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.
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)
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
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
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)
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 AZ CU
24. Design matrix
Z AZ CU
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)
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