Generation of action potentials in a mathematical model of corticotrophs

Fast dynamics in the HPA axis: Insight from mathematical and experimental studies

The activity of the hypothalamic-pituitary-adrenal (HPA) axis is characterised by complex dynamics spanning several timescales. This ranges from slow circadian rhythms in blood hormone concentration to faster ultradian pulses of hormone secretion and even more rapid oscillations in electrical and calcium activity in neuroendocrine cells of the hypothalamus and pituitary gland. Here, we focus on the system's oscillations on the short timescale. We highlight some of the mathematical modelling and experimental work that has been carried out to characterise the mechanisms regulating this highly dynamic mode of neuroendocrine signalling and discuss some future directions that may be explored to enhance understanding of HPA function.

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.

A computational model for gonadotropin releasing cells in the teleost fish medaka

Pituitary endocrine cells fire action potentials (APs) to regulate their cytosolic Ca²⁺ concentration and hormone secretion rate. Depending on animal species, cell type, and biological conditions, pituitary APs are generated either by TTX-sensitive Na⁺ currents (I_Na), high-voltage activated Ca²⁺ currents (I_Ca), or by a combination of the two. Previous computational models of pituitary cells have mainly been based on data from rats, where I_Na is largely inactivated at the resting potential, and spontaneous APs are predominantly mediated by I_Ca. Unlike in rats, spontaneous I_Na-mediated APs are consistently seen in pituitary cells of several other animal species, including several species of fish. In the current work we develop a computational model of gonadotropin releasing cells in the teleost fish medaka (Oryzias latipes). The model stands out from previous modeling efforts by being (1) the first model of a pituitary cell in teleosts, (2) the first pituitary cell model that fires sponateous APs that are predominantly mediated by I_Na, and (3) the first pituitary cell model where the kinetics of the depolarizing currents, I_Na and I_Ca, are directly fitted to voltage-clamp data. We explore the firing properties of the model, and compare it to the properties of previous models that fire I_Ca-based APs. We put a particular focus on how the big conductance K⁺ current (I_BK) modulates the AP shape. Interestingly, we find that I_BK can prolong AP duration in models that fire I_Ca-based APs, while it consistently shortens the duration of the predominantly I_Na-mediated APs in the medaka gonadotroph model. Although the model is constrained to experimental data from gonadotroph cells in medaka, it may likely provide insights also into other pituitary cell types that fire I_Na-mediated APs.

Excitable cells elicit electrical pulses called action potentials (APs), which are generated and shaped by a combination of ion channels in the cell membrane. Since one type of ion channels is permeable to Ca²⁺ ions, there is typically an influx of Ca²⁺ during an AP. Pituitary cells therefore use AP firing to regulate their cytosolic Ca²⁺ concentration, which in turn controls their hormone secretion rate. The amount of Ca²⁺ that enters during an AP depends strongly on how long it lasts, and it is therefore important to understand the mechanisms that control this. Pituitary APs are generally mediated by a combination of Ca²⁺ channels and Na⁺ channels, and the relative contributions of from the two vary between cell types, animal species and biological conditions. Previous computer models have predominantly been adapted to data from pituitary cells which tend to fire Ca²⁺-based APs. Here we develop a new model, adapted to data from pituitary cells in the fish medaka, which APs that are predominantly Na⁺-based, and compare its dynamical properties to the previous models that fire Ca²⁺-based APs.

Common and diverse elements of ion channels and receptors underlying electrical activity in endocrine pituitary cells

The pituitary gland contains six types of endocrine cells defined by hormones they secrete: corticotrophs, melanotrophs, gonadotrophs, thyrotrophs, somatotrophs, and lactotrophs. All these cell types are electrically excitable, and voltage-gated calcium influx is the major trigger for their hormone secretion. Along with hormone intracellular content, G-protein-coupled receptor and ion channel expression can also be considered as defining cell type identity. While many aspects of the developmental and activity dependent regulation of hormone and G-protein-coupled receptor expression have been elucidated, much less is known about the regulation of the ion channels needed for excitation-secretion coupling in these cells. We compare the spontaneous and receptor-controlled patterns of electrical signaling among endocrine pituitary cell types, including insights gained from mathematical modeling. We argue that a common set of ionic currents unites these cells, while differential expression of another subset of ionic currents could underlie cell type-specific patterns. We demonstrate these ideas using a generic mathematical model, showing that it reproduces many observed features of pituitary electrical signaling. Mapping these observations to the developmental lineage suggests possible modes of regulation that may give rise to mature pituitary cell types.

