The Statistics of EEG\xa0Unipolar References: Derivations and Properties

Shiang Hu,Pedro A Valdes-Sosa,Maria L Bringas-Vega,Yun Qin,Dezhong Yao

doi:10.1007/s10548-019-00706-y

Shiang Hu, Pedro A Valdes-Sosa + Show 3 more

Open Access

https://doi.org/10.1007/s10548-019-00706-y

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Abstract

In this brief communication, which complements the EEG reference review (Yao et al. in Brain Topogr, 2019), we provide the mathematical derivations that show: (1) any EEG reference admits the general form of a linear transformation of the ideal multichannel EEG potentials with reference to infinity; (2) the average reference (AR), the reference electrode standardization technique (REST), and its regularized version (rREST) are solving the linear inverse problems that can be derived from both the maximum likelihood estimate (MLE) and the Bayesian theory; however, REST is based on more informative prior/constraint of volume conduction than that of AR; (3) we show for the first time that REST is also a unipolar reference (UR), allowing us to define a general family of URs with unified notations; (4) some notable properties of URs are ‘no memory’, ‘rank deficient by 1’, and ‘orthogonal projector centering’; (5) we also point out here, for the first time, that rREST provides the optimal interpolating function that can be used when the reference channel is missing or the ‘bad’ channels are rejected. The derivations and properties imply that: (a) any two URs can transform to each other and referencing with URs multiple times will not accumulate artifacts; (b) whatever URs the EEG data was previously transformed with, the minimum norm solution to the reference problem will be REST and AR with and without modeling volume conduction, respectively; (c) the MLE and the Bayesian theory show the theoretical optimality of REST. The advantages and limitations of AR and REST are discussed to guide readers for their proper use.

Highlights

This brief communication provides the detailed mathemati‐ cal demonstrations as well as some new findings that com‐ plement the EEG reference review (Yao et al 2019)
We propose the general form of the EEG reference problem, demonstrate that reference electrode standardiza‐ tion technique (REST) is a special type of unipolar reference (UR), generalize the family of pos‐ sible URs, summarize the notable properties of them and derive the average reference (AR) and REST from the maximum likelihood estimate (MLE) and the Bayesian theory (Table 1)
The general form of the EEG reference electrodes problem is understood as a linear transformation to the potentials referenced at infinity, that maybe either the URs or non-URs e.g. the bipolar recordings and the scalp Laplacian; the common structure of URs is recognized with unified notations; it is the first time to show the interpolation function in solving the reference problem and demonstrate REST as an UR

Summary

Introduction

This brief communication provides the detailed mathemati‐ cal demonstrations as well as some new findings that com‐ plement the EEG reference review (Yao et al 2019). We propose the general form of the EEG reference problem, demonstrate that REST is a special type of UR, generalize the family of pos‐ sible URs, summarize the notable properties of them and derive the AR and REST from the MLE and the Bayesian theory (Table 1).

Results

Conclusion