Nonlinear Dynamics

, Volume 93, Issue 4, pp 1899–1910 | Cite as

Generalized multiscale Lempel–Ziv complexity of cyclic alternating pattern during sleep

  • Chien-Hung Yeh
  • Wenbin ShiEmail author
Original Paper


Increasing evidences show that multiscale complexity measure is an intuitive and effective measure in quantifying various physical and physiological states. In this study, we demonstrate that the classical algorithm of multiscale Lempel–Ziv complexity (multiscale LZC or MLZ) has a critical limitation in neglecting rapid rhythms in complex systems. To this end, simulations added with different levels of white noise are designed to examine whether or not MLZ calculation neglects the effects of high-frequency noise. In addition, an algorithm by obtaining coarse-grained multiscale LZC, so-called generalized multiscale LZC (gMLZ), is proposed to yield a spectrum of complexity. A series of simulated non-stationary signals are generated for comparing the performances between MLZ and gMLZ. Besides, cyclic alternating pattern (CAP), characterized by the excessive synchronization of neuronal activity, has been associated with its power and physiological states. To understand how the synchronization of neuronal activities in different phase-A subtypes in exerting an influence over its power and complexity, we analyze the gMLZ of the real CAP database and compare it to its power spectra as well as modified multiscale entropy (MMSE), which is one of the most well-known multiscale complexity-based measures. The novel algorithm reveals that the evaluated complexities in different phase-A subtypes are inversely related to both the power and excessive synchronization in different timescales in general. The impact of frequencies, sleep stages and pathophysiological conditions on these two complexity measures is also examined. The discerning abilities of different phase-A subtypes using coarse-grained complexity measures (gMLZ and MMSE) are more consistent than power across different time scales. Our approach makes up a deficiency in handling with high-frequency oscillations and enables us to examine complexities of nonlinear systems in a wide-range of timescales.


Lempel–Ziv complexity Multiscale complexity Cyclic alternating pattern Sleep 



This research is sponsored by the China Postdoctoral Science Foundation (Grant 043206005).

Compliance with ethical standard

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of NeurologyChang Gung Memorial Hospital and UniversityTaoyuan CityTaiwan
  2. 2.State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic EngineeringTsinghua UniversityBeijingChina

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