Source-linked AI summary
Phase Locking Value revisited: teaching new tricks to an old dog
Ricardo Bruña, Fernando Maestú, Ernesto Pereda
TL;DR
Brain-connectivity estimation becomes costly as the number of signals grows, motivating a faster formulation of PLV. The paper derives an efficient equivalent expression, relates PLV to coherence, and introduces iPLV and ciPLV for zero-lag-insensitive synchronization. The implementation achieves two orders of magnitude of speedup, while ciPLV preserves nonzero-lag connectivity under volume conduction.
Problem
PLV connectivity estimation becomes computationally costly because its cost grows with the square of the number of signals, which can number in the thousands.
Method
The paper reformulates PLV using an efficient expression related to coherence and derives the zero-lag-insensitive metrics iPLV and ciPLV.
Results
The reformulated PLV achieves a speedup by a factor of 100, while ciPLV removes zero-lag connectivity and retains nonzero-lag connectivity.
Takeaways & Limitations
The implementation supports substantially faster whole-brain PLV estimation, and ciPLV is effective for synchronization assessment in the presence of volume conduction or source leakage.
Takeaways & Limitations
The method assumes that real brain signals cannot travel instantaneously between different parts of the brain.
Abstract
from arXiv · showhide
Despite the increase in calculation power in the last decades, the estimation of brain connectivity is still a tedious task. The high computational cost of the algorithms escalates with the square of the number of signals evaluated, usually in the range of thousands. In this work we propose a re-formulation of a widely used algorithm that allows the estimation of whole brain connectivity in much smaller times. We start from the original implementation of Phase Locking Value (PLV) and re-formulated it in a highly computational efficient way. Besides, this formulation stresses its strong similarity with coherence, which we used to introduce two new metrics insensitive to zero lag synchronization, the imaginary part of PLV (iPLV) and its corrected counterpart (ciPLV). The new implementation of PLV avoids some highly CPU-expensive operations, and achieved a 100-fold speedup over the original algorithm. The new derived metrics were highly robust in the presence of volume conduction. ciPLV, in particular, proved capable of ignoring zero-lag connectivity, while correctly estimating nonzero-lag connectivity. Our implementation of PLV makes it possible to calculate whole-brain connectivity in much shorter times. The results of the simulations using ciPLV suggest that this metric is ideal to measure synchronization in the presence of volume conduction or source leakage effects.
I. Introduction
Functional connectivity estimates statistical dependence between activities at recording sites, while phase synchronization evaluates whether oscillatory phases evolve together. PLV is widely used but becomes computationally demanding for thousands of M/EEG signals and is sensitive to volume conduction and source leakage.
- Functional connectivity is defined as statistical dependence between activities at two or more recording sites above chance level.
- PLV evaluates the spread of phase differences, ranging from zero for no phase dependence to one for complete phase dependence.Narrower phase-difference distributions yield higher PLV values.
- The computational cost of PLV grows with the square of the number of signals, which can reach thousands in distributed M/EEG source-space analysis.
- The paper rewrites the original PLV formulation into an equivalent expression that reduces calculation time by up to a factor of 100.
- The reformulation relates PLV to coherence and motivates two zero-lag-insensitive measures, iPLV and ciPLV, for assessing direct phase synchronization from M/EEG.
II. Methods
The methods replace expensive phase extraction and exponentiation steps with an efficient formulation based on normalized analytical signals and vector algebra. The same PLV–coherence relationship supports iPLV and ciPLV, designed to reject zero-lag effects.
- A. Computational optimization: Instantaneous phase is extracted from a band-pass Hilbert analytical signal, assuming each signal contains one oscillator.Narrow-band filtering or a complex wavelet is used to obtain the analytical signal.
- A. Computational optimization: The reformulation avoids computationally expensive phase extraction and exponentiation operations by using properties of normalized analytical signals.
- A. Computational optimization: Vector algebra yields a computationally efficient PLV calculation with considerable speedup and low memory penalization.
- Coherence is described as a weighted average of unit phase vectors, whereas PLV averages phase relationships without joint-amplitude weighting.The weighting is determined by the joint amplitude of the signals at a given frequency.
- C. Zero-lag-insensitivity versions: The imaginary PLV and corrected imaginary PLV extend coherence-based zero-lag-insensitive measures to PLV.ciPLV corrects the lack of normalization in iPLV.
III. Results
The proposed PLV implementation substantially accelerates whole-brain connectivity estimation with only a small memory increase. Simulations and source-space analyses show that ciPLV preserves nonzero-lag synchronization while resisting volume-conduction effects.
