A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons

Modelling KNDy neurons and gonadotropin-releasing hormone pulse generation

The pulsatile release of gonadotropin-releasing hormone (GnRH) and its frequency are crucial for healthy reproductive function. To understand what drives GnRH pulses, a combination of experimental and mathematical modelling approaches has been used. Early work focussed on the possibility that GnRH pulse generation is an intrinsic feature of GnRH neurons, with autocrine feedback generating pulsatility. However, there is now ample evidence suggesting that a network of upstream neurons secreting kisspeptin, neurokinin-B and dynorphin are the source of this GnRH pulse generator. The interplay of slow positive and negative feedback via neurokinin-B and dynorphin, respectively, allows the network to act as a relaxation oscillator, driving pulsatile secretion of kisspeptin, and consequently, of GnRH and LH. Here, we review the mathematical modelling approaches exploring both scenarios and suggest that with pulsatile GnRH secretion driven by the KNDy pulse generator, autocrine feedback still has the potential to modulate GnRH output.

Proteomic Analysis of Hypothalamus and Pituitary Gland in Pre and Postpubertal Brahman Heifers

The hypothalamus and the pituitary gland are directly involved in the complex systemic changes that drive the onset of puberty in cattle. Here, we applied integrated bioinformatics to elucidate the critical proteins underlying puberty and uncover potential molecular mechanisms from the hypothalamus and pituitary gland of prepubertal (n = 6) and postpubertal (n = 6) cattle. Proteomic analysis in the hypothalamus and pituitary gland revealed 275 and 186 differentially abundant (DA) proteins, respectively (adjusted p-value < 0.01). The proteome profiles found herein were integrated with previously acquired transcriptome profiles. These transcriptomic studies used the same tissues harvested from the same heifers at pre- and post-puberty. This comparison detected a small number of matched transcripts and protein changes at puberty in each tissue, suggesting the need for multiple omics analyses for interpreting complex biological systems. In the hypothalamus, upregulated DA proteins at post-puberty were enriched in pathways related to puberty, including GnRH, calcium and oxytocin signalling pathways, whereas downregulated proteins were observed in the estrogen signalling pathway, axon guidance and GABAergic synapse. Additionally, this study revealed that ribosomal pathway proteins in the pituitary were involved in the pubertal development of mammals. The reported molecules and derived protein-protein networks are a starting point for future experimental approaches that might dissect with more detail the role of each molecule to provide new insights into the mechanisms of puberty onset in cattle.

Mathematical models in GnRH research

Mathematical modelling is an indispensable tool in modern biosciences, enabling quantitative analysis and integration of biological data, transparent formulation of our understanding of complex biological systems, and efficient experimental design based on model predictions. This review article provides an overview of the impact that mathematical models had on GnRH research. Indeed, over the last 20 years mathematical modelling has been used to describe and explore the physiology of the GnRH neuron, the mechanisms underlying GnRH pulsatile secretion, and GnRH signalling to the pituitary. Importantly, these models have contributed to GnRH research via novel hypotheses and predictions regarding the bursting behaviour of the GnRH neuron, the role of kisspeptin neurons in the emergence of pulsatile GnRH dynamics, and the decoding of GnRH signals by biochemical signalling networks. We envisage that with the advent of novel experimental technologies, mathematical modelling will have an even greater role to play in our endeavour to understand the complex spatiotemporal dynamics underlying the reproductive neuroendocrine system.

Gonadotrophin-releasing hormone and kisspeptin: It takes two to tango

Kisspeptin (Kp), a family of peptides comprising products of the Kiss1 gene, was discovered 20 years ago; it is recognised as the major factor controlling the activity of the gonadotrophin-releasing hormone (GnRH) neurones and thus the activation of the reproductive axis in mammals. It has been widely documented that the effects of Kp on reproduction through its action on GnRH neurones are mediated by the GPR54 receptor. Kp controls the activation of the reproductive axis at puberty, maintains reproductive axis activity in adults and is involved in triggering ovulation in some species. Although there is ample evidence coming from both conditional knockout models and conditional-induced Kp neurone death implicating the Kp/GPR54 pathway in the control of reproduction, the mechanism(s) underlying this process may be more complex than a sole direct control of GnRH neuronal activity by Kp. In this review, we provide an overview of the recent advances made in elucidating the interplay between Kp- and GnRH- neuronal networks with respect to regulating the reproductive axis. We highlight the existence of a possible mutual regulation between GnRH and Kp neurones, as well as the implication of Kp-dependent volume transmission in this process. We also discuss the capacity of heterodimerisation between GPR54 and GnRH receptor (GnRH-R) and its consequences on signalling. Finally, we illustrate the role of mathematical modelling that accounts for the synergy between GnRH-R and GPR54 in explaining the role of these two receptors when defining GnRH neuronal activity and GnRH pulsatile release.

