Who drives whom?
Symmetric measures tell you two regions are coupled; the harder, more interesting question is directionality — does region X drive Y, or the reverse? Answering it means moving from functional to effective/directed connectivity, and the workhorse framework is Granger causality built on autoregressive models.
Granger causality and VAR models
The idea is prediction: X Granger-causes Y if the past of X improves the prediction of Y beyond Y's own past. You fit a vector autoregressive (VAR) model and compare residual variances with and without the other channel's history.
\mathbf{x}(t) = \sum_{k=1}^{p} \mathbf{A}_k\,\mathbf{x}(t-k) + \mathbf{e}(t)A VAR(p) model: current activity is a linear function of the past p lags plus innovation e(t). Its coefficients A_k carry all directionality.
A vector autoregressive (VAR) model predicts each channel's current value from a weighted combination of all channels' recent past. Those weights capture how signals drive one another over time — the foundation for directed connectivity.
- \mathbf{x}(t)
- The multichannel signal at the present time.
- \mathbf{A}_k
- The coefficient matrix for lag k — who predicts whom.
- p
- The model order — how many past steps are used.
- \mathbf{e}(t)
- The innovation — the part that can't be predicted.
F_{X\to Y} = \ln\frac{\operatorname{var}\!\left(e_Y \mid Y^{-}\right)}{\operatorname{var}\!\left(e_Y \mid Y^{-},\,X^{-}\right)}Granger causality X→Y: the log-ratio of Y's prediction-error variance without versus with X's past. Zero means no directed influence.
X Granger-causes Y if knowing X's past helps predict Y beyond Y's own past. This compares Y's prediction-error variance with and without X's history; if adding X shrinks the error, the log-ratio is positive — evidence of directed influence.
- F_{X\to Y}
- Granger causality from X to Y.
- \operatorname{var}(e_Y\mid Y^{-})
- Y's prediction-error variance using only Y's own past.
- \operatorname{var}(e_Y\mid Y^{-},X^{-})
- Y's error variance when X's past is also used.
- \ln
- The log-ratio — zero means no directed influence.
Spectral directed measures: PDC and DTF
Neural drive is frequency-specific, so we want directionality per band. Transform the VAR coefficients into the frequency domain and you obtain partial directed coherence (PDC) and the directed transfer function (DTF). PDC isolates the direct j\to i influence; DTF captures the total (including indirect) influence.
\mathrm{PDC}_{j\to i}(f) = \frac{\lvert \bar{A}_{ij}(f)\rvert}{\sqrt{\sum_{k}\lvert \bar{A}_{kj}(f)\rvert^{2}}}, \qquad \bar{A}(f)=I-\sum_{k=1}^{p}\mathbf{A}_k\,e^{-\,i\,2\pi f k}PDC normalizes the frequency-domain VAR coefficients by outflow from the source channel j, giving a directed, per-frequency measure.
Partial directed coherence takes the VAR model into the frequency domain and asks, at each frequency, how much of channel j's outflow goes specifically to channel i. Normalizing by j's total outflow gives a directed, frequency-resolved connectivity measure.
- \mathrm{PDC}_{j\to i}(f)
- The directed influence from j to i at frequency f.
- \bar{A}(f)
- The VAR coefficients transformed into the frequency domain.
- \bar{A}_{ij}(f)
- The j\to i entry of that frequency-domain matrix.
- \sqrt{\sum_k|\bar{A}_{kj}(f)|^{2}}
- The normalizer — channel j's total outflow.
Source-space connectivity and beamformers
The deeper fix for field spread is to stop computing connectivity on sensors at all. Reconstruct source time courses with a spatial filter — a beamformer — then estimate connectivity between sources. The LCMV beamformer does this in the time domain by minimizing output power subject to unit gain at the target location; its frequency-domain twin, the DICS beamformer, targets a specific band for oscillatory coupling.
\mathbf{W} = \bigl(\mathbf{L}^{\top}\Sigma^{-1}\mathbf{L}\bigr)^{-1}\mathbf{L}^{\top}\Sigma^{-1}The LCMV weights: minimum output power (via the data covariance Σ) under unit gain at the sources defined by the lead field L.
The beamformer weights that build one virtual sensor per source: they minimize total output power while guaranteeing unit gain toward the target source's lead field. Doing connectivity on these source-space signals avoids much of the sensor-level leakage.
- \mathbf{W}
- The beamformer weight matrix.
- \mathbf{L}
- The lead field — the source-to-sensor map.
- \Sigma
- The sensor data covariance.
- (\mathbf{L}^{\top}\Sigma^{-1}\mathbf{L})^{-1}
- The normalizer that enforces unit gain at the sources.
This helps but does not cure the problem: the inverse is ill-posed, so a beamformer's reconstructed sources still bleed into one another — residual source leakage. State-of-the-art pipelines therefore add leakage correction, such as pairwise orthogonalization of the source signals before computing amplitude coupling.
Open problems and honest caveats
Where is the frontier heading? Toward connectivity fused with dynamical-systems models rather than static graphs; toward multimodal grounding (concurrent recording or stimulation to test a claimed link); and toward BCI applications — network biomarkers to drive closed-loop neuromodulation, and connectivity features that generalize across sessions better than raw power.
Two halves, one pipeline
Notice that this track's two themes are inseparable. The spatial filters of guides 1–3 — Laplacians, CSP, ICA, beamformers — are precisely the tools that make the connectivity of guides 4–5 trustworthy, by un-mixing the sources before you ask how they talk. Spatial filtering and functional connectivity are not two topics; they are the front and back of one pipeline from raw channels to a defensible brain network.