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Phase Reconstruction with Iterated Hilbert Transforms

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Physics of Biological Oscillators

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

We discuss theoretical and practical issues of data-driven phase reconstruction approaches for nonlinear oscillatory systems by means of the geometric technique of embeddings and protophase-to-phase transformation. In this chapter, we introduce a natural extension of the well-studied Hilbert transform by iteration. The novel approach, termed iterated Hilbert transform embeddings, implements central assumptions underlying phase reconstruction and allows for exact demodulation of purely phase modulated signals. Here, we examine the performance of the novel method for the more challenging situation of generic phase-amplitude modulated signals of a simple nonlinear oscillatory system. In particular we present the benefits of the approach for secondary phase analysis steps illustrated by reconstruction of the phase response curve. Limitations of the approach are disussed for a noise-driven phase dynamics.

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Acknowledgements

Both authors thank Aneta Stefanovska and Peter McClintock for the kind invitation to contribute to this interdisciplinary work. Erik Gengel thanks the Friedrich-Ebert-Stiftung for financial support. This paper was supported by the RSF grant 17-12-01534. The analysis in Sec. 3.3 was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931).

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Authors and Affiliations

  1. Institute for Physics and Astronomy, Universit of Potsdam, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany

    Erik Gengel & Arkady Pikovsky

  2. Friedrich-Ebert Stiftung, Bonn, Germany

    Erik Gengel

  3. Higher School of Economics, Nizhny Novgorod, Russia

    Arkady Pikovsky

  4. Department of Control Theory, Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia

    Arkady Pikovsky

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  1. Erik Gengel
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Correspondence to Arkady Pikovsky .

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Editors and Affiliations

  1. Department of Physics, Lancaster University, Lancaster, UK

    Aneta Stefanovska

  2. Department of Physics, Lancaster University, Lancaster, UK

    Peter V. E. McClintock

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Gengel, E., Pikovsky, A. (2021). Phase Reconstruction with Iterated Hilbert Transforms. In: Stefanovska, A., McClintock, P.V.E. (eds) Physics of Biological Oscillators. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-59805-1_12

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