What can fool it#

Each of these is demonstrated by a script that asserts the number, not merely described.

Unequal signal-to-noise ratio fabricates direction#

Two completely independent oscillators differing only in noise level:

noise A	noise B	dPTE[A→B]
0.2	0.2	0.492
0.2	0.4	0.684
0.2	0.8	0.797

The noisier channel is harder to predict from its own past, so the cleaner one looks like a driver. Real recordings always differ in SNR between channels.

A surrogate test removes it cleanly: for such a pair the null mean lands at 0.739 against an observed 0.736, and it is correctly rejected — while genuine coupling survives at p = 0.005. In one worked case the artefact even has a higher raw dPTE (0.719) than the real coupling (0.664), so ranking pairs by raw value picks the wrong one.

→ two_node_coupling.py, significance_testing.py

Indirect paths outrank direct ones#

In a chain a → b → c → d where only the one-hop links exist, with equal noise on every channel, band-passed, and time-reversal-differenced:

separation	mean dPTE	real edge?
1 hop	0.571	yes
2 hops	0.622	no
3 hops	0.656	no

dPTE rises monotonically with path length, because a longer path accumulates more phase lag, and PTE detects a longer lag more easily. Thresholding by dPTE returns close to the inverse of the true network. If noise also builds up along the chain, as it does when each node adds its own, the effect is stronger still, but that part is the signal-to-noise artefact, and time reversal removes it.

The lag effect is the fundamental limit of any bivariate measure, and more data makes it worse: in a neural-mass simulation, recall saturated at 60 s while precision fell from 0.50 to 0.36 by 200 s as indirect paths became detectable.

There is therefore an optimum recording length for network reconstruction, and it is not “as much as possible”.

→ epoched_analysis.py, neural_mass_network.py

Symmetric coupling is invisible — by design#

Strongly but symmetrically coupled channels sit at exactly 0.500. That is the correct answer, not a miss: dPTE measures net direction, and a balanced bidirectional link has none. Score it against raw connectivity and you will penalise it for being right.

→ kuramoto_network.py

On a real connectome, structural recovery largely fails#

Against The Virtual Brain’s directed 76-region tract-tracing connectome, pyPTE separates one-way edges from unconnected pairs at AUC 0.62 — above chance, but far from the 0.96–1.00 the same estimator reaches on sparse synthetic networks. At 27% density nearly every region pair is joined by a short indirect path, which is the previous caveat operating at whole-brain scale.

Scale bites twice. With m regions there are m * (m - 1) / 2 channel pairs — 2,850 here — so the strictest FDR threshold falls below 2e-5, while N surrogates cannot produce a p-value below 1 / (N + 1). Edgewise surrogate testing at this scale needs thousands of surrogates; cluster_permutation is the practical alternative.

→ tvb_connectome.py

Detection is model-dependent#

There is no universal sample requirement. Phase oscillators resolve in a few thousand samples; realistic Jansen-Rit columns needed 60 s or more, and were strongly sensitive to the transmission delay. Effect size depends on the dynamics, the SNR and the lag, which is why group-level testing is the robust answer rather than a rule of thumb.

Other things to know#

Very strong coupling reduces detectability. Oscillators synchronise and their phases become redundant, so dPTE peaks at intermediate coupling.
Common drivers create apparent links: if a drives both b and c, then b and c will look connected.
Volume conduction in sensor-space M/EEG produces zero-lag mixing that no phase-based measure can separate from genuine coupling. Prefer source space.
Cost is O(m^2) in channels. A 100-channel montage over 8,000 samples takes about 0.8 s; 100 surrogates on it takes about 1.4 minutes.