Mathematical theory of biological periodicities: Formulation of then-body case

Abstract

In a series of papers, L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956;19, 9–18, 1957) showed that a non-linear biochemical interaction between the anterior pituary gland and the thyroid gland may result under certain conditions in sustained periodical oscillations of the rates of production and of the blood level of the thyrotropic and of the thyroid hormone. They treated the systems, however, as a homogeneous one. N. Rashevsky (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1965;Bull. Math. Biophysics,29, 395–401, 1967) generalized the above results by taking into account the histological structures of the two glands as well as the diffusion coefficients and permeabilities of cells involved. The present paper is the first step toward the theory of interaction of any numbern of glands or, more generally,n components. The differential equations which govern the behavior of such a system represent a system of2n ²+n non-linear first order ordinary equations and involve a total of 7 n ²+3n parameters of partly histological, partly biochemical nature. The requirements of the existence of sustained oscillations demand 4n ²+2n+2 inequalities between those 7n ²+3n parameters.