Source-linked AI summary
Critical comments on EEG sensor space dynamical connectivity analysis
Frederik van de Steen, Luca Faes, Esin Karahan, Jitkomut Songsiri, Pedro Antonio Valdes Sosa, Daniele Marinazzo
TL;DR
MVAR-based causal connectivity analysis on EEG sensor-space time series does not allow interpretation in terms of anatomically interacting brain areas because sensor signals mix sources. The paper uses theoretical considerations and simulations to show that volume conduction can produce spurious connectivity, including bidirectional DTF interactions and nonzero DTF despite vanishing phase differences.
Problem
EEG time series are mixtures of activated sources, so sensor-space analysis cannot identify which source's past activity is involved or interpret connectivity as anatomically interacting brain areas.
Method
The paper focuses on MVAR-based connectivity analysis and examines how mixing of source activity affects causal connectivity measures, including DTF.
Results
Volume conduction can produce spurious connectivity: DTF of mixed time series can show bidirectional interaction, and vanishing phase difference does not imply vanishing DTF.
Takeaways & Limitations
Sensor-space causal connectivity results should not be interpreted as interactions between anatomically localized brain sources.
Takeaways & Limitations
The discussed conditions without measurement noise are arguably unrealistic for real EEG data, and many issues remain for future research.
Abstract
from arXiv · showhide
Many different analysis techniques have been developed and applied to EEG recordings that allow one to investigate how different brain areas interact. One particular class of methods, based on the linear parametric representation of multiple interacting time series, is widely used to study causal connectivity in the brain. However, the results obtained by these methods should be interpreted with great care. The goal of this paper is to show, both theoretically and using simulations, that results obtained by applying causal connectivity measures on the sensor (scalp) time series do not allow interpretation in terms of interacting brain sources. This is because 1) the channel locations cannot be seen as an approximation of a source's anatomical location and 2) spurious connectivity can occur between sensors. Although many measures of causal connectivity derived from EEG sensor time series are affected by the latter, here we will focus on the well-known time domain index of Granger causality (GC) and on the frequency domain directed transfer function (DTF). Using the state-space framework and designing two simulation studies we show that mixing effects caused by volume conduction can lead to spurious connections, detected either by time domain GC or by DTF. Therefore, GC/DTF causal connectivity measures should be computed at the source level, or derived within analysis frameworks that model the effects of volume conduction. Since mixing effects can also occur in the source space, it is advised to combine source space analysis with connectivity measures that are robust to mixing.
1. Introduction
MVAR-based connectivity measures are widely used to study directed interactions in neural time series, but applying them directly to EEG sensors complicates anatomical interpretation and can produce spurious connections. The paper examines these issues for GC and DTF using theoretical analysis and simulations.
- Scope: The paper focuses on MVAR-based causal connectivity analysis applied to EEG data rather than debating MVAR or GC methods generally.
- Motivation: MVAR-based methods quantify directed causal influence in neural time series using measures such as Granger causality and directed transfer function.They are widely used because of their flexibility and relatively easy implementation.
- Central problem: Sensor-space causal connectivity cannot be interpreted as interactions between anatomically located brain sources because sensor locations do not identify source anatomy.
- Central problem: Volume conduction mixes source activity across sensors, allowing direct sensor-space application of MVAR-based measures to detect spurious connections.
- Approach: The study investigates volume-conduction effects on time-domain GC and frequency-domain DTF using theoretical considerations and two simulations.The simulations include a simple toy model and more realistic generated EEG data.
- Approach: Three analysis strategies are outlined: direct sensor analysis, source reconstruction followed by connectivity analysis, and state-space analysis.
2. Causal Connectivity Analysis of EEG Time Series
The paper models interacting cortical sources and their projection to EEG sensors, then defines GC and DTF for directed connectivity. Because sensors contain mixtures of sources and sources are unobserved, sensor-level analysis can differ from source-level connectivity, motivating reconstruction or state-space approaches.
- Source and sensor modelling: EEG scalp potentials arise from synchronized neural activity and are modeled as measurements of cortical source time series.Volume conduction instantly mixes source activations in the measured scalp potentials.
- Source and sensor modelling: Source dynamics are represented with an MVAR model, where lagged coefficient matrices quantify influences of past source states on current states.The innovation process represents white-noise inputs to the source dynamics.
- Source and sensor modelling: A non-sparse lead-field matrix linearly projects sources to sensors, so each scalp channel generally contains information from multiple sources.
- Causal connectivity measures: Time-domain GC compares prediction-error variances from full and reduced MVAR models to quantify directed influence from a driver to a target.GC is positive when including the driver reduces prediction error variance.
- Causal connectivity measures: DTF uses the transfer function to quantify normalized frequency-domain influence from one innovation process to another observed process.It represents total directed influence, including direct and indirect connections, and complements GC rather than sharing its exact interpretation.
- EEG connectivity strategies: Direct sensor analysis cannot identify which source's past drives another because sensor series are mixtures of activated sources.Parameters and resulting GC or DTF values can therefore differ from those of the interacting sources.
- EEG connectivity strategies: A two-stage strategy reconstructs neural sources before applying connectivity analysis, whereas a state-space strategy combines source dynamics with the measurement process.The reconstruction approach is expected to recover true connectivity only when source reconstruction and MVAR estimation are sufficiently accurate.
3. Effect of Volume Conduction on Sensor-Space Causal Connectivity
Volume conduction mixes source activity into sensor recordings, so sensor-space GC and DTF can report connectivity that does not reflect interactions among brain sources. Theoretical analysis and simulations show that source-level or volume-conduction-aware analysis is needed for interpretation.
