Minimum‐Phase Property of the Hemodynamic Response Function, and Implications for Granger Causality in <scp>fMRI</scp>
Bibliographic record
Abstract
Granger causality (GC) is widely used in neuroimaging to estimate directed statistical dependence between brain regions using time series of brain activity. A known problem is that fMRI measures brain activity indirectly via the blood-oxygen-level-dependent (BOLD) signal, which can distort GC estimates by introducing different time-to-peak responses across brain regions. However, how these distortions affect the validity of inferred connections is not fully understood. Previous studies have shown that false positives are not introduced if the haemodynamic response function (HRF) is minimum-phase; but whether the HRF is actually minimum-phase has remained contentious. Here, we address this issue by studying the transfer functions of three realistic biophysical models. We find that the minimum-phase condition is met for a wide range of physiologically plausible parameter values. Therefore, statistical testing of GC can be viable even if the HRF varies across brain regions, with the following two limitations. First, the minimum-phase condition is violated for parameter combinations that generate an initial dip in the HRF. Second, slow sampling of the BOLD signal (seconds) compared to the timescales of neural signal propagation (milliseconds) may still introduce spurious GC inferences. Beyond GC analysis, the closed-form expressions for the transfer functions of these popular HRF models are valuable for modeling fMRI time series since they balance mathematical tractability with biological plausibility.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.045 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".