Physiophysiological Interaction (PPI)

Granger Causality Analysis (GCA)and Dynamic Causal Modeling

(DCM) for resting-state fMRI

Xin Di, PhDNew Jersey Institute of Technology

Definition by Friston (1994): “temporal correlations between spatially remote neurophysiological events”

Regular methods:Correlation, coherence, PCA/ICA…

A simple linear model

Connectivity is stable over timeNo causality information

Functional connectivity

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Modulation of connectivity by a third regionPhysiophysiological 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)

Go beyond simple correlations

Modulatory interaction

X1

Y

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+ or x ?

Linear relationship

Model interaction between the two seeds

The relationship between y and x2 is:

Models for modulatory interaction

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Modulatory interaction

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Voxel-wise general linear model (GLM)

• Defining two seeds• Calculating PPI term• Defining individual PPI GLM model for• Group-level GLM analysis

Analysis of modulatory interaction

Defining seeds• Two seeds• Hypothesis-driven• The two seeds should be

somehow connected

Analysis of modulatory interaction

Two mains nodes of each resting-state networks obtained from ICA resultsDi and Biswal, 2013, in PLoS One

Analysis of modulatory interaction

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Deconvolve

Multiply

Convolve

PPI

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Analysis of modulatory interaction

Statistical analysis: Design

parameters

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parameter estimability

(gray not uniquely specified)

Design description...

Basis functions : hrfNumber of sessions : 1

Trials per session : 0 Interscan interval : 2.00 {s}

High pass Filter : Cutoff: 100 {s}Global calculation : mean voxel value

Grand mean scaling : session specificGlobal normalisation : None

An example design matrix

Main effects:time series of two ROIs

Interaction

Covariates:WM/CSFHead motion

Analysis of modulatory interaction

Group analysis: one sample t-test

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

A brief summary of PPI analysis

Based on prediction “whether one time series is useful in forecasting

another”

Granger causality

From wikipedia

Granger Causality model (model 1)

Autoregressive model (model 2)

Equations for Granger Causality

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tmtmttmtmttt xbxbxbyayayaay ...... 221122110

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.

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)

Model order

Implementation of Granger Causality

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

Exploratory - seed-based analysis

Implementation of Granger Causality

Granger causality among nodes of the DMN  (Uddin et al., 2008)

ROI-based analysis

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

Pitfalls of Granger Causality

HRF for different subjects and different regions (Handwerker et al., 2004)

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

Pitfalls of Granger Causality

BOLD Granger Causality reflects vascular anatomy (Webb et al., 2013, in PLoS One)

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

A brief summary of Granger causality

DCM was originally developed for fMRI data (Friston et al., 2003)

Generative model Making inference by comparing models Hypothesis-driven

Dynamic causal modeling (DCM)

Differential equation model

Matrix form of the model

Dynamic causal modeling (DCM)

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Modeling low frequency fluctuations

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

Modeling low frequency fluctuations

Design matrix

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Di & Biswal, 2013

DCM model

Stochastic DCM (Daunizeau et al., 2009) Deterministic DCM based on crossed spectra but

not time series (Friston et al., 2014) Available in SPM12b

Recent advances on resting-state DCM

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Making inference by comparing models Hypothesis-driven

Defining ROIs (up to 8) Constructing model space Model comparisons Parameter testing

DCM in practice

DCM model definition

All possible models: 46 = 4096Hypothesis constrained models: 3 x 2 x 5 = 30

Model families Critical comments on dynamic causal modelling (Lohmann et al., 2012)

DCM results

Model family Comparison

Model comparison Model parameters results

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

DCM analysis

A brief summary of DCM

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

Concluding remarks

Thank you for your attention

Acknowledgement: our lab membersDr. Bharat BiswalSuril GohelRui YuanKeerthana Karunakaran…