Modeling circadian regulation of ovulation timing: age-related disruption of estrous cyclicity

The circadian clocks within the hypothalamic–pituitary–gonadal axis control estrous cycles in female rodents. The suprachiasmatic nucleus (SCN), where the central clock is located, generates daily signals to trigger surge release of luteinizing hormone (LH), which in turn induces ovulation. It has been observed in aged rodents that output from the SCN such as neuronal firing activity is declined, and estrous cycles become irregular and finally stop. Circadian clock mutants display accelerated reproductive aging, suggesting the complicated interplay between the circadian system and the endocrine system. To investigate such circadian regulation of estrous cycles, we construct a mathematical model that describes dynamics of key hormones such as LH and of circadian clocks in the SCN and in the ovary, and simulate estrous cycles for various parameter values. Our simulation results demonstrate that reduction of the amplitude of the SCN signal, which is a symptom of aging, makes estrous cycles irregular. We also show that variation in the phase of the SCN signal and changes in the period of ovarian circadian clocks exacerbates the aging effect on estrous cyclicity. Our study suggests that misalignment between the SCN and ovarian circadian oscillations is one of the primary causes of the irregular estrous cycles.

Synchronization of active rotators interacting with environment

Multiple organs in a living system respond to environmental changes, and the signals from the organs regulate the physiological environment. Inspired by this biological feedback, we propose a simple autonomous system of active rotators to explain how multiple units are synchronized under a fluctuating environment. We find that the feedback via an environment can entrain rotators to have synchronous phases for specific conditions. This mechanism is markedly different from the simple entrainment by a common oscillatory external stimulus that is not interacting with systems. We theoretically examine how the phase synchronization depends on the interaction strength between rotators and environment. Furthermore, we successfully demonstrate the proposed model by realizing an analog electric circuit with microelectronic devices. This bioinspired platform can be used as a sensor for monitoring varying environments and as a controller for amplifying signals by their feedback-induced synchronization.

How Pulsatile Kisspeptin Stimulation and GnRH Autocrine Feedback Can Drive GnRH Secretion: A Modeling Investigation

Pulsatile secretion of GnRH from hypothalamic GnRH neurons tightly regulates the release of mammalian reproductive hormones. Although key factors such as electrical activity and stimulation by kisspeptin have been extensively studied, the underlying mechanisms that regulate GnRH release are still not fully understood. Previously developed mathematical models studied hormonal release and electrical properties of GnRH neurons separately, but they never integrated both components. Herein, we present a more complete biophysical model to investigate how electrical activity and hormonal release interact. The model consists of two components: an electrical submodel comprised of a modified Izhikevich formalism incorporating several key ionic currents to reproduce GnRH neuronal bursting behavior, and a hormonal submodel that incorporates pulsatile kisspeptin stimulation and a GnRH autocrine feedback mechanism. Using the model, we examine the electrical activity of GnRH neurons and how kisspeptin affects GnRH pulsatility. The model reproduces the noise-driven bursting behavior of GnRH neurons as well as the experimentally observed electrophysiological effects induced by GnRH and kisspeptin. Specifically, the model reveals that external application of GnRH causes a transient hyperpolarization followed by an increase in firing frequency, whereas administration of kisspeptin leads to long-lasting depolarization of the neuron. The model also shows that GnRH release follows a pulsatile profile similar to that observed experimentally and that kisspeptin and GnRH exhibit ∼7-1 locking in their pulsatility. These results suggest that external kisspeptin stimulation with a period of ∼8 minutes drives the autocrine mechanism beyond a threshold to generate pronounced GnRH pulses every hour.

Modeling the dynamic behavior of biochemical regulatory networks

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.