Modeling the diversity of spontaneous and agonist-induced electrical activity in anterior pituitary corticotrophs

Pituitary corticotrophs fire action potentials spontaneously and in response to stimulation with corticotropin-releasing hormone (CRH) and arginine vasopressin (AVP), and such electrical activity is critical for calcium signaling and calcium-dependent adrenocorticotropic hormone secretion. These cells typically fire tall, sharp action potentials when spontaneously active, but a variety of other spontaneous patterns have also been reported, including various modes of bursting. There is variability in reports of the fraction of corticotrophs that are electrically active, as well as their patterns of activity, and the sources of this variation are not well understood. The ionic mechanisms responsible for CRH- and AVP-triggered electrical activity in corticotrophs are also poorly characterized. We use electrophysiological measurements and mathematical modeling to investigate possible sources of variability in patterns of spontaneous and agonist-induced corticotroph electrical activity. In the model, variation in as few as two parameters can give rise to many of the types of patterns observed in electrophysiological recordings of corticotrophs. We compare the known mechanisms for CRH, AVP, and glucocorticoid actions and find that different ionic mechanisms can contribute in different but complementary ways to generate the complex time courses of CRH and AVP responses. In summary, our modeling suggests that corticotrophs have several mechanisms at their disposal to achieve their primary function of pacemaking depolarization and increased electrical activity in response to CRH and AVP.NEW & NOTEWORTHY We and others recently demonstrated that the electrical activity and calcium dynamics of corticotrophs are strikingly diverse, both spontaneously and in response to the agonists CRH and AVP. Here we demonstrate this diversity with electrophysiological measurements and use mathematical modeling to investigate its possible sources. We compare the known mechanisms of agonist-induced activity in the model, showing how the context of ionic conductances dictates the effects of agonists even when their target is fixed.

The relative contributions of store-operated and voltage-gated Ca2+ channels to the control of Ca2+ oscillations in airway smooth muscle

KEY POINTS

Agonist-dependent oscillations in the concentration of free cytosolic calcium are a vital mechanism for the control of airway smooth muscle contraction and thus are a critical factor in airway hyper-responsiveness. Using a mathematical model, closely tied to experimental work, we show that the oscillations in membrane potential accompanying the calcium oscillations have no significant effect on the properties of the calcium oscillations. In addition, the model shows that calcium entry through store-operated calcium channels is critical for calcium oscillations, but calcium entry through voltage-gated channels has much less effect. The model predicts that voltage-gated channels are less important than store-operated channels in the control of airway smooth muscle tone.

ABSTRACT

Airway smooth muscle contraction is typically the key mechanism underlying airway hyper-responsiveness, and the strength of muscle contraction is determined by the frequency of oscillations of intracellular calcium (Ca²⁺ ) concentration. In airway smooth muscle cells, these Ca²⁺ oscillations are caused by cyclic Ca²⁺ release from the sarcoplasmic reticulum, although Ca²⁺ influx via plasma membrane channels is also necessary to sustain the oscillations over longer times. To assess the relative contributions of store-operated and voltage-gated Ca²⁺ channels to this Ca²⁺ influx, we generated a comprehensive mathematical model, based on experimental Ca²⁺ measurements in mouse precision-cut lung slices, to simulate Ca²⁺ oscillations and changes in membrane potential. Agonist-induced Ca²⁺ oscillations are accompanied by oscillations in membrane potential, although the membrane potential oscillations are too small to generate large Ca²⁺ currents through voltage-gated Ca²⁺ channels, and thus have little effect on the Ca²⁺ oscillations. Ca²⁺ entry through voltage-gated channels only becomes important when the cell is depolarized (e.g. by a high external K⁺ concentration). As a result, agonist-induced Ca²⁺ oscillations are critically dependent on Ca²⁺ entry through store-operated channels but do not depend strongly on Ca²⁺ entry though voltage-gated channels.

