Onset, timing, and exposure therapy of stress disorders: mechanistic insight from a mathematical model of oscillating neuroendocrine dynamics

A probabilistic model of relapse in drug addiction

More than 60% of individuals recovering from substance use disorder relapse within one year. Some will resume drug consumption even after decades of abstinence. The cognitive and psychological mechanisms that lead to relapse are not completely understood, but stressful life experiences and external stimuli that are associated with past drug-taking are known to play a primary role. Stressors and cues elicit memories of drug-induced euphoria and the expectation of relief from current anxiety, igniting an intense craving to use again; positive experiences and supportive environments may mitigate relapse. We present a mathematical model of relapse in drug addiction that draws on known psychiatric concepts such as the "positive activation; negative activation" paradigm and the "peak-end" rule to construct a relapse rate that depends on external factors (intensity and timing of life events) and individual traits (mental responses to these events). We analyze which combinations and ordering of stressors, cues, and positive events lead to the largest relapse probability and propose interventions to minimize the likelihood of relapse. We find that the best protective factor is exposure to a mild, yet continuous, source of contentment, rather than large, episodic jolts of happiness.

Major depressive disorder and bistability in an HPA-CNS toggle switch

Major depressive disorder (MDD) is the most common psychiatric disorder. It has a complex and heterogeneous etiology. Most treatments take weeks to show effects and work well only for a fraction of the patients. Thus, new concepts are needed to understand MDD and its dynamics. One of the strong correlates of MDD is increased activity and dysregulation of the hypothalamic-pituitary-adrenal (HPA) axis which produces the stress hormone cortisol. Existing mathematical models of the HPA axis describe its operation on the scale of hours, and thus are unable to explore the dynamic on the scale of weeks that characterizes many aspects of MDD. Here, we propose a mathematical model of MDD on the scale of weeks, a timescale provided by the growth of the HPA hormone glands under control of HPA hormones. We add to this the mutual inhibition of the HPA axis and the hippocampus and other regions of the central nervous system (CNS) that forms a toggle switch. The model shows bistability between euthymic and depressed states, with a slow timescale of weeks in its dynamics. It explains why prolonged but not acute stress can trigger a self-sustaining depressive episode that persists even after the stress is removed. The model explains the weeks timescale for drugs to take effect, as well as the dysregulation of the HPA axis in MDD, based on gland mass changes. This understanding of MDD dynamics may help to guide strategies for treatment.

Electroacupuncture Ameliorates Hypothalamic‒Pituitary‒Adrenal Axis Dysfunction Induced by Surgical Trauma in Mice Through the Hypothalamic Oxytocin System

Electroacupuncture (EA) can effectively reduce surgical stress reactions and promote postoperative recovery, but the mechanisms remain unclear. The present study aims to examine the effects of EA on the hyperactivity of the hypothalamic‒pituitary‒adrenal (HPA) axis and investigate its potential mechanisms. Male C57BL/6 mice were subjected to partial hepatectomy (HT). The results showed that HT increased the concentrations of corticotrophin-releasing hormone (CRH), corticosterone (CORT), and adrenocorticotropic hormone (ACTH) in the peripheral blood and upregulated the expression of CRH and glucocorticoid receptors (GR) proteins in the hypothalamus. EA treatment significantly inhibited the hyperactivity of the HPA axis by decreasing the concentration of CRH, CORT, and ACTH in peripheral blood and downregulating the expression of CRH and GR in the hypothalamus. Moreover, EA treatment reversed the HT-induced downregulation of oxytocin (OXT) and oxytocin receptor (OXTR) in the hypothalamus. Furthermore, intracerebroventricular injection of the OXTR antagonist atosiban blocked the effects of EA. Thus, our findings implied that EA mitigated surgical stress-induced HPA axis dysfunction by activating the OXT/OXTR signaling pathway.