GnRH in the Human Female Reproductive Axis

Gonadotropin-releasing hormone (GnRH) is recognized as the central regulator of the functions of the pituitary-gonadal axis. The increasing knowledge on the mechanisms controlling the development and the function of GnRH-producing neurons is leading to a better diagnostic and therapeutic approach for hypogonadotropic hypogonadisms and for alterations of the puberty onset. During female life span, the function of the GnRH pulse generator may be affected by a number of inputs from other neuronal systems, offering alternative strategies for diagnostic and therapeutic interventions. Moreover, the identification of a GnRH/GnRH receptor system in both human ovary and endometrium has widened the spectrum of action of the peptide outside its hypothalamic functions. The pharmacological use of GnRH itself or its synthetic analogs (agonists and antagonists) provides a valid tool to either stimulate or block gonadotropin secretion and to modulate the female fertility in several reproductive disorders and in assisted reproduction technology. The use of GnRH agonists in young female patients undergoing chemotherapy is also considered a promising therapeutic approach to counteract iatrogenic ovarian failure.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

The gonadotropin-releasing hormone system: Perspectives from reproduction to cancer (Review)

Recently, an increasing amount of evidence indicates that human gonadotropin-releasing hormone (hGnRH) and its receptor (hGnRHR) are important regulatory components not only to the reproduction process but also in the regulation of some cancer cell functions such as cell proliferation, in both hormone-dependent and -independent types of tumors. The hGnRHR is a naturally misfolded protein that is retained mostly in the endoplasmic reticulum; however, this mechanism can be overcome by treatment with several pharmacoperones, therefore, increasing the amount of receptors in the cell membrane. In addition, several reports indicate that the expression level of hGnRHR in tumor cells is even lower than in pituitary or gonadotrope cells. The signal transduction pathways activated by hGnRH in both gonadotrope and different cancer cell types are described in the present review. We also discuss how the rescue of misfolded receptors in tumor cells could be a promising strategy for cancer therapy.

Synchronization of cells with activator-inhibitor pathways through adaptive environment-mediated coupling

In this paper, we report the synchronized dynamics of cells with activator-inhibitor pathways via an adaptive environment-mediated coupling scheme with feedbacks and control mechanisms. The adaptive character of the extracellular medium is modeled via its damping parameter as a physiological response aiming at annihilating the cellular differentiation existing between the chaotic biochemical pathways of the cells, in order to preserve homeostasis. We perform an investigation on the existence and stability of the synchronization manifold of the coupled system under the proposed coupling pattern. Both mathematical and computational tools suggest the accessibility of conducive prerequisites (conditions) for the emergence of a robust synchronous regime. The relevance of a phase-synchronized dynamics is appraised and several numerical indicators advocate for the prevalence of this fascinating phenomenon among the interacting cells in the phase space.

A model for the fast synchronous oscillations of firing rate in rat suprachiasmatic nucleus neurons cultured in a multielectrode array dish

When dispersed and cultured in a multielectrode dish (MED), suprachiasmatic nucleus (SCN) neurons express fast oscillations of firing rate (FOFR; fast relative to the circadian cycle), with burst duration ∼10 min, and interburst interval varying from 20 to 60 min in different cells but remaining nevertheless rather regular in individual cells. In many cases, separate neurons in distant parts of the 1 mm recording area of a MED exhibited correlated FOFR. Neither the mechanism of FOFR nor the mechanism of their synchronization among neurons is known. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as a key intercellular synchronizing agent, we built a model in which VIP acts as both a feedback regulator to generate FOFR in individual neurons, and a diffusible synchronizing agent to produce coherent electrical output of a neuronal network. In our model, VIP binding to its (VPAC₂) receptors acts through G_s G-proteins to activate adenylyl cyclase (AC), increase intracellular cAMP, and open cyclic-nucleotide-gated (CNG) cation channels, thus depolarizing the cell and generating neuronal firing to release VIP. In parallel, slowly developing homologous desensitization and internalization of VPAC₂ receptors terminates elevation of cAMP and thereby provides an interpulse silent interval. Through mathematical modeling, we show that this VIP/VPAC₂/AC/cAMP/CNG-channel mechanism is sufficient for generating reliable FOFR in single neurons. When our model for FOFR is combined with a published model of synchronization of circadian rhythms based on VIP/VPAC₂ and Per gene regulation synchronization of circadian rhythms is significantly accelerated. These results suggest that (a) auto/paracrine regulation by VIP/VPAC₂ and intracellular AC/cAMP/CNG-channels are sufficient to provide robust FOFR and synchrony among neurons in a heterogeneous network, and (b) this system may also participate in synchronization of circadian rhythms.