BK Channels and the Control of the Pituitary

The pituitary gland provides the important link between the nervous system and the endocrine system and regulates a diverse range of physiological functions. The pituitary is connected to the hypothalamus by the pituitary stalk and is comprised primarily of two lobes. The anterior lobe consists of five hormone-secreting cell types which are electrically excitable and display single-spike action potentials as well as complex bursting patterns. Bursting is of particular interest as it raises intracellular calcium to a greater extent than spiking and is believed to underlie secretagogue-induced hormone secretion. BK channels have been identified as a key regulator of bursting in anterior pituitary cells. Experimental data and mathematical modeling have demonstrated that BK activation during the upstroke of an action potential results in a prolonged depolarization and an increase in intracellular calcium. In contrast, the posterior lobe is primarily composed of axonal projections of magnocellular neurosecretory cells which extend from the supraoptic and paraventricular nuclei of the hypothalamus. In these neuroendocrine cells, BK channel activation results in a decrease in excitability and hormone secretion. The opposite effect of BK channels in the anterior and posterior pituitary highlights the diverse role of BK channels in regulating the activity of excitable cells. Further studies of pituitary cell excitability and the specific role of BK channels would lead to a greater understanding of how pituitary cell excitability is regulated by both hypothalamic secretagogues and negative feedback loops, and could ultimately lead to novel treatments to pituitary-related disorders.

Mathematical modeling of the hypothalamic-pituitary-adrenal gland (HPA) axis, including hippocampal mechanisms

This paper presents a mathematical model of the HPA axis. The HPA axis consists of the hypothalamus, the pituitary and the adrenal glands in which the three hormones CRH, ACTH and cortisol interact through receptor dynamics. Furthermore, it has been suggested that receptors in the hippocampus have an influence on the axis. A model is presented with three coupled, non-linear differential equations, with the hormones CRH, ACTH and cortisol as variables. The model includes the known features of the HPA axis, and includes the effects from the hippocampus through its impact on CRH in the hypothalamus. The model is investigated both analytically and numerically for oscillating solutions, related to the ultradian rhythm seen in data, and for multiple fixed points related to hypercortisolemic and hypocortisolemic depression. The existence of an attracting trapping region guarantees that solution curves stay non-negative and bounded, which can be interpreted as a mathematical formulation of homeostasis. No oscillating solutions are present when using physiologically reasonable parameter values. This indicates that the ultradian rhythm originate from different mechanisms. Using physiologically reasonable parameters, the system has a unique fixed point, and the system is globally stable. Therefore, solutions converge to the fixed point for all initial conditions. This is in agreement with cortisol levels returning to normal, after periods of mild stress, in healthy individuals. Perturbing parameters lead to a bifurcation, where two additional fixed points emerge. Thus, the system changes from having a unique stable fixed point into having three fixed points. Of the three fixed points, two are stable and one is unstable. Further investigations show that solutions converge to one of the two stable fixed points depending on the initial conditions. This could explain why healthy people becoming depressed usually fall into one of two groups: a hypercortisolemic depressive group or a hypocortisolemic depressive group.

Control of hypothalamic-pituitary-adrenal stress axis activity by the intermediate conductance calcium-activated potassium channel, SK4

The anterior pituitary corticotroph is a major control point for the regulation of the hypothalamic-pituitary-adrenal (HPA) axis and the neuroendocrine response to stress. Although corticotrophs are known to be electrically excitable, ion channels controlling the electrical properties of corticotrophs are poorly understood. Here, we exploited a lentiviral transduction system to allow the unequivocal identification of live murine corticotrophs in culture. We demonstrate that corticotrophs display highly heterogeneous spontaneous action-potential firing patterns and their resting membrane potential is modulated by a background sodium conductance. Physiological concentrations of corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) cause a depolarization of corticotrophs, leading to a sustained increase in action potential firing. A major component of the outward potassium conductance was mediated via intermediate conductance calcium-activated (SK4) potassium channels. Inhibition of SK4 channels with TRAM-34 resulted in an increase in corticotroph excitability and exaggerated CRH/AVP-stimulated ACTH secretion in vitro. In accordance with a physiological role for SK4 channels in vivo, restraint stress-induced plasma ACTH and corticosterone concentrations were significantly enhanced in gene-targeted mice lacking SK4 channels (Kcnn4(-/-)). In addition, Kcnn4(-/-) mutant mice displayed enhanced hypothalamic c-fos and nur77 mRNA expression following restraint, suggesting increased neuronal activation. Thus, stress hyperresponsiveness observed in Kcnn4(-/-) mice results from enhanced secretagogue-induced ACTH output from anterior pituitary corticotrophs and may also involve increased hypothalamic drive, thereby suggesting an important role for SK4 channels in HPA axis function.

