Ring Populations

  • Arthur T. Winfree
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 12)


My intent in this chapter is to direct your attention to several idealizations of rhythmic behavior in collections of many similar ring devices. It turns out that some of the peculiar limitations on the behavior of simple clocks do not apply to populations of simple clocks. Here we also encounter our first example in which a phase singularity emerges from an idealized model of the structure and mechanism of a rhythmic system. The chapter is divided into four sections:
  1. A.

    Collective rhythmicity in a population without interactions among constituent clocks. This is mainly about phase resetting by a stimulus.

  2. B.

    Collective rhythmicity in a population whose individuals are all influenced by the aggregate rhythmicity of the community. This is mainly about mutual synchronization and opposition to it.

  3. C.

    Spatially distributed simple clocks without interactions. This is mostly about patterns of phase in space.

  4. D.

    Ring devices interacting locally in space. This is mostly about waves.



Ambiguous Phase Phase Singularity Phase Ring Physical Ring Mutual Synchronization 
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  1. 1.
    You might expect more fine structure in these maps, especially for small numbers of clocks. It is there. The number of clocks can be counted by counting repeated features in the contour maps. But these features are smaller the greater the number of clocks so they are easily lost in the variability of data. Although they are plainly enough revealed by mathematical analysis, very fine-grained computation is required to map them out numerically. The maps I present in Figures 9 and 12 and in Chapter 8 are deliberately smoothed to emphasize only their gross qualitative structure.Google Scholar
  2. 2.
    This is the case with two identical siphon oscillators such as are commonly assumed to imitate the biochemical regulation of mitosis in blobs of Physarum (Scheffey, 1975, pers. comm.; see Chapter 22). In fact it is usual for the more realistically complicated oscillators taken up in Chapters 5 and 6 (e.g., van der Pol oscillators: Linkens, 1976, 1977) to have multiple stable modes of pairwise entrainment (Ruelle, 1972), and usual physiologically (Block and Page, 1980). See Box F of this chapter and Box A of Chapter 8.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Arthur T. Winfree
    • 1
  1. 1.Department of Ecology and Evolutionary BiologyUniversity of ArizonaTucsonUSA

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