Cell-to-cell coordination for the spontaneous cAMP oscillation in Dictyostelium

We propose a new cellular dynamics scheme for the spontaneous cAMP oscillations in Dictyostelium discoideum. Our scheme seamlessly integrates both receptor dynamics and G-protein dynamics into our previously developed cellular dynamics scheme. Extensive computer simulation studies based on our new cellular dynamics scheme were conducted in mutant cells to evaluate the molecular network. The validity of our proposed molecular network as well as the controversial PKA-dependent negative feedback mechanism was supported by our simulation studies. Spontaneous cAMP oscillations were not observed in a single mutant cell. However, multicellular states of various mutant cells consistently initiated spontaneous cAMP oscillations. Therefore, cell-to-cell coordination via the cAMP receptor is essential for the robust initiation of spontaneous cAMP oscillations.

A mathematical model of pulse-coded hormone signal responses in pituitary gonadotroph cells

Cells in the pituitary that synthesize luteinizing and follicle-stimulating hormones regulate the relative production of these two key reproductive hormones in response to signals from the hypothalamus. These signals are encoded in the frequency of gonadotrophin-releasing-hormone pulses. In vitro experiments with a murine-derived cell line have identified key elements of the processes that decode the signal to regulate transcription of the subunits encoding these hormones. The mathematical model described in this paper is based on the results of those experiments and advances quantitative understanding of the biochemical decoder. The model consists of non-linear differential equations for each of six processes that lead to the synthesis of follicle-stimulating hormone. Simulations of the model exhibit key characteristics found in the experiments, including a preference for follicle-stimulating hormone synthesis at low pulse frequencies and a loss of this characteristic when a mutation is introduced.

Fast synchronous oscillations of firing rate in cultured rat suprachiasmatic nucleus neurons: possible role in circadian synchronization in the intact nucleus

The coherent circadian rhythm of the brain's master circadian pacemaker, the suprachiasmatic nucleus (SCN), is a result of synchronization of electrical activity of many SCN neurons possessing their own circadian oscillators. However, how the activity of these neurons is synchronized is not precisely known. By plotting the electrical firing rates of dispersed rat SCN neurons in multi-electrode array dishes with 20-s averaging of action-potential activity, we have investigated a novel phenomenon: fast (relative to the circadian cycle) oscillations of firing rate (FOFR) with duration of bursts ∼10min and interburst interval varying in a range from 20 to 60min in different cells, remaining nevertheless rather regular in individual cells. In many cases, separate neurons in distant parts of the 1mm recording area of an array exhibited correlated FOFR. FOFR of individual cells were positively or negatively correlated with those of other cells in a functioning neural network. Intriguingly, in occasional neuron pairs, transformation of their irregular firing to circadian peaks was accompanied by appearance of FOFR and an increase in the magnitude of firing correlation. We hypothesize that this FOFR observed in cultured SCN neurons contribute to synchronization of the circadian rhythm in the intact SCN.

Systems biology of cellular rhythms

Rhythms abound in biological systems, particularly at the cellular level where they originate from the feedback loops present in regulatory networks. Cellular rhythms can be investigated both by experimental and modeling approaches, and thus represent a prototypic field of research for systems biology. They have also become a major topic in synthetic biology. We review advances in the study of cellular rhythms of biochemical rather than electrical origin by considering a variety of oscillatory processes such as Ca++ oscillations, circadian rhythms, the segmentation clock, oscillations in p53 and NF-κB, synthetic oscillators, and the oscillatory dynamics of cyclin-dependent kinases driving the cell cycle. Finally we discuss the coupling between cellular rhythms and their robustness with respect to molecular noise.

Pituitary network connectivity as a mechanism for the luteinising hormone surge

Ovulation in vertebrates is caused by a surge of luteinising hormone (LH) from the pituitary. The LH surge is initiated by rising oestradiol concentration, although the precise mechanism of oestradiol action in humans and primates is not yet understood. Recent advances in labelling and three-dimensional imaging have revealed a rich pituitary structure of interwoven networks of different cell types. In the present study, we develop a mathematical model to test the hypothesis that oestradiol modulation of connectivity between pituitary cells can underlie the LH surge. In the model, gonadotrophin-releasing hormone (GnRH) pulses stimulate LH secretion by two independent mechanisms. The first mechanism corresponds to the well known direct action of GnRH on gonadotrophs, which is inhibited by the rising oestradiol concentration. The second mechanism of GnRH action is to stimulate a recurrent network of pituitary cells; in this case, the folliculostellate cells, which in turn stimulate LH secretion from the gonadotrophs. The network activity is modelled by a one-dimensional ordinary differential equation. The key to the LH surge in the model lies in the assumption that oestradiol modulates network connectivity. When the circulating oestradiol concentration is low, the network is barely connected, and cannot maintain a recurrent signal. When the oestradiol concentration is high, the network is highly connected, and maintains a high level of activity even after GnRH stimulation, thereby leading to a surge of LH secretion.