A feedback control model for cortisol secretion

Existing mathematical models for cortisol secretion do not describe the entire cortisol secretion process, from the neural firing of corticotropin releasing hormone (CRH) in the hypothalamus to cortisol concentration in the plasma. In this paper, we lay the groundwork to construct a more comprehensive model, relating CRH, Adrenocorticotropic hormone (ACTH), and cortisol. We start with an existing mathematical model for cortisol secretion, and combine it with a simplified neural firing model that describes CRH and ACTH release. This simplified neural firing model is obtained using the extended FitzHugh-Nagumo (FHN) model, which includes a time-varying spiking threshold [3]. A key feature of our model is the presence of a feedback loop from cortisol secretion to ACTH secretion.

Ion channels and signaling in the pituitary gland

Endocrine pituitary cells are neuronlike; they express numerous voltage-gated sodium, calcium, potassium, and chloride channels and fire action potentials spontaneously, accompanied by a rise in intracellular calcium. In some cells, spontaneous electrical activity is sufficient to drive the intracellular calcium concentration above the threshold for stimulus-secretion and stimulus-transcription coupling. In others, the function of these action potentials is to maintain the cells in a responsive state with cytosolic calcium near, but below, the threshold level. Some pituitary cells also express gap junction channels, which could be used for intercellular Ca(2+) signaling in these cells. Endocrine cells also express extracellular ligand-gated ion channels, and their activation by hypothalamic and intrapituitary hormones leads to amplification of the pacemaking activity and facilitation of calcium influx and hormone release. These cells also express numerous G protein-coupled receptors, which can stimulate or silence electrical activity and action potential-dependent calcium influx and hormone release. Other members of this receptor family can activate calcium channels in the endoplasmic reticulum, leading to a cell type-specific modulation of electrical activity. This review summarizes recent findings in this field and our current understanding of the complex relationship between voltage-gated ion channels, ligand-gated ion channels, gap junction channels, and G protein-coupled receptors in pituitary cells.

A mathematical analysis of agonist- and KCl-induced Ca(2+) oscillations in mouse airway smooth muscle cells

Airway hyperresponsiveness is a major characteristic of asthma and is generally ascribed to excessive airway narrowing associated with the contraction of airway smooth muscle cells (ASMCs). ASMC contraction is initiated by a rise in intracellular calcium concentration ([Ca(2+)](i)), observed as oscillatory Ca(2+) waves that can be induced by either agonist or high extracellular K(+) (KCl). In this work, we present a model of oscillatory Ca(2+) waves based on experimental data that incorporate both the inositol trisphosphate receptor and the ryanodine receptor. We then combined this Ca(2+) model and our modified actin-myosin cross-bridge model to investigate the role and contribution of oscillatory Ca(2+) waves to contractile force generation in mouse ASMCs. The model predicts that: 1), the difference in behavior of agonist- and KCl-induced Ca(2+) waves results principally from the fact that the sarcoplasmic reticulum is depleted during agonist-induced oscillations, but is overfilled during KCl-induced oscillations; 2), regardless of the order in which agonist and KCl are added into the cell, the resulting [Ca(2+)](i) oscillations will always be the short-period, agonist-induced-like oscillations; and 3), both the inositol trisphosphate receptor and the ryanodine receptor densities are higher toward one end of the cell. In addition, our results indicate that oscillatory Ca(2+) waves generate less contraction than whole-cell Ca(2+) oscillations induced by the same agonist concentration. This is due to the spatial inhomogeneity of the receptor distributions.

