State-transition approaches, including EEG microstate analysis and related fMRI methods such as hidden Markov models (HMMs) and co-activation pattern (CAP) analysis, provide widely used tools for coarse-graining neural dynamics into a small set of quasi-stable states. Its utility has been demonstrated across resting-state and task paradigms, with broad applications ranging from cognitive neuroscience to candidate biomarkers for psychiatric and neurological disorders. A fundamental limitation remains, however: nearly all downstream temporal measures are conditional on the template maps defined at the outset. In the conventional pipeline, templates are derived from polarity-invariant clustering of voltage maps at global field power (GFP) peaks, making the resulting state definitions sensitive to preprocessing, sampling, initialization, clustering algorithms, and the choice of cluster number. Consequently, the method captures coarse regularities in EEG dynamics, while only weakly constraining the larger geometric organization from which those states emerge. This template dependence poses a major challenge for reproducibility and for comparisons across studies and EEG caps. Here, we revisit this problem from a topological-geometric perspective. We treat templates not as cluster centroids extracted from GFP-peak maps, but as landmarks embedded in the global structure of a state space constructed from mutual similarities among scalp voltage maps. In this formulation, microstate templates are rediscovered as discrete representatives of dominant axes that organize continuous neural-state topography. This reformulation preserves polarity as a meaningful geometric relation instead of eliminating it at the outset as analytical redundancy. It also shifts attention from isolated state labels to the terrain of the state space itself: the broader relational structure within which local states become interpretable. Using this approach, we show that landmark-based state definitions outperform conventional templates in capturing state structure and improving analytical performance. These findings suggest that the central problem in EEG microstate analysis is broader than clustering optimization: it concerns how to define valid nodes for coarse-graining continuous dynamics without discarding the topology that organizes them. By shifting the conceptual basis of microstate analysis from templates to landmarks, the present approach provides a more principled and potentially more stable foundation for state definition, including in fMRI. This topolo-geometric reappraisal extends conventional microstate analysis and opens a path toward more unified comparisons across datasets, paradigms, and recording systems.
Asai, T., Kashihara, S., Chiyohara, S.
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