Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Shingo Gibo, Hiroshi Ito

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)


    Many biological rhythms are generated by negative feedback regulation. Griffith (1968) proved that a negative feedback model with two variables expressed by ordinary differential equations do not generate self-sustained oscillations. Kurosawa et al. (2002) expanded Griffith's result to the general type of negative feedback model with two variables. In this paper, we propose discrete and ultradiscrete feedback models with two variables that exhibit self-sustained oscillations. To obtain the model, we applied tropical discretization and ultradiscretization to a continuous model with two variables and then investigated its bifurcation structures and the conditions of parameters for oscillations. We found that when the degradation rate of the variables is lower than their synthesis rate, the proposed models generate oscillations by Neimark-Sacker bifurcation. We further demonstrate that the ultradiscrete model can be reduced to a Boolean system under some conditions.

    Original languageEnglish
    Pages (from-to)89-95
    Number of pages7
    JournalJournal of Theoretical Biology
    Publication statusPublished - Aug 7 2015

    All Science Journal Classification (ASJC) codes

    • Statistics and Probability
    • Modelling and Simulation
    • Biochemistry, Genetics and Molecular Biology(all)
    • Immunology and Microbiology(all)
    • Agricultural and Biological Sciences(all)
    • Applied Mathematics


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