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["Friston, 2011","Friston et al., 1997","Friston et al., 2003","Bullmore & Sporns, 2009","O'Reilly et al., 2012","McLaren et al., 2012","Watts & Strogatz, 1998"]
Brain connectivity analysis goes beyond mapping where activation occurs to ask how brain regions interact. This requires choosing among fundamentally different analytical frameworks: functional connectivity (statistical associations), effective connectivity (directed causal influences), and network topology (graph-theoretic properties). Each framework answers different questions and makes different assumptions.
A competent programmer without neuroscience training would not know the critical distinction between functional and effective connectivity, would likely confuse correlation with causation in brain networks, and would not appreciate why motion artifacts are particularly devastating for connectivity analyses. This skill encodes the domain judgment required to select and correctly implement brain connectivity methods.
When to Use This Skill
Choosing a connectivity method for task-based or resting-state fMRI
Evaluating whether Granger causality is appropriate for fMRI
Reviewing or troubleshooting existing connectivity analyses
Research Planning Protocol
Before executing the domain-specific steps below, you MUST:
State the research question — What specific question is this analysis/paradigm addressing?
Justify the method choice — Why is this approach appropriate? What alternatives were considered?
Declare expected outcomes — What results would support vs. refute the hypothesis?
Note assumptions and limitations — What does this method assume? Where could it mislead?
Present the plan to the user and WAIT for confirmation before proceeding.
For detailed methodology guidance, see the research-literacy skill.
⚠️ Verification Notice
This skill was generated by AI from academic literature. All parameters, thresholds, and citations require independent verification before use in research. If you find errors, please open an issue.
Functional vs. Effective Connectivity
This is the most fundamental distinction in brain connectivity analysis (Friston, 2011):
Property
Functional Connectivity
Effective Connectivity
Definition
Statistical dependency between time series
Directed causal influence of one region on another
Directionality
Undirected
Directed
Causality
Cannot infer causality
Attempts to infer causal structure
Model requirement
Model-free (correlations)
Model-based (requires explicit neural model)
Hypothesis type
Exploratory or confirmatory
Strictly confirmatory
Primary methods
Correlation, partial correlation, ICA
DCM, PPI, structural equation modeling
Source
Friston, 2011
Friston, 2011; Friston et al., 2003
Domain warning: Functional connectivity (correlation) does not imply direct anatomical or causal connection. Two regions can show high functional connectivity because they share a common input, not because they directly influence each other (Friston, 2011).
Method Selection Decision Tree
What is your connectivity question?
|
+-- "Do regions A and B covary in activity?"
| --> Functional connectivity (correlation, partial correlation)
|
+-- "Does the coupling between A and B change with task context?"
| --> PPI (Friston et al., 1997; O'Reilly et al., 2012)
|
+-- "Does region A causally influence region B, and does this
| change with experimental manipulation?"
| --> DCM (Friston et al., 2003)
| Requires: strong prior hypotheses, limited model space
|
+-- "What is the network topology (hubs, modules, efficiency)?"
| --> Graph theory (Bullmore & Sporns, 2009)
|
+-- "Does activity in A temporally precede and predict B?"
--> Granger causality (with STRONG caveats for fMRI;
see Granger causality section below)
Functional Connectivity Methods
Seed-Based Correlation
Extract mean time series from a seed ROI
Compute Pearson correlation with all other voxels (or ROIs)
Apply Fisher z-transformation: z = 0.5 * ln((1+r)/(1-r)) before group statistics (Fisher, 1921)
Enter z-maps into second-level group analysis
Critical preprocessing:
Apply stringent motion correction: FD threshold < 0.2 mm for scrubbing or censoring (Power et al., 2014)
Band-pass filter to 0.01-0.1 Hz for resting-state (Biswal et al., 1995; but see Niazy et al., 2011 for caveats)
Regress out motion parameters (24-parameter model), WM/CSF signals (aCompCor; Behzadi et al., 2007)
Partial Correlation
Partial correlation between regions A and B controls for shared variance with all other regions in the model. This reduces the influence of common inputs but requires careful regularization when the number of regions is large relative to the number of time points (Smith et al., 2011).