Modeling pulsativity in the hypothalamic-pituitary-adrenal hormonal axis

A new mathematical model for biological rhythms in the hypothalamic–pituitary–adrenal (HPA) axis is proposed. This model takes the form of a system of impulsive time-delay differential equations which include pulsatile release of adrenocorticotropin (ACTH) by the pituitary gland and a time delay for the release of glucocorticoid hormones by the adrenal gland. Numerical simulations demonstrate that the model’s response to periodic and circadian inputs from the hypothalamus are consistent with those generated by recent models which do not include a pulsatile pituitary. In contrast the oscillatory phenomena generated by the impulsive delay equation mode occur even if the time delay is zero. The observation that the time delay merely introduces a small phase shift suggesting that the effects of the adrenal gland are “downstream” to the origin of pulsativity. In addition, the model accounts for the occurrence of ultradian oscillations in an isolated pituitary gland. These observations suggest that principles of pulse modulated control, familiar to control engineers, may have an increasing role to play in understanding the HPA axis.

Understanding virtual patients efficiently and rigorously by combining machine learning with dynamical modelling

Individual biological organisms are characterized by daunting heterogeneity, which precludes describing or understanding populations of ‘patients’ with a single mathematical model. Recently, the field of quantitative systems pharmacology (QSP) has adopted the notion of virtual patients (VPs) to cope with this challenge. A typical population of VPs represents the behavior of a heterogeneous patient population with a distribution of parameter values over a mathematical model of fixed structure. Though this notion of VPs is a powerful tool to describe patients’ heterogeneity, the analysis and understanding of these VPs present new challenges to systems pharmacologists. Here, using a model of the hypothalamic–pituitary–adrenal axis, we show that an integrated pipeline that combines machine learning (ML) and bifurcation analysis can be used to effectively and efficiently analyse the behaviors observed in populations of VPs. Compared with local sensitivity analyses, ML allows us to capture and analyse the contributions of simultaneous changes of multiple model parameters. Following up with bifurcation analysis, we are able to provide rigorous mechanistic insight regarding the influences of ML-identified parameters on the dynamical system’s behaviors. In this work, we illustrate the utility of this pipeline and suggest that its wider adoption will facilitate the use of VPs in the practice of systems pharmacology.

Chronic Stress in Bipolar Disorders Across the Different Clinical States: Roles of HPA Axis and Personality

INTRODUCTION

Chronic stress has been linked to the pathophysiology of bipolar disorder (BD); however, the underlying mechanism remains unclear. In BD patients, hypothalamic-pituitary-adrenal (HPA) axis activity is associated with stress. This study aimed to examine the relationship between HPA axis activity and BD symptoms in various clinical states, as well as how personality influences the process.

METHODS

This study investigated the differences in HPA axis activity among four BD states. We enrolled 813 BD patients in an 8-week longitudinal study to examine the relationship between HPA axis activity and symptom trajectories using dynamic temporal warping (DTW) analysis and an unsupervised machine learning technique. Furthermore, using mediation analyses, the relationship between the HPA axis, personality, and BD symptoms was investigated.

RESULTS

Analysis of variance (ANOVA) analysis showed that glucocorticoid cortisol (CORT) and adrenocorticotropin (ACTH) did not differ significantly among the four clinical states of BD. The DTW integrating clustering analysis revealed that the two clusters were optimal, with cluster 1 characterized by severe manic symptoms, which then improved, and cluster 2, characterized by milder manic severity, which also improved. The two clusters showed different ACTH levels (t = 2.289, p = 0.022), and logistic regression analysis revealed a slight positive association between ACTH levels and cluster 1. Furthermore, the mediation analysis indicated that ACTH influences curative efficacy via conscientiousness (βc =0.103, p=0.001).

DISCUSSION

In conclusion, we found that a higher level of ACTH is associated with severe manic symptoms, indicating a chronic stress response in BD patients. Additionally, the ACTH levels affect short-term BD curative efficacy via the mediation of conscientiousness, providing a psychotherapeutic strategy direction for BD.

A Modeling and Machine Learning Pipeline to Rationally Design Treatments to Restore Neuroendocrine Disorders in Heterogeneous Individuals

Heterogeneity among individual patients presents a fundamental challenge to effective treatment, since a treatment protocol working for a portion of the population often fails in others. We hypothesize that a computational pipeline integrating mathematical modeling and machine learning could be used to address this fundamental challenge and facilitate the optimization of individualized treatment protocols. We tested our hypothesis with the neuroendocrine systems controlled by the hypothalamic–pituitary–adrenal (HPA) axis. With a synergistic combination of mathematical modeling and machine learning (ML), this integrated computational pipeline could indeed efficiently reveal optimal treatment targets that significantly contribute to the effective treatment of heterogeneous individuals. What is more, the integrated pipeline also suggested quantitative information on how these key targets should be perturbed. Based on such ML revealed hints, mathematical modeling could be used to rationally design novel protocols and test their performances. We believe that this integrated computational pipeline, properly applied in combination with other computational, experimental and clinical research tools, can be used to design novel and improved treatment against a broad range of complex diseases.