- A. Speedup achieved using the proposed algorithm: 100-fold speedup was achieved over the optimized PLV implementation with only a marginal increase in memory use.The original implementation was slowest but most memory-efficient, while the optimized implementation achieved almost a 3-fold speedup.
- A. Speedup achieved using the proposed algorithm: 2459 time-series were processed into a full connectivity matrix in 113 seconds using around 8 GB of RAM.The corresponding optimized implementation was estimated to require 3 hours.
- B. Comparison of PLV and coherence: PLV and coherence increased similarly with coupling, while coherence slightly overestimated low-coupling synchronization.The overestimation was attributed to the smaller number of phases used for coherence.
- B. Comparison of PLV and coherence: Shorter coherence windows reduced low-coupling overestimation but lowered maximal coherence to 0.85 through frequency smoothing.
- With 10% volume conduction, PLV increased at lower couplings, whereas iPLV and ciPLV remained unaffected; ciPLV stayed almost identical to the no-mixing case.
- ciPLV continued to extract the correct synchronization value under conditions producing errors in PLV and iPLV.
IV. Discussion
The reformulated PLV enables much faster whole-brain connectivity estimation and clarifies PLV’s relationship to coherence. Its iPLV and ciPLV variants reduce sensitivity to zero-lag effects, although source leakage remains a limitation and parcellation sacrifices inter-area variability.
- Computational efficiency: Two orders of magnitude speedup makes source-to-source PLV feasible in minutes per subject and whole-study connectivity feasible within hours.The approach reduces computational burden while supporting whole-brain analyses.
- Computational efficiency: The algorithm’s practical whole-brain workflow uses representative parcel time courses, reducing thousands of source signals to hundreds but discarding inter-area variability.Parcellation imposes homogeneous parcels defined by one unique time series.
- PLV and coherence: PLV and coherence are closely related, with PLV better suited to synchronization over a whole band and coherence suited to fixed frequencies.The comparison used coupled chaotic systems and distinguished whole-band from fixed-frequency evaluation.
- Zero-lag-insensitive metrics: iPLV and ciPLV were introduced as PLV analogues of imaginary coherency to reduce sensitivity to volume conduction and source leakage.Both algorithms were tested under instantaneous linear mixing of signals from two systems.
- Zero-lag-insensitive metrics: ciPLV corrects iPLV’s phase-dependent weakness and was unbiased and insensitive to volume conduction or source-leakage effects while retaining nonzero-lag connectivity.iPLV approaches its maximum at a phase difference of π/2 and can fall near zero for small but consistently nonzero differences.
- Scope and limitations: No bivariate connectivity index is free from spurious connectivity caused by source leakage, so zero-lag-insensitive measures remain imperfect.The paper distinguishes removal of zero-lag interference from complete immunity to leakage-related false connectivity.
V. Conclusion
The paper reformulates PLV for much faster estimation and uses its similarity to coherence to define zero-lag-insensitive measures robust to volume conduction and source leakage.
- 100-fold speedup is achieved with the vector-optimized implementation, although comparable further gains are unlikely.The optimization is mathematical rather than hardware-specific, and the implementation can still be improved.
- A new PLV formulation replaces phase extraction and exponentiation with matrix multiplication, enabling implementation in any language.The provided Matlab code is machine-independent.
- The paper addresses the mathematical definition of PLV, while additional computational improvements may remain possible.The authors distinguish the reformulation from broader implementation optimization.
- The reformulation highlights PLV–coherence similarities and motivates iPLV and ciPLV as measures insensitive to zero-lag synchronization.These measures are derived from the real and imaginary parts of PLV and are analogous to corrected imaginary coherence.
- iPLV and ciPLV are robust against volume conduction and source leakage effects.Their construction follows the corrected imaginary part of coherence.
Appendix
The appendix compares three Matlab PLV implementations, progressing from nested loops to vectorized and matrix-based calculations. The proposed implementation evaluates source pairs simultaneously while avoiding several expensive operations.
- Inputs: All three implementations receive Hilbert analytic signals organized by signals, samples, and trials.The appendix presents Matlab code for evaluating their behavior.
- Implementations: The first implementation evaluates each signal pair independently with two nested loops, making it memory-conservative but slow.Only two signals are evaluated per interaction.
- Implementations: The second implementation compares every signal with all others using vectorization, achieving a 2.5-fold speedup over the original with slightly higher memory use.It is described as vectorized and memory efficient.
- Proposed algorithm: The proposed implementation normalizes analytic data and computes all source-pair PLVs through matrix multiplication.It avoids angle and exponential functions and removes the loop over signals.