Use regularized partial correlation (e.g., L1-penalized precision matrix) when the number of ROIs exceeds ~20% of the number of time points (Varoquaux et al., 2010)
Independent Component Analysis (ICA)
ICA decomposes the whole-brain fMRI signal into spatially independent components, each representing a network (Beckmann & Smith, 2004).
Model order (number of components): 20-25 for major networks; 70-100 for fine-grained parcellation (Smith et al., 2009)
Group ICA: Use temporal concatenation or dual regression (Beckmann et al., 2009; Nickerson et al., 2017)
Artifact identification: Classify components as signal or noise based on spatial pattern, power spectrum, and temporal properties (Griffanti et al., 2017)
Psychophysiological Interactions (PPI)
PPI tests whether the functional coupling between a seed region and other brain areas changes as a function of psychological context (e.g., task condition; Friston et al., 1997).
PPI Model Specification
The PPI GLM includes three regressors:
Physiological regressor: Deconvolved time series from the seed region (the neural signal estimate)
Psychological regressor: The task regressor (e.g., condition A vs. condition B)
Interaction term: Element-wise product of the physiological and psychological regressors, reconvolved with the HRF
The interaction term is the regressor of interest. A significant PPI effect means that functional coupling with the seed region differs between task conditions (O'Reilly et al., 2012).
Generalized PPI (gPPI)
Standard PPI tests one psychological contrast at a time. Generalized PPI (gPPI) includes all task conditions simultaneously (McLaren et al., 2012):
Create separate PPI terms for each condition
Include all main-effect task regressors and the seed time series
More sensitive than standard PPI because it properly accounts for all conditions
Recommended over standard PPI for designs with more than two conditions (McLaren et al., 2012)
PPI Practical Considerations
Parameter
Recommendation
Source
Seed ROI definition
Functional (from activation map) or anatomical atlas; 6-10 mm sphere typical
O'Reilly et al., 2012
Deconvolution
Required for interaction term (SPM default); extract neural signal before computing interaction
Gitelman et al., 2003
Minimum events per condition
20+ per condition for stable PPI estimates
O'Reilly et al., 2012
Multiple seed regions
Run separate PPI analyses per seed, then correct for multiple comparisons
O'Reilly et al., 2012
Dynamic Causal Modeling (DCM)
DCM models the causal architecture of brain regions using a biophysical generative model of neural dynamics and hemodynamic coupling (Friston et al., 2003).
DCM Model Specification
A DCM is defined by three matrices:
Matrix
Function
Interpretation
A (intrinsic)
Fixed connections between regions (always on)
Baseline coupling strength
B (modulatory)
Condition-dependent modulation of connections
How task context changes coupling (the key experimental question)
C (driving input)
Which regions receive direct task input
Where external stimuli enter the network
DCM Requirements and Constraints
Requirement
Guideline
Source
Number of regions
3-8 regions maximum
Stephan et al., 2010
Prior hypotheses
Strong prior hypotheses about network architecture required
Stephan et al., 2010
Model space size
Keep total models < 20-30 for reliable BMS
Stephan et al., 2009
Task design
Event-related or block designs with clear experimental manipulation
Friston et al., 2003
Time series extraction
From first eigenvariate of activated voxels within ROI
Stephan et al., 2010
Data quality
Strong task activation in all modeled regions
Stephan et al., 2010
Bayesian Model Selection (BMS)
After estimating all candidate models, use BMS to identify the winning model (Stephan et al., 2009):
Fixed-effects BMS (FFX): Assumes all subjects have the same model. Use when sample is homogeneous
Random-effects BMS (RFX): Allows different subjects to have different optimal models. Use for heterogeneous populations (Stephan et al., 2009)
Family-level inference: Group models into families sharing a feature (e.g., all models with a particular connection), then compare families. More robust than individual model comparison (Penny et al., 2010)
Report exceedance probability (probability that a model/family is more likely than all others) and protected exceedance probability (corrected for chance; Rigoux et al., 2014)
DCM Pitfalls
Too many models: Exhaustive model comparison with > 30 models leads to unreliable BMS (Stephan et al., 2010)
No activation in modeled regions: DCM requires that included regions show significant task-related activation. Including silent regions produces unreliable estimates
Confusing DCM with Granger causality: DCM models neural dynamics, not temporal precedence in BOLD signals
Granger Causality
Granger causality tests whether past values of time series X improve prediction of time series Y beyond Y's own past (Granger, 1969).
