 2 y a a x (a a x ) x
 The relationship between y and x2 is:
2 3 1 a  a  x

6. X1
Y
X2

7.  Voxel-wise general linear model (GLM)
• Defining two seeds
• Calculating PPI term
• Defining individual PPI GLM model for
• Group-level GLM analysis

8.  Defining seeds
• Two seeds
• Hypothesis-driven
• The two seeds should be
somehow connected
Two mains nodes of each resting-state
networks obtained from ICA results
Di and Biswal, 2013, in PLoS One

9. 3
2
1
0
ROI 1 ROI 2
0 50 100 150 200 250 300 350 400 450 500
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
PPI
0 50 100 150 200 250 300 350 400 450 500
4
3
2
1
0
-1
-2
-1
-2
-3
-4
0 50 100 150 200 250 300 350 400 450 500
-2.5
Deconvolve
Multiply
Convolve
Deconvolve

10. Statistical analysis: Design
An example design matrix
parameters
images
Sn(1) PPI.Y1
Sn(1) PPI.Y2
Sn(1) PPI.ppi
Sn(1) WM PCA 1
Sn(1) CSF PCA 1
Sn(1) R1
Sn(1) R2
Sn(1) R3
Sn(1) R4
Sn(1) R5
Sn(1) R6
Sn(1) constant
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parameter estimability
(gray   not uniquely specified)
Design description...
Main effects:
time series of two ROIs
Interaction
Covariates:
WM/CSF
Head motion

11. Group analysis: one sample t-test

12.  Modulatory interaction involves three regions
 Two regions need to be defined as seeds
(combination problem)
 Reliability of the interaction is lower than the
reliability of the two main effects of time series
 No causality information

13.  Based on prediction
 “whether one time series is useful in forecasting
another”
From wikipedia

14.  Granger Causality model (model 1)
t t t m t m t t m t m t y  a  a  y  a  y   a  y  b  x  b  x   b  x        ... ... 0 1 1 2 2 1 1 2 2
 Autoregressive model (model 2)
t t t m t m t y  a  a  y  a  y   a  y     ... 0 1 1 2 2
Statistical inference:
• F test: var(model 1)/var(model 2)
Whether including history of time series x can significantly
explain time series y?
• One sample t-test of each b parameters.
Causal effects on specific time points.

15.  Neuronal transmission delay: 50 – 100 ms
 Typical sampling rate (TR) of fMRI data: 1 – 3 s
 Model order can be determined by model
comparison (e.g. AIC)

16. Exploratory - seed-based analysis
Regions that are significantly influenced by the right
frontal-insular cortex (rFIC) (Zang et al., 2012)

17. ROI-based analysis
Granger causality among nodes of the
DMN (Uddin et al., 2008)

18.  Granger causality is based
on BOLD delays of 1 – 3 s,
while neuronal delays are
about 50 – 100 ms
 Hemodynamic response is
much longer (6s to peak)
 Hemodynamic response
varied across brain regions
 Cerebral blood flow →
vascular anatomy
HRF for different subjects and
different regions (Handwerker
et al., 2004)

19.  Granger causality is based
on BOLD delays of 1 – 3 s,
while neuronal delays are
about 50 – 100 ms
 Hemodynamic response is
much longer (6s to peak)
 Hemodynamic response
varied across brain regions
 Cerebral blood flow →
vascular anatomy
BOLD Granger Causality reflects
vascular anatomy (Webb et al.,
2013, in PLoS One)

20.  Granger causality analysis is based on predictability
of BOLD signals in 1 – 3 seconds order
 Regional variations of hemodynamic responses
may mislead Granger causal effects
 Granger causality results should be compared with
previous neurophysiology studies

21.  DCM was originally developed for fMRI data (Friston
et al., 2003)
 Generative model
 Making inference by comparing models
 Hypothesis-driven

22.  Differential equation model
1 11 1 12 2 1 11 1 z a z a z ... a z c u m m          
2 21 1 22 2 2 22 2 z a z a z ... a z c u m m          
 Matrix form of the model
Z  AZ CU

23. Fourier series at frequencies:
0.01, 0.02, 0.04, and 0.08 Hz

24. Design matrix
Z  AZ CU
Di & Biswal, 2013
C  U  c    x  c   
x
cos(0.01 2  ) sin(0.01 2 
)
c x c x
       
cos(0.02 2  ) sin(0.02 2 
)
c x c x
       
cos(0.04 2  ) sin(0.04 2 
)
sin
4
cos(0.08 2 ) sin(0.08 2 )
cos
4
sin
3
cos
3
sin
2
cos
2
sin
1
cos
1
c x c x
        


25.  DCM model
z  A z
 Stochastic DCM (Daunizeau et al., 2009)
 Deterministic DCM based on crossed spectra but
not time series (Friston et al., 2014)
Available in SPM12b

26.  Making inference by comparing models
 Hypothesis-driven
 Defining ROIs (up to 8)
 Constructing model space
 Model comparisons
 Parameter testing

27. All possible models: 46 = 4096
Hypothesis constrained models: 3 x 2 x 5 = 30
Model families Critical comments on dynamic causal modelling
(Lohmann et al., 2012)

28. Model family
Comparison
Model comparison Model parameters results

29.  DCM analysis is highly hypothesis-driven
 Appropriately defined model space is critical for
DCM analysis

30.  Higher order models can help to address questions
like modulation of connectivity and causality
 Each model has pros and cons
 Hypotheses are important
 Results should be grounded on anatomical
connections and neurophysiological results

31. Acknowledgement: our lab members
Dr. Bharat Biswal
Suril Gohel
Rui Yuan
Keerthana Karunakaran
…