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Background The hypothalamic-pituitary-adrenal axis (HPA axis) is a major part of the neuroendocrine system responsible for the regulation of the response to physical or mental stress and for the control of the synthesis of the stress hormone cortisol. Dysfunctions of the HPA axis characterized by either low (hypocortisolism) or increased (hypercortisolism) cortisol levels are implicated in various pathological conditions. Their understanding and therapeutic correction may be supported by mathematical modeling and simulation of the HPA axis. Methods Mass action and Michaelis Menten enzyme kinetics were used to provide a mechanistic description of the feedback mechanisms within the pituitary gland cells by which cortisol inhibits its own production. A separation of the nucleus from the cytoplasm by compartments enabled a differentiation between slow genomic and fast non-genomic processes. The model in parts was trained against time resolved ACTH stress response data from an in vitro cell culture of murine AtT-20 pituitary tumor cells and analyzed by bifurcation discovery tools. Results A recently found pituitary gland cell membrane receptor that mediates rapid non-genomic actions of glucocorticoids has been incorporated into our model of the HPA axis. As a consequence of the distinction between genomic and non-genomic feedback processes our model possesses an extended dynamic repertoire in comparison to existing HPA models. In particular, our model exhibits limit cycle oscillations and bistable behavior associated to hypocortisolism but also features a (second) bistable switch which captures irreversible transitions in hypercortisolism to elevated cortisol levels. Conclusions Model predictive control and inverse bifurcation analysis have been previously applied in the simulation-based design of therapeutic strategies for the correction of hypocortisolism. Given the HPA model extension presented in this paper, these techniques may also be used in the study of hypercortisolism. As an example, we show how sparsity enforcing penalization may suggest network interventions that allow the return from elevated cortisol levels back to nominal ones.