[Paper Review] Complex tensor factorisation with PARAFAC2 for the estimation of brain connectivity from the EEG

Loukianos Spyrou, Mario A. Parra+1 more|Edinburgh Research Explorer|May 2, 2017

Tensor decomposition and applications22 references3 citations

TL;DR

This paper proposes complex PARAFAC2 tensor factorisation to estimate brain connectivity from EEG data by jointly modelling spatial, spectral, and complex trial profiles. It demonstrates superior performance over traditional PARAFAC in estimating connectivity, especially in noisy conditions and in detecting altered connectivity in mild cognitive impairment and Alzheimer’s patients, validating its ability to reveal biologically plausible coupling patterns not captured by standard methods.

ABSTRACT

Objective: The coupling between neuronal populations and its magnitude have been shown to be informative for various clinical applications. One method to estimate brain connectivity is with electroencephalography (EEG) from which the cross-spectrum between different sensor locations is derived. We wish to test the efficacy of tensor factorisation in the estimation of brain connectivity. Methods: Complex tensor factorisation based on PARAFAC2 is used to decompose the EEG into scalp components described by the spatial, spectral, and complex trial profiles. An EEG model in the complex domain was derived that shows the suitability of PARAFAC2. A connectivity metric was also derived on the complex trial profiles of the extracted components. Results: Results on a benchmark EEG dataset confirmed that PARAFAC2 can estimate connectivity better than traditional tensor analysis such as PARAFAC within a range of signal-to-noise ratios. The analysis of EEG from patients with mild cognitive impairment or Alzheimer's disease showed that PARAFAC2 identifies loss of brain connectivity better than traditional approaches and agreeing with prior pathological knowledge. Conclusion: The complex PARAFAC2 algorithm is suitable for EEG connectivity estimation since it allows to extract meaningful coupled sources and provides better estimates than complex PARAFAC. Significance: A new paradigm that employs complex tensor factorisation has demonstrated to be successful in identifying brain connectivity and the location of couples sources for both a benchmark and a real-world EEG dataset. This can enable future applications and has the potential to solve some the issues that deteriorate the performance of traditional connectivity metrics.

Motivation & Objective

• To develop a tensor factorisation framework that captures coupled neural sources in EEG data by jointly modelling spatial, spectral, and temporal dynamics.
• To address limitations of traditional connectivity metrics—such as volume conduction, noise, and reference electrode effects—by leveraging complex-valued tensor decomposition.
• To derive a connectivity metric based on the complex trial profiles of extracted components to estimate functional coupling between brain regions.
• To validate the method on both benchmark and real-world EEG datasets, including patients with mild cognitive impairment and Alzheimer’s disease.
• To demonstrate that complex PARAFAC2 can better estimate connectivity than standard PARAFAC, especially under low signal-to-noise ratios.

Proposed method

• Formulate an EEG model in the complex domain that accounts for phase shifts and non-stationarities across trials.
• Apply complex PARAFAC2 decomposition to factorise the EEG cross-spectrum tensor into spatial, spectral, and complex trial components.
• Use the complex trial profiles as a basis for computing a novel connectivity metric that quantifies coupling strength between components.
• Incorporate phase-synchronization principles by focusing on the imaginary part of coherence within the tensor framework to reduce volume conduction effects.
• Ensure model validity by deriving theoretical and empirical support for PARAFAC2's suitability over PARAFAC in the complex domain.
• Validate the method using a benchmark dataset with known connectivity structure and real EEG data from patients with MCI and Alzheimer’s disease.

Experimental results

Research questions

• RQ1 Can complex PARAFAC2 outperform standard PARAFAC in estimating brain connectivity from EEG data under varying signal-to-noise ratios?
• RQ2 Does the proposed method effectively detect altered connectivity patterns in patients with mild cognitive impairment and Alzheimer’s disease compared to healthy controls?
• RQ3 Can the complex trial profiles derived from PARAFAC2 provide a more accurate and robust estimate of neural coupling than traditional connectivity metrics?
• RQ4 To what extent does the method preserve the spatial and spectral characteristics of underlying neural sources while estimating connectivity?
• RQ5 Is the connectivity metric derived from complex PARAFAC2 components consistent with known pathological and physiological mechanisms in neurodegenerative disorders?

Key findings

• PARAFAC2 achieved higher connectivity estimation accuracy (CONN) than PARAFAC across all signal-to-noise ratios, with a clear performance gain at low to moderate SNR.
• At a phase-relevant ratio (PR) of 0.6, PARAFAC2 reached a performance ceiling with CONN ≈ 0.8 for R=2 and R=8, outperforming prior methods that required prior frequency band knowledge.
• In real EEG data, the power-adjusted connectivity metric showed significantly higher connectivity in MCI patients compared to age-matched controls, consistent with the Scaffolding Theory of Cognitive Aging and Decline.
• The MCI group exhibited increased synchronisation in alpha, beta, and theta bands despite lower power, indicating compensatory neural mechanisms, while the MCI-FAD group did not show this pattern.
• The binding task induced more distributed activations than the shape task, supporting the role of long-range connectivity in complex cognitive processing, as captured by the method.
• The method successfully identified the true source locations with higher accuracy (LOC) than prior approaches, even without prior knowledge of frequency bands or SNR thresholds.