Models for Rhythmic Phenomena for Living Bodies and Their Applications

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Sleep-wake rhythm and periodic body temperature change persist with period close to 24-hour even in isolation from 24-hour environmental change, which shows that these periodic changes are generated by some endogenous mechanism. These periodic changes observed in living things are called circadian rhythms. In order to help finding the mechanism of circadian rhythms, we set up mathematical models, analyze them, and verify analytical results with numerical analysis. Besides circadian rhythms, oscillations of some glycolytic intermediates are other periodic phenomena observed in living things. The period of biochemical oscillations ranges a few minutes to 10 minutes, and the period of circadian rhythms are more than 100 times as long. In spite of the significant difference, one can not deny some sort of relation between circadian rhythms and biochemical reactions, because circadian rhythms are observed in variety of living things, and because biochemical reactions are essential for living things to obtain energy. In this study, we suppose that a part of a living thing is surrounded by another part. In case there is no transport of substances between them, the change of concentrations of substances in the inner part is governed by a nonlinear oscillator. Then we assume that there is transport of the substances between the parts. The assumption leads to a system in which a system governing the concentrations in the inner part and a system governing the concentrations in the outer part are coupled. Assuming that a non oscillator has some characteristics, we show that periodic solutions of the coupled system can exist. Moreover, we analyze the system to find conditions under which oscillations of long period can be generated from oscillations of relatively short period. We analyze a model for a electrical circuit anda model for a biochemical reactions, numerically, and verify the results of analysis.