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Research Article | DOI: https://doi.org/10.31579/2692-9406/222
*Corresponding Author: Lakshmi. N. Sridhar, Chemical Engineering Department, University of Puerto Rico, Mayaguez, PR 00681.
Citation: Lakshmi. N. Sridhar, (2025), Analysis and Control of a Dynamic Model Involving Hypothalamic-Pituitary-adrenal (HPA) Axis, J. Biomedical Research and Clinical Reviews, 10(5); DOI:10.31579/2692-9406/222
Copyright: © 2025, Lakshmi. N. Sridhar. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Received: 30 June 2025 | Accepted: 07 July 2025 | Published: 15 July 2025
Keywords: bifurcation; optimization; control; hypothalamic-pituitary-adrenal (HPA) axis
The hypothalamic-pituitary-adrenal (HPA) axis links the nervous and endocrine systems’ functions, which is of vital importance for maintaining homeostasis in mammalian organisms both under normal conditions and during stress. The integration of the nervous and endocrine systems’ functions is very nonlinear and exhibits oscillatory behavior. Bifurcation analysis is a powerful mathematical tool used to deal with the nonlinear dynamics of any process. Several factors must be considered, and multiple objectives must be met simultaneously. Bifurcation analysis and multiobjective nonlinear model predictive control (MNLMPC) calculations are performed on a model of the hypothalamic-pituitary-adrenal (HPA) axis. The MATLAB program MATCONT was used to perform the bifurcation analysis. The MNLMPC calculations were performed using the optimization language PYOMO in conjunction with the state-of-the-art global optimization solvers IPOPT and BARON. The bifurcation analysis Hopf bifurcation points that lead to limit cycles. These Hopf points were eliminated using an activation factor that involves the tanh function. The multiobjective nonlinear model predictive control calculations converge to the Utopia point, which is the best solution.
Introduction
Tsigos et al (2002) [1] investigated neuroendocrine factors and stress factors in Hypothalamic–pituitary–adrenal axis problems. Savic and Jelic (2005) [2] performed a theoretical study of hypothalamo-pituitary adrenocortical axis dynamics. Jelic et al (2005) [3] mathematically modelled the hypothalamic–pituitary–adrenal system activity. Lenbury et al (2005) [4] developed a delay-differential equation model of the feedback-controlled hypothalamus-pituitary-adrenal axis in humans. Kyrylov et al (2005) [5] modelled the oscillatory behavior of the hypothalamic–pituitary–adrenal axis. Savic et al (2006) [6] discussed the stability of a general delay differential model of the hypothalamo-pituitary-adrenocortical system. Smith et al (2006) [7] studied the role of the hypothalamic–pituitary–adrenal axis in neuroendocrine responses to stress. Gupta et al (2007) [8] showed that the inclusion of the glucocorticoid receptor in a hypothalamic pituitary adrenal axis model reveals bistability.
Bairagi et al (2008) [9] discussed the variability in the secretion of corticotropin-releasing hormone, adrenocorticotropic hormone and cortisol and understandability of the hypothalamic–pituitary–adrenal axis dynamics. Vinther et al (2011) [10] developed a minimal model of the hypothalamic–pituitary–adrenal axis. Jelic et al (2011) [11] discussed the predictive modeling of the hypothalamic–pituitary–adrenal (HPA) function. Markovic et al (2011) [12], performed predictive modeling studies of the hypothalamic–pituitary–adrenal (HPA) axis response to acute and chronic stress. Markovic et al (2011) [13] investigated the, the stability of the extended model of the hypothalamic–pituitary–adrenal axis examined by stoichiometric network analysis. Andersen et al (2013) [14] performed mathematical modeling studies of the hypothalamic–pituitary–adrenal gland (HPA) axis, including hippocampal mechanisms. Postnova et al (2013) [15] developed a minimal physiologically based model of the HPA axis under influence of the sleep-wake cycles. Gudmand-Hoeyer et al (2014) [16] performed a Patient-specific modeling of the neuroendocrine HPA-axis and studied its relation to depression. Hosseinichimeh et al (2015) [17] performed additional modeling the hypothalamus-pituitary-adrenal axis. Malek et al (2015) [18] discussed the dynamics of the HPA axis and inflammatory cytokines.