Mathematical modeling of depressive disorders: Circadian driving, bistability and dynamical transitions

The hypothalamus-pituitary-adrenal (HPA) axis is a key neuroendocrine system implicated in stress response, major depression disorder, and post-traumatic stress disorder. We present a new, compact dynamical systems model for the response of the HPA axis to external stimuli, representing stressors or therapeutic intervention, in the presence of a circadian input. Our work builds upon previous HPA axis models where hormonal dynamics are separated into slow and fast components. Several simplifications allow us to derive an effective model of two equations, similar to a multiplicative-input FitzHugh-Nagumo system, where two stable states, a healthy and a diseased one, arise. We analyze the effective model in the context of state transitions driven by external shocks to the hypothalamus, but also modulated by circadian rhythms. Our analyses provide mechanistic insight into the effects of the circadian cycle on input driven transitions of the HPA axis and suggest a circadian influence on exposure or cognitive behavioral therapy in depression, or post-traumatic stress disorder treatment.

An integrate-and-fire model for pulsatility in the neuroendocrine system

A model for pulsatility in neuroendocrine regulation is proposed which combines Goodwin-type feedback control with impulsive input from neurons located in the hypothalamus. The impulsive neural input is modeled using an integrate-and-fire mechanism; namely, inputs are generated only when the membrane potential crosses a threshold, after which it is reset to baseline. The resultant model takes the form of a functional-differential equation with continuous and impulsive components. Despite the impulsive nature of the inputs, realistic hormone profiles are generated, including ultradian and circadian rhythms, pulsatile secretory patterns, and even chaotic dynamics.

Mathematical Modelling of Endocrine Systems

Hormone rhythms are ubiquitous and essential to sustain normal physiological functions. Combined mathematical modelling and experimental approaches have shown that these rhythms result from regulatory processes occurring at multiple levels of organisation and require continuous dynamic equilibration, particularly in response to stimuli. We review how such an interdisciplinary approach has been successfully applied to unravel complex regulatory mechanisms in the metabolic, stress, and reproductive axes. We discuss how this strategy is likely to be instrumental for making progress in emerging areas such as chronobiology and network physiology. Ultimately, we envisage that the insight provided by mathematical models could lead to novel experimental tools able to continuously adapt parameters to gradual physiological changes and the design of clinical interventions to restore normal endocrine function.

High-fidelity discrete modeling of the HPA axis: a study of regulatory plasticity in biology

BACKGROUND

The hypothalamic-pituitary-adrenal (HPA) axis is a central regulator of stress response and its dysfunction has been associated with a broad range of complex illnesses including Gulf War Illness (GWI) and Chronic Fatigue Syndrome (CFS). Though classical mathematical approaches have been used to model HPA function in isolation, its broad regulatory interactions with immune and central nervous function are such that the biological fidelity of simulations is undermined by the limited availability of reliable parameter estimates.

METHOD

Here we introduce and apply a generalized discrete formalism to recover multiple stable regulatory programs of the HPA axis using little more than connectivity between physiological components. This simple discrete model captures cyclic attractors such as the circadian rhythm by applying generic constraints to a minimal parameter set; this is distinct from Ordinary Differential Equation (ODE) models, which require broad and precise parameter sets. Parameter tuning is accomplished by decomposition of the overall regulatory network into isolated sub-networks that support cyclic attractors. Network behavior is simulated using a novel asynchronous updating scheme that enforces priority with memory within and between physiological compartments.

