From single sources to networks
So far we sharpened one source at a time. Functional connectivity asks the next question: which regions are statistically coupled, and how? The vocabulary matters — functional, effective and directed connectivity are distinct: functional connectivity is a symmetric statistical dependence, effective connectivity implies a causal influence, and directed connectivity estimates its direction. This guide covers the symmetric, undirected measures; the next covers direction.
Coherence and the cross-spectrum
The foundational measure is coherence, the frequency-resolved correlation between two channels. It is built from the cross-spectral density S_{xy}(f) normalized by each channel's own spectrum, giving a value in [0,1] per frequency.
C_{xy}(f) = \frac{\lvert S_{xy}(f)\rvert^{2}}{S_{xx}(f)\,S_{yy}(f)}Magnitude-squared coherence: the fraction of variance at frequency f that two channels share.
The same magnitude-squared coherence as before: the fraction of variance two channels share at frequency f, on a 0-to-1 scale. Here it's used as a functional-connectivity measure between brain signals.
- C_{xy}(f)
- Coherence between channels x and y at frequency f.
- |S_{xy}(f)|^{2}
- The shared (cross) power, squared.
- S_{xx}(f),\,S_{yy}(f)
- Each channel's own power.
- [0,1]
- The range — from unrelated (0) to perfectly shared (1).
High magnitude-squared coherence means the two signals keep a consistent amplitude-and-phase relationship across segments. It is intuitive and cheap — and, as we are about to see, dangerously easy to inflate.
Phase synchrony: the PLV
Coherence blends amplitude and phase. Often you want only phase consistency, independent of amplitude. The phase-locking value (PLV) does exactly that: extract the instantaneous phase difference (via the Hilbert or wavelet transform) and measure how tightly it clusters on the unit circle.
\mathrm{PLV}(f) = \left\lvert \frac{1}{N}\sum_{n=1}^{N} e^{\,i\,\Delta\phi_n(f)} \right\rvertPLV: the length of the mean phase-difference vector over N trials/segments — 1 is perfect locking, 0 is uniform phase.
PLV is the length of the average phase-difference arrow across trials or segments. Near 1 means the two signals keep a steady phase relationship; near 0 means their phase difference is essentially random.
- \mathrm{PLV}(f)
- The phase-locking value at frequency f.
- \Delta\phi_n(f)
- The phase difference on segment n at that frequency.
- e^{i\Delta\phi_n}
- A unit arrow for that phase difference.
- \left|\tfrac{1}{N}\sum\right|
- The length of the average over N segments.
The confound that ruins naive connectivity
The insight that rescues the field: a true interaction between two separate regions takes time — axonal conduction and synapses impose a nonzero delay, hence a nonzero phase lag. Instantaneous, zero-lag coupling is the fingerprint of the artifact, not of physiology.
Confound-robust measures
The fix is to keep only the part of the cross-spectrum that carries a phase lag. Imaginary coherency discards the real (zero-lag) part and keeps the imaginary part; the phase-lag index (PLI, and its weighted cousin wPLI) uses only the sign of the imaginary cross-spectrum, making it robust to source strength and to the worst of the field-spread bias.
\mathrm{PLI}(f) = \bigl\lvert \langle \operatorname{sign}\!\left(\operatorname{Im}[S_{xy}(f)]\right) \rangle \bigr\rvertPLI: the consistency of the sign of the phase lag — insensitive to zero-lag (volume-conduction) coupling by construction.
A key problem: volume conduction makes distant sensors look synchronized at zero lag even with no real coupling. PLI dodges this by looking only at whether the phase lag is consistently positive or negative — zero-lag leakage contributes nothing.
- \mathrm{PLI}(f)
- The phase-lag index at frequency f.
- \operatorname{Im}[S_{xy}(f)]
- Imaginary part of the cross-spectrum — nonzero only for a true lag.
- \operatorname{sign}(\cdot)
- Keeps only the direction (sign) of the lag.
- \langle\cdot\rangle
- Averages over trials or segments.
Sensor space still lies — go to source space
Even imaginary-part measures are computed on mixed sensor signals, so they remain biased by field spread. The principled next step is to reconstruct sources first and estimate connectivity there, applying explicit leakage correction. That, together with directed measures, is the subject of the final guide.