- Interpretive boundary: Sensor locations cannot generally identify the anatomical sources contributing to a channel’s past values or another source’s prediction.Channel time series are mixtures of activated sources unless restrictive prior assumptions hold.
- Theoretical mechanism: Even without interacting sources, significant sensor interactions can arise from standard time-domain or frequency-domain connectivity analysis.The paper specifically discusses TGC and DTF as affected measures.
- Theoretical mechanism: A non-diagonal leadfield mixes source signals into sensor time series, even when the sources themselves are modeled as non-interacting.The resulting sensor representations combine past values from multiple sources with sensor-specific weights.
- Theoretical mechanism: Different sensor mixtures let one sensor’s past improve prediction of another, producing spurious connectivity measured by both TGC and DTF.The effect appears as non-diagonal coefficient or transfer-function matrices in sensor-space analyses.
- Special cases: Non-interacting sensors can occur under narrower conditions, including proportional source power spectra, no measurement noise, and no more sensors than sources.With more sensors than sources, the resulting coefficient matrix can still be non-diagonal depending on the leadfield.
- Simulation I: For two non-interacting sources, source TGC is zero in both directions, whereas mixed sensors show TGC12=0.1064 and TGC21=0.1417.The mixed-source analysis also showed bidirectional interaction in the DTF profiles.
- Simulation II: In the realistic three-source simulation, sensor-space DTF showed non-zero connections from O2 to AF8, O2 to T8, and AF8 to T8.Source-reconstructed and state-space trends were almost identical to the originally generated time series.
4. Discussion
The discussion concludes that sensor-space causal connectivity cannot be interpreted as anatomical interaction because sensor locations do not proxy source locations and volume-conduction mixing creates spurious connectivity. Source-space analysis can recover causal structure under some conditions, but residual mixing and reconstruction errors require caution and mixing-robust measures.
- Main conclusion: EEG channel locations cannot generally be treated as approximations of the anatomical locations of contributing sources.Source depth and orientation affect how activity projects to scalp channels, so nearby channels need not represent nearby sources.
- Volume conduction: Volume-conduction mixing can produce spurious sensor connections even when sources do not causally interact.Differential mixing can create apparent connections without additive measurement noise, and sensor analysis can show significant connectivity when sources of interest are non-interacting.
- Volume conduction: A zero phase difference does not guarantee zero DTF after volume-conduction mixing.In the first simulation, source DTF was zero whereas mixed time series showed nonzero, bidirectional DTF; related effects also affect time-domain measures such as TGC.
- Source-space analysis: Source-space DTF and GC can recover the original causal structure under suitable reconstruction conditions, but inverse solutions leave spatial leakage and mixing.Head-model errors, uncertain source locations, source-selection constraints, and incomplete separation of nearby sources limit anatomical interpretation.
- Main conclusion: Sensor-space GC and DTF do not support interpretation in terms of anatomically interacting brain areas.The conclusion applies to causal connectivity analysis performed in either the time or frequency domain.
- Practical implications: The authors advise combining source-space analysis with connectivity measures robust to volume conduction.Mixing effects can remain after source reconstruction, especially for nearby sources, and future source-space connectivity analysis still requires methodological development.
Appendix A
The appendix derives how linear mixing affects source and sensor interactions under different leadfield dimensions. With no measurement noise and scalar source dynamics, non-interacting sources generally remain non-interacting at sensors for square or fat mixing, but skinny mixing can create spurious sensor interactions.
- Setup: The source model is linearly mixed into sensor observations through a leadfield matrix, with source reconstruction analyzed using its pseudo-inverse.The derivation assumes the observations lie in the range of the leadfield and allows a nullspace component in the reconstructed sources.
- Leadfield cases: Assuming identical scalar autoregressive dynamics across sources, the analysis separates square, fat, and skinny leadfield matrices.These cases determine whether the mixing transformation preserves or introduces interaction structure.
- Square leadfield: For square leadfields, the nullspace component vanishes and source reconstruction uses the inverse leadfield.The appendix explicitly obtains Z(t) = 0 and L† = L^-1 in this case.
- Results: With no measurement noise and Ad = adI, non-interacting sources produce non-interacting sensors when the leadfield is fat or square.This conclusion follows from the appendix’s derivations for the corresponding leadfield geometries.
- Results: When the leadfield is skinny, spurious interaction can occur at the sensors even when the sources do not interact.Thus, the dimensional relationship between sensors and sources is a condition governing whether mixing creates apparent connectivity.
Appendix B
The appendix examines how linear superposition changes the phase difference between two signals. It identifies special cases where phase is preserved and shows by simulation that volume conduction can otherwise shift the observed phase.
- Definition: Phase difference is defined as the phase of the cross-spectrum between two time series.The appendix formulates the phase difference between the mixed signals from their cross-spectrum.
- Linear mixing: The cross-spectrum of mixed signals is derived from the source cross-spectra and the coefficients of their linear superposition.This provides the analytical basis for assessing phase changes under mixing.
- Conditions for preservation: Linear superposition preserves phase difference when the leadfield is diagonal or anti-diagonal, or when the source phase difference is 0 or π.The latter condition makes the imaginary part of the source cross-spectrum zero.
- Conclusion: Under conditions outside the preservation cases, volume conduction can change the observed phase difference.The appendix illustrates this effect by plotting mixed-signal phase difference against k.
- Simulation: Two signals with a source phase difference of π/3 were mixed while the off-diagonal leadfield terms varied as L12 = L11k and L21 = L22k.The mixed cross-spectrum phase was calculated for k = 0, ..., 2.