RESULTS

Consistent with much more complex conventional models of the HPA axis, this parsimonious framework supports two cyclic attractors, governed by higher and lower levels of cortisol respectively. Importantly, results suggest that stress may remodel the stability landscape of this system, favoring migration from one stable circadian cycle to the other. Access to each regime is dependent on HPA axis tone, captured here by the tunable parameters of the multi-valued logic. Likewise, an idealized glucocorticoid receptor blocker alters the regulatory topology such that maintenance of persistently low cortisol levels is rendered unstable, favoring a return to normal circadian oscillation in both cortisol and glucocorticoid receptor expression.

CONCLUSION

These results emphasize the significance of regulatory connectivity alone and how regulatory plasticity may be explored using simple discrete logic and minimal data compared to conventional methods.

Perturbing the Hypothalamic-Pituitary-Adrenal Axis: A Mathematical Model for Interpreting PTSD Assessment Tests

We use a dynamical systems model of the hypothalamic-pituitary-adrenal (HPA) axis to understand the mechanisms underlying clinical protocols used to probe patient stress response. Specifically, we address dexamethasone (DEX) and ACTH challenge tests, which probe pituitary and adrenal gland responses, respectively. We show that some previously observed features and experimental responses can arise from a bistable mathematical model containing two steady-states, rather than relying on specific and permanent parameter changes due to physiological disruption. Moreover, we show that the timing of a perturbation relative to the intrinsic oscillation of the HPA axis can affect challenge test responses. Conventional mechanistic hypotheses supported and refuted by the challenge tests are reexamined by varying parameters in our mathematical model associated with these hypotheses. We show that (a) adrenal hyposensitivity can give rise to the responses seen in ACTH challenge tests and (b) enhanced cortisol-mediated suppression of the pituitary in subjects with PTSD is not necessary to explain the responses observed in DEX stress tests. We propose a new two-stage DEX/external stressor protocol to more clearly distinguish between the conventional hypothesis of enhanced suppression of the pituitary and bistable dynamics hypothesized in our model.

Blunted basal corticosterone pulsatility predicts post-exposure susceptibility to PTSD phenotype in rats

The basal activity of the hypothalamic-pituitary-adrenal axis is highly dynamic and is characterized by both circadian and ultradian (pulsatile) patterns of hormone secretion. Pulsatility of glucocorticoids has been determined to be critical for optimal transcriptional, neuroendocrine, and behavioral responses. We used an animal model of post-traumatic stress disorder (PTSD) to assess whether stress-induced impairment of behavioral responses is correlated with aberrant secretion of corticosterone. Serial blood samples were collected manually via the jugular vein cannula during the light-(inactive)-phase in conscious male rats at 20-min intervals for a period of 5h before and 6.5h after exposure to predator scent stress. The outcome measures included behavior in an elevated plus-maze and acoustic startle response 7days after exposure. Individual animals were retrospectively classified as having "extreme", "partial", or "minimal" behavioral responses according to pre-set cut-off criteria for behavioral response patterns. Corticosterone secretion patterns were analyzed retrospectively. Under basal conditions, the amplitude of ultradian oscillations of corticosterone levels, rather than the mean corticosterone level or the frequency of corticosterone pulsatility, was significantly reduced in individuals who displayed PTSD-phenotype 8days later. In addition, extreme disruption of behavior on day 8 post-exposure was also characterized by a blunting of corticosterone response to the stressor. Animals with behavior that was only partially affected or unaffected displayed none of the above changes. Blunted basal corticosterone pulse amplitude is a pre-existing susceptibility or risk factor for PTSD, which originates from prior (life) experiences and may therefore predict post-exposure PTSD-phenotype in rats.

The evolution of the molecular response to stress and its relevance to trauma and stressor-related disorders

The experience of "stress", in its broadest meaning, is an inevitable part of life. All living creatures have evolved multiple mechanisms to deal with such threats and challenges and to avoid damage to the organism that may be incurred from these stress responses. Trauma and stressor-related disorders are psychiatric conditions that are caused specifically by the experience of stress, though depression, anxiety and some other disorders may also be unleashed by stress. Stress, however, is not a mandatory criterion of these diagnoses. This article focuses on the evolution of the neurochemicals involved in the response to stress and the systems in which they function. This includes the skin and gut, and the immune system. Evidence suggests that responses to stress are evolutionarily highly conserved, have wider involvement than the hypothalamic pituitary adrenal stress axis alone, and that excessive stress responses can produce stressor-related disorders in both humans and animals.