GnRH signaling, the gonadotrope and endocrine control of fertility

Mammalian reproductive cycles are controlled by an intricate interplay between the hypothalamus, pituitary and gonads. Central to the function of this axis is the ability of the pituitary gonadotrope to appropriately respond to stimulation by gonadotropin-releasing hormone (GnRH). This review focuses on the role of cell signaling and in particular, mitogen-activated protein kinase (MAPK) activities regulated by GnRH that are necessary for normal fertility. Recently, new mouse models making use of conditional gene deletion have shed new light on the relationships between GnRH signaling and fertility in both male and female mice. Within the reproductive axis, GnRH signaling is initiated through discrete membrane compartments in which the receptor resides leading to the activation of the extracellular signal-regulated kinases (ERKs 1/2). As defined by gonadotrope-derived cellular models, the ERKs appear to play a central role in the regulation of a cohort of immediate early genes that regulate the expression of late genes that, in part, define the differentiated character of the gonadotrope. Recent data would suggest that in vivo, conditional, pituitary-specific disruption of ERK signaling by GnRH leads to a gender-specific perturbation of fertility. Double ERK knockout in the anterior pituitary leads to female infertility due to LH biosynthesis deficiency and a failure in ovulation. In contrast, male mice are modestly LH deficient; however, this does not have an appreciable impact on fertility.

A design principle underlying the synchronization of oscillations in cellular systems

Biological oscillations are found ubiquitously in cells and are widely variable, with periods varying from milliseconds to months, and scales involving subcellular components to large groups of organisms. Interestingly, independent oscillators from different cells often show synchronization that is not the consequence of an external regulator. What is the underlying design principle of such synchronized oscillations, and can modeling show that the complex consequences arise from simple molecular or other interactions between oscillators? When biological oscillators are coupled with each other, we found that synchronization is induced when they are connected together through a positive feedback loop. Increasing the coupling strength of two independent oscillators shows a threshold beyond which synchronization occurs within a few cycles, and a second threshold where oscillation stops. The positive feedback loop can be composed of either double-positive (PP) or double-negative (NN) interactions between a node of each of the two oscillating networks. The different coupling structures have contrasting characteristics. In particular, PP coupling is advantageous with respect to stability of period and amplitude, when local oscillators are coupled with a short time delay, whereas NN coupling is advantageous for a long time delay. In addition, PP coupling results in more robust synchronized oscillations with respect to amplitude excursions but not period, with applied noise disturbances compared to NN coupling. However, PP coupling can induce a large fluctuation in the amplitude and period of the resulting synchronized oscillation depending on the coupling strength, whereas NN coupling ensures almost constant amplitude and period irrespective of the coupling strength. Intriguingly, we have also observed that artificial evolution of random digital oscillator circuits also follows this design principle. We conclude that a different coupling strategy might have been selected according to different evolutionary requirements.

The hypothalamic GnRH pulse generator: multiple regulatory mechanisms

Pulsatile secretion of gonadotropin-releasing hormone (GnRH) release is an intrinsic property of hypothalamic GnRH neurons. Pulse generation has been attributed to multiple specific mechanisms, including spontaneous electrical activity of GnRH neurons, calcium and cAMP signaling, a GnRH receptor autocrine regulatory component, a GnRH concentration-dependent switch in GnRH receptor (GnRH-R) coupling to specific G proteins, the expression of G protein-coupled receptors (GPCRs) and steroid receptors, and homologous and heterologous interactions between cell membrane receptors expressed in GnRH neurons. The coexistence of multiple regulatory mechanisms for pulsatile GnRH secretion provides a high degree of redundancy in maintaining this crucial component of the mammalian reproductive process. These studies provide insights into the basic cellular and molecular mechanisms involved in GnRH neuronal function.

An integrated model of electrical spiking, bursting, and calcium oscillations in GnRH neurons

The plasma membrane electrical activities of neurons that secrete gonadotropin-releasing hormone (GnRH) have been studied extensively. A couple of mathematical models have been developed previously to explain different aspects of these activities. The goal of this article is to develop a single model that accounts for the previously modeled experimental results and some more recent results that have not been accounted for. The latter includes two types of membrane potential bursting mechanisms and their associated cytosolic calcium oscillations. One bursting mechanism has not been reported in experiments and is thus regarded as a model prediction. Although the model is mainly based on data collected in immortalized GnRH cell lines, it is capable of explaining some properties of GnRH neurons observed in several other preparations including mature GnRH neurons in hypothalamic slices. We present a spatial model that incorporates a detailed description of calcium dynamics in a three-dimensional cell body with the ion channels evenly distributed on the cell surface. A phenomenological reduction of the spatial model into a simplified form is also presented. The simplified model will facilitate the study of the roles of plasma membrane electrical activities in the pulsatile release of GnRH.