Markovic et al (2016) [19] investigated the cholesterol effects on the dynamics of the hypothalamic–pituitary–adrenal (HPA) axis. Cupic et al (2016) [20] studied the dynamic transitions in a model of the hypothalamic–pituitary–adrenal axis. Pierre et al (2016) [21] investigated the role of the hypothalamic–pituitary–adrenal axis in modulating seasonal changes in immunity. Abulseoud et al (2017) [22] demonstrated the existence of corticosterone oscillations during mania induction in the lateral hypothalamic kindled rat-Experimental observations. Stanojevic et al (2017) [23] performed kinetic modelling of testosterone-related differences in the hypothalamic–pituitary–adrenal axis response to stress. Bangsgaard et al (2017) [24] performed patient-specific modelling studies of the HPA axis related to the clinical diagnosis of depression. Kim et al (2017) [25] performed mathematical modeling to improve diagnosis of post-traumatic and related stress disorders by perturbing the hypothalamic–pituitary–adrenal stress response system. Stanojevic et al (2017) [26] modelled the hypothalamic–pituitary–adrenal axis perturbations by externally induced cholesterol pulses of finite duration and with asymmetrically distributed concentration profiles. Kim LU et al (2017) [27] perturbed the hypothalamic pituitary–adrenal axis and developed a mathematical model for interpreting PTSD assessment tests. Comput Psychiatry 2017, 2:28-49. Kaslik et al (2018) [28] discussed the stability and demonstrated the existence of Hopf bifurcations for the hypothalamic–pituitary–adrenal axis model with memory. The aim of this paper is to 1) perform bifurcation studies on the hypothalamic–pituitary–adrenal axis model described in Cupic et al (2016) [20], demonstrate the existence of Hopf bifurcations and provide a strategy to eliminate them and 2) to perform multiobjective nonlinear model predictive control calculations on the same hypothalamic–pituitary–adrenal axis model.This document is organized as follows. The model equations for the hypothalamic–pituitary–adrenal axis model Cupic et al (2016) [20]is first described. This is followed by a description of the numerical methods (bifurcation analysis and MNLMPC). The results and discussion are then presented, followed by the conclusions.
Model Equations Cupic et al (2016) [20]
The model equations are
Where;
The base parameters are
represent cholesterol, CRH, ACTH, cortisol, and aldosterone.
Bifurcation analysis
The MATLAB software MATCONT is used to perform the bifurcation calculations. Bifurcation analysis deals with multiple steady-states and limit cycles. Multiple steady states occur because of the existence of branch and limit points. Hopf bifurcation points cause limit cycles. A commonly used MATLAB program that locates limit points, branch points, and Hopf bifurcation points is MATCONT (Dhooge Govearts, and Kuznetsov, 2003[29]; Dhooge Govearts, Kuznetsov, Mestrom and Riet, 2004[30]). This program detects Limit points (LP), branch points (BP), and Hopf bifurcation points(H) for an ODE system
(3)
Let the bifurcation parameter be 
Since the gradient is orthogonal to the tangent vector,
The tangent plane at any point 
must satisfy
Where A is
where 
is the Jacobian matrix. For both limit and branch points, the matrix 
must be singular. The n+1 th component of the tangent vector 
= 0 for a limit point (LP)and for a branch point (BP) the matrix 
must be singular. At a Hopf bifurcation point,
@ Indicates the bialternate product while 
is the n-square identity matrix. Hopf bifurcations cause limit cycles and should be eliminated because limit cycles make optimization and control tasks very difficult. More details can be found in Kuznetsov (1998 [31]; 2009 [32]) and Govaerts [2000] [33]
Hopf bifurcations cause unwanted oscillatory behavior and limit cycles. The tanh activation function (where a control value u is replaced by) 
is commonly used in neural nets (Dubey et al 2022[34]; Kamalov et al, 2021[35] and Szandała, 2020[36]) and optimal control problems (Sridhar 2023[37]) to eliminate spikes in the optimal control profile. Hopf bifurcation points cause oscillatory behavior. Oscillations are similar to spikes, and the results in Sridhar (2024) [37] demonstrate that the tanh factor also eliminates the Hopf bifurcation by preventing the occurrence of oscillations. Sridhar (2024) [38] explained with several examples how the activation factor involving the tanh function successfully eliminates the limit cycle causing Hopf bifurcation points. This was because the tanh function increases the time period of the oscillatory behavior, which occurs in the form of a limit cycle caused by Hopf bifurcations.
