A Quantitative Comparison of Two Methods for Higher-Order EEG Microstate Syntax Analysis

Entropy rate (ER) and sample entropy (SE) are two metrics that have been used to quantify the syntactic complexity of electroencephalography (EEG) microstate sequences. We here present a theoretical and numerical comparison of these two metrics and apply them to a resting-state EEG dataset from individuals with Alzheimer's disease (AD) and a control group. We first derive theoretical ER and SE estimates for first-order discrete Markov processes, providing a null hypothesis for statistical testing of higher-order syntax properties. Under the first-order syntax null hypothesis, we find a close mathematical relationship between both metrics that can be expressed by the microstate transition probability matrix. An inequality is derived that shows ER to be an upper bound to SE under the Markov approximation. We quantify accuracy and precision of the theoretical ER and SE estimates on EEG microstate sequences from the healthy control group. We then show that ER and SE identify significant higher-order syntax properties in microstate sequences from the control and AD groups. We investigate continuous and jump microstate sequences. In the former, each time point is labelled with the best matching microstate label, and in the latter, duplicate labels are removed, exclusively retaining transitions between non-identical microstates. Group comparison demonstrates that continuous microstate sequences from the AD group have lower entropy values (ER, SE), whereas jump sequences from the AD group have higher entropy values compared to control. Finally, we introduce a new syntax metric that normalizes ER and SE values with respect to their first-order syntax levels, to assess differences that only depend on syntax order. This metric revealed no differences between control and AD groups for either continuous or jump microstate sequences. This study provides further insights into higher-order microstate syntax and how it can be quantified with respect to the underlying first-order syntax. Similarities and differences between ER and SE as syntax metrics are highlighted and exemplified on experimental data. Our results show that (i) EEG microstate sequences from control and AD subjects show higher-order syntax properties across the tested syntax levels, (ii) continuous and jump sequences from control and AD groups are syntactically different, and (iii) differences between the control and AD groups disappear when higher-order syntax properties are normalized to the group-specific Markov level.