Mathematical models of endocrine systems

Abstract

There is proposed a generalized mathematical model of endocrine systems, consisting of a set of differential equations which describe a chain of chemical reactions. The product of each reaction stimulates or inhibits some other reaction in the chain except possibly the last, which may or may not influence the system. At least one reaction must be independent and able to proceed without stimulation or inhibition by the products of other reactions.

If only two reactions of the type assumed constitute a closed chain, sustained periodic variations in the concentrations of the reaction products cannot occur. If the chain consists of three or more reactions forming a closed loop, sustained oscillations, such as are observed in the menstrual cycle or in the mental disorder called periodic catatonia, can occur under suitable conditions. In this case, the concentrations of the system components exhibit relaxation oscillations characterized by periodic degeneration of the system when an independent reaction becomes completely inhibited by other reaction products. A set of conditions sufficient to produce periodicities in component concentrations is presented.

Application of the model to the normally periodic system of the menstrual cycle and to the abnormal endocrine system which causes periodic catatonia is discussed.