Multiobjective Nonlinear Model Predictive Control (MNLMPC)
Flores Tlacuahuaz et al (2012) [39] developed a multiobjective nonlinear model predictive control (MNLMPC) method that is rigorous and does not involve weighting functions or additional constraints. This procedure is used for performing the MNLMPC calculations Here 
(j=12.n) represents the variables that need to be minimized/maximized simultaneously for a problem involving a set of ODE
being the final time value, and n the total number of objective variables and. u the control parameter. This MNLMPC procedure first solves the single objective optimal control problem independently optimizing each of the variables 
individually. The minimization/maximization of 
will lead to the values 
Then the optimization problem that will be solved is
This will provide the values of u at various times. The first obtained control value of u is implemented and the rest are discarded. This procedure is repeated until the implemented and the first obtained control values are the same or if the Utopia point where ( 
for all j) is obtained. Pyomo (Hart et al, 2017) [40] is used for these calculations. Here, the differential equations are converted to a Nonlinear Program (NLP) using the orthogonal collocation method The NLP is solved using IPOPT (Wächter And Biegler, 2006) [41]and confirmed as a global solution with BARON (Tawarmalani, M. and N. V. Sahinidis 2005) [42].
The steps of the algorithm are as follows
and obtain 
at various time intervals ti. The subscript i is the index for each time step. 
and get the control values for various times.
for all j. When the bifurcation analysis was performed with k4 as the bifurcation parameter, a Hopf bifurcation point was located at (x1, x2x3, x4, k4) values of (0.003067, 0, 0, 0, 0, 0.815058) as shown in curve AB in Fig. 1 When k4 was modified to k4 tanh(k4)/100 the Hopf bifurcation disappears. This is shown in curve AC in Fig. 1. When the bifurcation analysis was performed with k4 as the bifurcation parameter, a Hopf bifurcation point was located at (x1, x2x3, x4, k4) values of (0.003067, 0, 0, 0, 0, 12.539816) as shown in curve AB in Fig. 2. When k5 was modified to k5 tanh(k5)/100 the Hopf bifurcation disappears. This is shown in curve AC in Fig. 2. When k6 was the bifurcation parameter, a Hopf bifurcation point was located at (x1, x2x3, x4, k6) values of (0.003067, 0, 0, 0, 0, 2.869237) as shown in curve ABC in Fig. 3 When k6 was modified to k6 tanh(k6)/100 the Hopf bifurcation disappears. This is shown in the curve ABD in Fig. 3.
When k11 was the bifurcation parameter, a Hopf bifurcation point was located at (x1, x2x3, x4, k11) values of (0.003067, 0, 0, 0, 0, 0.062997) as shown in curve ABC in Fig. 4 When k11 was modified to k11 tanh(k11)/650 the Hopf bifurcation disappears. This is shown in the curve ABD in Fig. 4.
When k12 was the bifurcation parameter, two Hopf bifurcation points were located at (x1, x2x3, x4, k12) values of (0.003067, 0, 0, 0, 0, 0.400456) and (0.003067, 0, 0, 0, 0, 0.517213) as shown in curve ABC in Fig. 5 When k12 was modified to k12 tanh(k12)/0.005 the Hopf bifurcation disappears. This is shown in the curve ADC in Fig. 5. With all the bifurcation parameters, the use of the tanh activation factor was successful in eliminating the Hopf bifurcation points that cause unwanted limit cycles and oscillatory behavior validating the analysis of Sridhar (2024) [38]
For the MNLMPC calculations,
were minimized individually and led to values of 0.135435E-02, 0, 0, and 0. k4 was the control parameter. The multiobjective optimal control problem will involve the minimization of 
subject to the equations governing the model. This led to a value of zero (the Utopia solution). The control value k4 stayed constant at 0.05. Figure 6, 7 amd 8 show the various MNLMPC profiles.