Severe Limitations for fMRI
Limitation
Explanation
Source
Hemodynamic confound
The HRF varies across brain regions by 1-2 seconds. A region with a faster HRF will appear to "Granger-cause" a region with a slower HRF, regardless of true neural timing
David et al., 2008; Seth et al., 2013
Low temporal resolution
fMRI TR (0.5-3 seconds) is far slower than neural dynamics (milliseconds), aliasing true causal structure
Seth et al., 2013
Downsampling artifacts
Subsampling a continuous process can reverse the apparent causal direction
Seth et al., 2013
Domain warning: Granger causality applied to fMRI BOLD signals is widely considered unreliable for inferring neural causal direction due to hemodynamic variability (David et al., 2008; Seth et al., 2013). If causal inference is needed, use DCM (which explicitly models the hemodynamic delay) or consider EEG/MEG (which has millisecond temporal resolution).
Graph Theory for Brain Networks
Graph theory characterizes the topological organization of brain networks (Bullmore & Sporns, 2009).
Node Definition
Method
Description
Example Atlases
Source
Anatomical parcellation
Predefined atlas regions
AAL (116 regions), Harvard-Oxford
Tzourio-Mazoyer et al., 2002
Functional parcellation
Data-driven boundaries
Schaefer (100-1000 parcels), Gordon (333 parcels)
Schaefer et al., 2018; Gordon et al., 2016
Voxel-level
Each voxel is a node
~100,000 nodes
Computationally expensive; rarely used for full graph metrics
Recommendation: Use functional parcellation (Schaefer or Gordon atlases) for most analyses. Functional parcels better respect the true boundaries of functional regions than anatomical atlases (Schaefer et al., 2018).
Edge Definition
Method
Description
Threshold Approach
Source
Correlation
Pearson r between time series
Proportional (top 10-20% of connections) or absolute (r > 0.2-0.3)
Bullmore & Sporns, 2009
Partial correlation
Controls for indirect connections
Same thresholding options
Varoquaux et al., 2010
Coherence
Frequency-domain coupling
Magnitude squared coherence > threshold
Sun et al., 2004
Domain warning: Thresholding strategy dramatically affects graph metrics. Always report results across a range of thresholds or use proportional thresholding to control for differences in overall connectivity strength across subjects (van den Heuvel et al., 2017).
Key Graph Metrics
Metric
Meaning
Normalization
Source
Clustering coefficient
Proportion of a node's neighbors that are also connected to each other
Normalization requirement: Raw graph metrics are meaningless without comparison to null models. Always normalize against random networks (preserving degree distribution) or lattice networks (Watts & Strogatz, 1998; Rubinov & Sporns, 2010). Generate at least 1,000 random networks for stable null distributions.