Regulation of Ca v 3.1 channels by glucocorticoids

The activity of low voltage-activated Ca(2+) (Ca(V)3) channels is tightly coupled to neurotransmitter and hormone secretion. Previous studies have shown that Ca(V)3 channels are regulated by glucocorticoids (GCs), though the mechanism underlying channel regulation remains unclear. Here, using the pituitary GH(3) cell line as a model, we investigated whether Ca(V)3 channel expression is under the control of GCs, and if their actions are mediated by transcriptional and/or post-transcriptional mechanisms. RT-PCR and western blot analyses showed that Ca(V)3.1 but not Ca(V)3.2 and Ca(V)3.3 channels is expressed in the GH(3) cells, and patch clamp recordings confirmed that Ca(2+) currents through low voltage-activated channels were decreased after chronic treatment with GCs. Consistent with this, total plasma membrane expression of Ca(V)3.1 protein as analyzed by cell-surface biotinylation assays and semi-quantitative western blotting was also down-regulated, while quantitative real-time RT-PCR analysis revealed a significant decrease of Ca(V)3.1 mRNA expression in the treated cells. In contrast, patch-clamp recordings on HEK-293 cells stably expressing recombinant Ca(V)3.1 channels showed that Ca(2+) currents were not affected by GC treatment. These results suggest that decreased transcription is a likely mechanism to explain the inhibitory actions of GCs on the functional expression of native Ca(V)3.1 channels.

Role of L-type Ca2+ channels in iron transport and iron-overload cardiomyopathy

Excessive body iron or iron overload occurs under conditions such as primary (hereditary) hemochromatosis and secondary iron overload (hemosiderosis), which are reaching epidemic levels worldwide. Primary hemochromatosis is the most common genetic disorder with an allele frequency greater than 10% in individuals of European ancestry, while hemosiderosis is less common but associated with a much higher morbidity and mortality. Iron overload leads to iron deposition in many tissues especially the liver, brain, heart and endocrine tissues. Elevated cardiac iron leads to diastolic dysfunction, arrhythmias and dilated cardiomyopathy, and is the primary determinant of survival in patients with secondary iron overload as well as a leading cause of morbidity and mortality in primary hemochromatosis patients. In addition, iron-induced cardiac injury plays a role in acute iron toxicosis (iron poisoning), myocardial ischemia–reperfusion injury, Friedreich ataxia and neurodegenerative diseases. Patients with iron overload also routinely suffer from a range of endocrinopathies, including diabetes mellitus and anterior pituitary dysfunction. Despite clear connections between elevated iron and clinical disease, iron transport remains poorly understood. While low-capacity divalent metal and transferrin-bound transporters are critical under normal physiological conditions, L-type Ca²⁺ channels (LTCC) are high-capacity pathways of ferrous iron (Fe²⁺) uptake into cardiomyocytes especially under iron overload conditions. Fe²⁺ uptake through L-type Ca²⁺ channels may also be crucial in other excitable cells such as pancreatic beta cells, anterior pituitary cells and neurons. Consequently, LTCC blockers represent a potential new therapy to reduce the toxic effects of excess iron.

CRH-induced electrical activity and calcium signalling in pituitary corticotrophs

Pituitary corticotroph cells generate repetitive action potentials and associated Ca2+ transients in response to the agonist corticotropin releasing hormone (CRH). There is indirect evidence suggesting that the agonist, by way of complex intracellular mechanisms, modulates the voltage sensitivity of the L-type Ca2+ channels embedded in the plasma membrane. We have previously constructed a Hodgkin-Huxley-type model of this process, which indicated that an increase in the L-type Ca2+ current is sufficient to generate repetitive action potentials (LeBeau et al. (1997). Biophys. J.73, 1263-1275). CRH is also believed to inhibit an inwardly rectifying K+ current. In this paper, we have found that a CRH-induced inhibition of the inwardly rectifying K+ current increases the model action potential firing frequency, [Ca2+]i transients and membrane excitability. This dual modulatory action of CRH on inward rectifier and voltage-gated Ca2+ channels better describes the observed CRH-induced effects. This structural alteration to the model along with parameter changes bring the model firing frequency in line with experimental data. We also show that the model exhibits experimentally observed bursting behaviour, where the depolarization spike is followed by small oscillations in the membrane potential.