Figure 1: Bifurcation analysis with k4 as bifurcation parameter
Figure 2: Bifurcation analysis with k5 as bifurcation parameter
Figure 3: Bifurcation analysis with k6 as bifurcation parameter
Figure 4: Bifurcation analysis with k11 as bifurcation parameter
Figure 5: Bifurcation analysis with k12 as bifurcation parameter
Figure 6: MNLMPC with x1 vs t
Figure 7: MNLMPC with x2 x3 x4 vs t
Figure 8: MNLMPC with x5 vs t
Bifurcation analysis and multiobjective nonlinear model predictive control calculations were performed on a dynamic hypothalamic-pituitary-adrenal (HPA) axis model. The bifurcation analysis revealed the existence of a limit cycle causing Hopf bifurcation points, which are eliminated using an activation factor involving the tanh function. The multi-objective nonlinear model predictive calculations converged to the Utopia point (the best possible solution). The integration of the bifurcation analysis and multiobjective nonlinear model predictive control for a dynamic hypothalamic-pituitary-adrenal (HPA) axis model is the main contribution of this paper.
All data used is presented in the paper
The author, Dr. Lakshmi N Sridhar has no conflict of interest.
Dr. Sridhar thanks Dr. Carlos Ramirez and Dr. Suleiman for encouraging him to write single-author papers.
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We found the peer review process quick and positive in its input. The support from the editorial officer has been very agile, always with the intention of improving the article and taking into account our subsequent corrections.
My article, titled 'No Way Out of the Smartphone Epidemic Without Considering the Insights of Brain Research,' has been republished in the International Journal of Clinical Case Reports and Reviews. The review process was seamless and professional, with the editors being both friendly and supportive. I am deeply grateful for their efforts.
To Dear Erin Aust – Editorial Coordinator of Journal of General Medicine and Clinical Practice! I declare that I am absolutely satisfied with your work carried out with great competence in following the manuscript during the various stages from its receipt, during the revision process to the final acceptance for publication. Thank Prof. Elvira Farina
Dear Jessica, and the super professional team of the ‘Clinical Cardiology and Cardiovascular Interventions’ I am sincerely grateful to the coordinated work of the journal team for the no problem with the submission of my manuscript: “Cardiometabolic Disorders in A Pregnant Woman with Severe Preeclampsia on the Background of Morbid Obesity (Case Report).” The review process by 5 experts was fast, and the comments were professional, which made it more specific and academic, and the process of publication and presentation of the article was excellent. I recommend that my colleagues publish articles in this journal, and I am interested in further scientific cooperation. Sincerely and best wishes, Dr. Oleg Golyanovskiy.
Dear Ashley Rosa, Editorial Coordinator of the journal - Psychology and Mental Health Care. " The process of obtaining publication of my article in the Psychology and Mental Health Journal was positive in all areas. The peer review process resulted in a number of valuable comments, the editorial process was collaborative and timely, and the quality of this journal has been quickly noticed, resulting in alternative journals contacting me to publish with them." Warm regards, Susan Anne Smith, PhD. Australian Breastfeeding Association.
Dear Jessica Magne, Editorial Coordinator, Clinical Cardiology and Cardiovascular Interventions, Auctores Publishing LLC. I appreciate the journal (JCCI) editorial office support, the entire team leads were always ready to help, not only on technical front but also on thorough process. Also, I should thank dear reviewers’ attention to detail and creative approach to teach me and bring new insights by their comments. Surely, more discussions and introduction of other hemodynamic devices would provide better prevention and management of shock states. Your efforts and dedication in presenting educational materials in this journal are commendable. Best wishes from, Farahnaz Fallahian.
Dear Maria Emerson, Editorial Coordinator, International Journal of Clinical Case Reports and Reviews, Auctores Publishing LLC. I am delighted to have published our manuscript, "Acute Colonic Pseudo-Obstruction (ACPO): A rare but serious complication following caesarean section." I want to thank the editorial team, especially Maria Emerson, for their prompt review of the manuscript, quick responses to queries, and overall support. Yours sincerely Dr. Victor Olagundoye.
Dear Ashley Rosa, Editorial Coordinator, International Journal of Clinical Case Reports and Reviews. Many thanks for publishing this manuscript after I lost confidence the editors were most helpful, more than other journals Best wishes from, Susan Anne Smith, PhD. Australian Breastfeeding Association.
Dear Agrippa Hilda, Editorial Coordinator, Journal of Neuroscience and Neurological Surgery. The entire process including article submission, review, revision, and publication was extremely easy. The journal editor was prompt and helpful, and the reviewers contributed to the quality of the paper. Thank you so much! Eric Nussbaum, MD
Dr Hala Al Shaikh This is to acknowledge that the peer review process for the article ’ A Novel Gnrh1 Gene Mutation in Four Omani Male Siblings, Presentation and Management ’ sent to the International Journal of Clinical Case Reports and Reviews was quick and smooth. The editorial office was prompt with easy communication.