Common Pitfalls
Confusing functional connectivity with effective connectivity: Correlation between two regions does not imply direct or causal connection. Two regions may correlate because they receive common input from a third region (Friston, 2011)
Motion artifacts in connectivity: Head motion creates spurious short-distance positive correlations and spurious long-distance negative correlations, mimicking real connectivity patterns (Power et al., 2012). Use stringent motion thresholds (FD < 0.2 mm; Power et al., 2014) and aggressive confound regression
Not Fisher z-transforming correlations: Pearson r values are bounded and non-normally distributed. Always apply Fisher z-transformation before parametric group statistics (Fisher, 1921)
Arbitrary graph thresholds: Reporting graph metrics at a single threshold without sensitivity analysis across thresholds can produce unreliable or threshold-dependent conclusions (van den Heuvel et al., 2017)
Too many DCM models: Exhaustive model comparison across > 30 models leads to unreliable Bayesian model selection. Constrain the model space based on prior hypotheses (Stephan et al., 2010)
Using Granger causality with fMRI: Regional HRF variability can reverse apparent causal direction in BOLD signals (David et al., 2008; Seth et al., 2013)
Confounding connectivity with coactivation: Two regions that are both activated by a task may show correlation due to shared task-driven variance, not intrinsic neural coupling. Regress out task effects before computing resting-state-style connectivity on task data (Cole et al., 2019)
Ignoring global signal effects: Global signal regression introduces artifactual anticorrelations. If using it, acknowledge this limitation (Murphy & Fox, 2017)
Minimum Reporting Checklist
Connectivity type (functional, effective, or graph-theoretic) and justification
ROI/node definition method, atlas used, number of regions
Time series extraction method (mean, first eigenvariate, PCA)
Preprocessing for connectivity (band-pass filter frequencies, confound model, motion threshold)
Edge definition (correlation type, thresholding method and values)
For PPI: seed region definition, deconvolution method, conditions in PPI model
For DCM: model space (A, B, C matrices for all models), BMS method (FFX/RFX), model evidence
For graph theory: normalization method, number of null networks, range of thresholds tested
Fisher z-transformation applied (yes/no)
Motion exclusion criteria and mean FD of included subjects
Software package and version
References
Beckmann, C. F., & Smith, S. M. (2004). Probabilistic independent component analysis for functional magnetic resonance imaging. IEEE Transactions on Medical Imaging, 23(2), 137-152.
Beckmann, C. F., Mackay, C. E., Filippini, N., & Smith, S. M. (2009). Group comparison of resting-state fMRI data using multi-subject ICA and dual regression. NeuroImage, 47(S1), S148.
Behzadi, Y., Restom, K., Liau, J., & Liu, T. T. (2007). A component based noise correction method (CompCor) for BOLD and perfusion based fMRI. NeuroImage, 37(1), 90-101.
Biswal, B., Yetkin, F. Z., Haughton, V. M., & Hyde, J. S. (1995). Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magnetic Resonance in Medicine, 34(4), 537-541.
Bullmore, E., & Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3), 186-198.
Cole, M. W., Ito, T., Bassett, D. S., & Schultz, D. H. (2019). Activity flow over resting-state networks shapes cognitive task activations. Nature Neuroscience, 19, 1718-1726.
David, O., Guillemain, I., Saillet, S., et al. (2008). Identifying neural drivers with functional MRI: An electrophysiological validation. PLoS Biology, 6(12), e315.
Fisher, R. A. (1921). On the "probable error" of a coefficient of correlation deduced from a small sample. Metron, 1, 3-32.
Friston, K. J. (2011). Functional and effective connectivity: A review. Brain Connectivity, 1(1), 13-36.
Friston, K. J., Buechel, C., Fink, G. R., et al. (1997). Psychophysiological and modulatory interactions in neuroimaging. NeuroImage, 6(3), 218-229.
Friston, K. J., Harrison, L., & Penny, W. (2003). Dynamic causal modelling. NeuroImage, 19(4), 1273-1302.
Gitelman, D. R., Penny, W. D., Ashburner, J., & Friston, K. J. (2003). Modeling regional and psychophysiologic interactions in fMRI: The importance of hemodynamic deconvolution. NeuroImage, 19(1), 200-207.
Gordon, E. M., Laumann, T. O., Adeyemo, B., et al. (2016). Generation and evaluation of a cortical area parcellation from resting-state correlations. Cerebral Cortex, 26(1), 288-303.
Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37(3), 424-438.
Griffanti, L., Douaud, G., Bijsterbosch, J., et al. (2017). Hand classification of fMRI ICA noise components. NeuroImage, 154, 188-205.
