Pulse-modulated model of testosterone regulation subject to exogenous signals

Dynamical behaviors arising due to continuous exogenous signals in a previously developed pulse-modulated mathematical model of non-basal testosterone regulation in the human male are studied. In the context of endocrine regulation, exogenous signals represent e.g. the drugs used in hormone replacement therapies, the effects due to circadian rhythm, and interactions with other endocrine loops of the organism. The model dynamics are hybrid since the hormone metabolism is suitably captured by a continuous description and the control feedback is implemented in a discrete (i.e. event-based) manner by the hypothalamus of the brain. It is demonstrated that the endocrine loop with an exogenous signal entering the continuous part can be equivalently described by proper modifications in the pulse-modulation functions of the autonomous model. This extends the scope of the autonomous pulse-modulated models of endocrine regulation to a broader class of problems, such as therapy optimization, where an exogenous signal is involved. The cases of constant and harmonic exogenous signals are treated in detail and illustrated by bifurcation analysis.