Dear Erin Aust, Editorial Coordinator, Journal of General Medicine and Clinical Practice. We are pleased to share our experience with the “Journal of General Medicine and Clinical Practice”, following the successful publication of our article. The peer review process was thorough and constructive, helping to improve the clarity and quality of the manuscript. We are especially thankful to Ms. Erin Aust, the Editorial Coordinator, for her prompt communication and continuous support throughout the process. Her professionalism ensured a smooth and efficient publication experience. The journal upholds high editorial standards, and we highly recommend it to fellow researchers seeking a credible platform for their work. Best wishes By, Dr. Rakhi Mishra.
Dear Jessica Magne, Editorial Coordinator, Clinical Cardiology and Cardiovascular Interventions, Auctores Publishing LLC. The peer review process of the journal of Clinical Cardiology and Cardiovascular Interventions was excellent and fast, as was the support of the editorial office and the quality of the journal. Kind regards Walter F. Riesen Prof. Dr. Dr. h.c. Walter F. Riesen.
Dear Ashley Rosa, Editorial Coordinator, International Journal of Clinical Case Reports and Reviews, Auctores Publishing LLC. Thank you for publishing our article, Exploring Clozapine's Efficacy in Managing Aggression: A Multiple Single-Case Study in Forensic Psychiatry in the international journal of clinical case reports and reviews. We found the peer review process very professional and efficient. The comments were constructive, and the whole process was efficient. On behalf of the co-authors, I would like to thank you for publishing this article. With regards, Dr. Jelle R. Lettinga.
Dear Clarissa Eric, Editorial Coordinator, Journal of Clinical Case Reports and Studies, I would like to express my deep admiration for the exceptional professionalism demonstrated by your journal. I am thoroughly impressed by the speed of the editorial process, the substantive and insightful reviews, and the meticulous preparation of the manuscript for publication. Additionally, I greatly appreciate the courteous and immediate responses from your editorial office to all my inquiries. Best Regards, Dariusz Ziora
Dear Chrystine Mejia, Editorial Coordinator, Journal of Neurodegeneration and Neurorehabilitation, Auctores Publishing LLC, We would like to thank the editorial team for the smooth and high-quality communication leading up to the publication of our article in the Journal of Neurodegeneration and Neurorehabilitation. The reviewers have extensive knowledge in the field, and their relevant questions helped to add value to our publication. Kind regards, Dr. Ravi Shrivastava.
Dear Clarissa Eric, Editorial Coordinator, Journal of Clinical Case Reports and Studies, Auctores Publishing LLC, USA Office: +1-(302)-520-2644. I would like to express my sincere appreciation for the efficient and professional handling of my case report by the ‘Journal of Clinical Case Reports and Studies’. The peer review process was not only fast but also highly constructive—the reviewers’ comments were clear, relevant, and greatly helped me improve the quality and clarity of my manuscript. I also received excellent support from the editorial office throughout the process. Communication was smooth and timely, and I felt well guided at every stage, from submission to publication. The overall quality and rigor of the journal are truly commendable. I am pleased to have published my work with Journal of Clinical Case Reports and Studies, and I look forward to future opportunities for collaboration. Sincerely, Aline Tollet, UCLouvain.
Dear Ms. Mayra Duenas, Editorial Coordinator, International Journal of Clinical Case Reports and Reviews. “The International Journal of Clinical Case Reports and Reviews represented the “ideal house” to share with the research community a first experience with the use of the Simeox device for speech rehabilitation. High scientific reputation and attractive website communication were first determinants for the selection of this Journal, and the following submission process exceeded expectations: fast but highly professional peer review, great support by the editorial office, elegant graphic layout. Exactly what a dynamic research team - also composed by allied professionals - needs!" From, Chiara Beccaluva, PT - Italy.
Dear Maria Emerson, Editorial Coordinator, we have deeply appreciated the professionalism demonstrated by the International Journal of Clinical Case Reports and Reviews. The reviewers have extensive knowledge of our field and have been very efficient and fast in supporting the process. I am really looking forward to further collaboration. Thanks. Best regards, Dr. Claudio Ligresti