Humphries, M. D., & Gurney, K. (2008). Network "small-world-ness": A quantitative method for determining canonical network equivalence. PLoS ONE, 3(4), e0002051.
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McLaren, D. G., Ries, M. L., Xu, G., & Johnson, S. C. (2012). A generalized form of context-dependent psychophysiological interactions (gPPI). NeuroImage, 61(4), 1277-1286.
Murphy, K., & Fox, M. D. (2017). Towards a consensus regarding global signal regression for resting state functional connectivity MRI. NeuroImage, 154, 169-173.
Newman, M. E. J. (2006). Modularity and community structure in networks. PNAS, 103(23), 8577-8582.
Nickerson, L. D., Smith, S. M., Ongur, D., & Beckmann, C. F. (2017). Using dual regression to investigate network shape and amplitude in functional connectivity analyses. Frontiers in Neuroscience, 11, 115.
Niazy, R. K., Xie, J., Miller, K., Beckmann, C. F., & Smith, S. M. (2011). Spectral characteristics of resting state networks. Progress in Brain Research, 193, 259-276.
O'Reilly, J. X., Woolrich, M. W., Behrens, T. E. J., et al. (2012). Tools of the trade: Psychophysiological interactions and functional connectivity. Social Cognitive and Affective Neuroscience, 7(5), 604-609.
Penny, W. D., Stephan, K. E., Daunizeau, J., et al. (2010). Comparing families of dynamic causal models. PLoS Computational Biology, 6(3), e1000709.
Power, J. D., Barnes, K. A., Snyder, A. Z., et al. (2012). Spurious but systematic correlations in functional connectivity MRI networks arise from subject motion. NeuroImage, 59(3), 2142-2154.
Power, J. D., Mitra, A., Laumann, T. O., et al. (2014). Methods to detect, characterize, and remove motion artifact in resting state fMRI. NeuroImage, 84, 320-341.
Rigoux, L., Stephan, K. E., Friston, K. J., & Daunizeau, J. (2014). Bayesian model selection for group studies revisited. NeuroImage, 84, 971-985.
Rubinov, M., & Sporns, O. (2010). Complex network measures of brain connectivity: Uses and interpretations. NeuroImage, 52(3), 1059-1069.
Schaefer, A., Kong, R., Gordon, E. M., et al. (2018). Local-global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI. Cerebral Cortex, 28(9), 3095-3114.
Seth, A. K., Barrett, A. B., & Barnett, L. (2015). Granger causality analysis in neuroscience and neuroimaging. Journal of Neuroscience, 35(8), 3293-3297.
Smith, S. M., Fox, P. T., Miller, K. L., et al. (2009). Correspondence of the brain's functional architecture during activation and rest. PNAS, 106(31), 13040-13045.
Smith, S. M., Miller, K. L., Salimi-Khorshidi, G., et al. (2011). Network modelling methods for FMRI. NeuroImage, 54(2), 875-891.
Stephan, K. E., Penny, W. D., Daunizeau, J., et al. (2009). Bayesian model selection for group studies. NeuroImage, 46(4), 1004-1017.
Stephan, K. E., Penny, W. D., Moran, R. J., et al. (2010). Ten simple rules for dynamic causal modelling. NeuroImage, 49(4), 3099-3109.
Sun, F. T., Miller, L. M., & D'Esposito, M. (2004). Measuring interregional functional connectivity using coherence and partial coherence analyses of fMRI data. NeuroImage, 21(2), 647-658.
Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., et al. (2002). Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. NeuroImage, 15(1), 273-289.
van den Heuvel, M. P., de Lange, S. C., Zalesky, A., et al. (2017). Proportional thresholding in resting-state fMRI functional connectivity networks and consequences for patient-control connectome studies. NeuroImage, 159, 437-449.
Varoquaux, G., Gramfort, A., Poline, J. B., & Thirion, B. (2010). Brain covariance selection: Better individual functional connectivity models using population prior. NIPS, 2334-2342.
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of "small-world" networks. Nature, 393(6684), 440-442.
See references/ for detailed method implementation guides and parameter lookup tables.
