https://doi.org/10.1140/epjs/s11734-026-02269-z
Regular Article
Examination of runs entropy: theoretical insights and practical applications in polysomnographic analysis
1
Institute of Physics, University of Zielona Góra, Chopina 15, 65-069, Zielona Góra, Poland
2
Faculty of Medicine and Health Sciences, University of Zielona Góra, Chopina 15, 65-069, Zielona Góra, Poland
3
Institute of Pedagogy, University of Zielona Góra, Chopina 15, 65-069, Zielona Góra, Poland
4
Department of Cardiology–Intensive Therapy, Poznan University of Medical Sciences, ul. Fredry 10, 61-701, Poznań, Poland
a
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Received:
17
November
2025
Accepted:
13
March
2026
Published online:
20
March
2026
Abstract
Runs entropy summarizes the distribution of monotonic accelerations and decelerations in RR-interval series. We formalize runs entropy as the Shannon entropy of the joint run category—type (acceleration/deceleration) and length—and show that the type-specific parts are conditional entropies. Within this framework, we define one derivative information measure, namely, mutual information (MI). As an illustration, we analyze polysomnographic ECG from 31 subjects in 5-min windows with expert sleep staging. We demonstrate that HRA run-category distributions carry stage-specific information beyond chance, even after adjusting for within-subject stage imbalance. MI heatmaps reveal that stage discrimination depends on both run type and length: very short accelerations (length 1) are most informative for Wake; medium-length decelerations (3–6 beats) characterize N2/N3, medium-length accelerations N2; exceptionally long runs carry disproportionate information at transitions (notably REM
N2, N1Wake) despite their rarity. These results consolidate the theoretical basis of runs entropy and demonstrate that runs-based information measures capture physiologically meaningful patterns in sleep.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
Bartosz Biczuk, Szymon Jurga, Elżbieta Turska, and Przemysław Guzik have contributed equally to this work.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
