Nonlinear dynamics and entrainment in a continuously forced pulse-modulated model of testosterone regulation

Dynamical behaviors arising in a previously developed pulse-modulated mathematical model of non-basal testosterone regulation in the human male due to continuous exogenous signals are studied. In the context of endocrine regulation, exogenous signals represent, e.g., the influx of a hormone replacement therapy drug, the influence of the circadian rhythm, and interactions with other endocrine loops. This extends the scope of the autonomous pulse-modulated models of endocrine regulation to a broader class of problems, such as therapy optimization, but also puts it in the context of biological rhythms studied in chronobiology. 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. The cases of a constant and a harmonic exogenous signal are treated in detail and illustrated by the results of bifurcation analysis. According to the model, adding a constant exogenous signal only reduces the mean value of testosterone, which result pertains to the effects of hormone replacement therapies under intact endocrine feedback regulation. Further, for the case of a single-tone harmonic positive exogenous signal, bistability and quasiperiodicity arise in the system. The convergence to either of the stationary solutions in a bistable regime is shown to be controlled by the phase of the exogenous signal thus relating this transition to the phenomenon of jet lag.