EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition

Abstract

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

Keywords: EEG cross‐spectrum; Riemannian manifold; causality; matrix regression; mediation analysis.

FIGURE 1

Main variables in a Mediation Analysis. The independent variable ($x$) might influence the dependent variable ($y$), either directly or through the mediator ($\mathbf{M}$). Here, $\mathbf{A}$, $\mathbf{B}$ and $c$ are path variables describing the relationships between the variables.

FIGURE 2

Illustration of the data transformation. Starting with a 3D object, after performing Riemannian Geometry a 2D object containing all the frequencies for each subject is obtained.

FIGURE 3

Illustration of the $\mathbf{A}$ regression coefficient in the mediation model.

FIGURE 4

Illustration of the $\mathbf{B}$ regression coefficient in the mediation model.

FIGURE 5

The parameter $\mathbf{A}$ is chosen to generate the simulation data. Here, j and k represent the number of elements of the simulation figure.

FIGURE 6

Probability of the coefficients $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{A}\circ \mathbf{B}$ for the four simulations. (a) Simulation 1: Here the outcome depends only on the stimulus. (b) Simulation 2: The mediator and the outcome depend on the stimulus, but the outcome does not depend on the mediator. (c) Simulation 3: The output depends on the mediator and the treatment, but the mediator does not depend on the treatment. (d) Simulation 4: The mediator depends on the input and the outcome of the mediator.

FIGURE 7

Power curves for the mediation effect $\sum _i\sum _j{a}_{\mathit{ij}}{b}_{\mathit{ij}}$.

FIGURE 8

Kernel distribution of the t values resulting from the test.

FIGURE 9

(a) Estimated coefficients for the BNS study and (b) significance of coefficients from Barbados malnutrition study for a significance level of $-{\log }_{10}\left(\text{threshold}p‐\text{value}\right)$.

FIGURE 10

The distribution and significance of the mediation effect are seen in the combination of electrodes conforming to the 10–20 system (less the last electrode due to the average reference). Only the low diagonal is shown due to the matrix being symmetric.

FIGURE 11

Connections in the scalp were found to be mediators between the malnutrition effect and cognition. The alpha band is the most prominent, showing 5 long (front‐occipital) and short connections. In contrast, the Beta and Delta bands show only two.