Mathematical analysis of the role of pituitary-adrenal interactions in ultradian rhythms of the HPA axis.

The hypothalamic-pituitary-adrenal (HPA) axis is a biological system in the human body that plays an important role in controlling stress and regulating various physiological elements, including the immune system, emotions, and moods in tense situations. Over the past two decades, several ordinary or delay differential equations models of the HPA axis have been proposed. In the majority of studies presented so far, corticotropin-releasing hormone (CRH), adrenocorticotropic hormone (ACTH), and cortisol are among the main variables employed to build the HPA axis models. In the present study, based on a previously introduced hypothesis which asserts that ultradian rhythms in the HPA axis are produced by the pituitary-adrenal network alone and these rhythms can endure in the absence of CRH secretion, a simple two-dimensional delayed dynamical model of the HPA axis based on only ACTH and cortisol is introduced. The model is shown to be able to capture the ultradian (low frequency) rhythms of ACTH and cortisol released into the bloodstream. By mathematical analysis of the model using the Hopf bifurcation theorem, it is also demonstrated how oscillating solutions can emerge. Also, the model employs physiologically reasonable parameter values to exhibit how in the absence of CRH secretion, a simple model of the pituitary-adrenal interaction can be used to produce ultradian rhythms of both cortisol and ACTH hormones.