Kisspeptin-10 facilitates a plasma membrane-driven calcium oscillator in gonadotropin-releasing hormone-1 neurons

Kisspeptins, the natural ligands of the G-protein-coupled receptor (GPR)-54, are the most potent stimulators of GnRH-1 secretion and as such are critical to reproductive function. However, the mechanism by which kisspeptins enhance calcium-regulated neuropeptide secretion is not clear. In the present study, we used GnRH-1 neurons maintained in mice nasal explants to examine the expression and signaling of GPR54. Under basal conditions, GnRH-1 cells exhibited spontaneous baseline oscillations in intracellular calcium concentration ([Ca(2+)](i)), which were critically dependent on the operation of voltage-gated, tetrodotoxin (TTX)-sensitive sodium channels and were not coupled to calcium release from intracellular pools. Activation of native GPR54 by kisspeptin-10 initiated [Ca(2+)](i) oscillations in quiescent GnRH-1 cells, increased the frequency of calcium spiking in oscillating cells that led to summation of individual spikes into plateau-bursting type of calcium signals in a subset of active cells. These changes predominantly reflected the stimulatory effect of GPR54 activation on the plasma membrane oscillator activity via coupling of this receptor to phospholipase C signaling pathways. Both components of this pathway, inositol 1,3,4-trisphosphate and protein kinase C, contributed to the receptor-mediated modulation of baseline [Ca(2+)](i) oscillations. TTX and 2-aminoethyl diphenylborinate together abolished agonist-induced elevation in [Ca(2+)](i) in almost all cells, whereas flufenamic acid was less effective. Together these results indicate that a plasma membrane calcium oscillator is spontaneously operative in the majority of prenatal GnRH-1 neurons and is facilitated by kisspeptin-10 through phosphatidyl inositol diphosphate hydrolysis and depolarization of neurons by activating TTX-sensitive sodium channels and nonselective cationic channels.

Motivations and methods for analyzing pulsatile hormone secretion

Endocrine glands communicate with remote target cells via a mixture of continuous and intermittent signal exchange. Continuous signaling allows slowly varying control, whereas intermittency permits large rapid adjustments. The control systems that mediate such homeostatic corrections operate in a species-, gender-, age-, and context-selective fashion. Significant progress has been made in understanding mechanisms of adaptive interglandular signaling in vivo. Principal goals are to understand the physiological origins, significance, and mechanisms of pulsatile hormone secretion. Key analytical issues are: 1) to quantify the number, size, shape, and uniformity of pulses, nonpulsatile (basal) secretion, and elimination kinetics; 2) to evaluate regulation of the axis as a whole; and 3) to reconstruct dose-response interactions without disrupting hormone connections. This review will focus on the motivations driving and the methodologies used for such analyses.

Robust synchrony and rhythmogenesis in endocrine neurons via autocrine regulations in vitro and in vivo

Episodic pulses of gonadotropin-releasing hormone (GnRH) are essential for maintaining reproductive functions in mammals. An explanation for the origin of this rhythm remains an ultimate goal for researchers in this field. Some plausible mechanisms have been proposed among which the autocrine-regulation mechanism has been implicated by numerous experiments. GnRH binding to its receptors in cultured GnRH neurons activates three types of G-proteins that selectively promote or inhibit GnRH secretion (Krsmanovic et al. in Proc. Natl. Acad. Sci. 100:2969-2974, 2003). This mechanism appears to be consistent with most data collected so far from both in vitro and in vivo experiments. Based on this mechanism, a mathematical model has been developed (Khadra and Li in Biophys. J. 91:74-83, 2006) in which GnRH in the extracellular space plays the roles of a feedback regulator and a synchronizing agent. In the present study, we show that synchrony between different neurons through sharing a common pool of GnRH is extremely robust. In a diversely heterogeneous population of neurons, the pulsatile rhythm is often maintained when only a small fraction of the neurons are active oscillators (AOs). These AOs are capable of recruiting nonoscillatory neurons into a group of recruited oscillators while forcing the nonrecruitable neurons to oscillate along. By pointing out the existence of the key elements of this model in vivo, we predict that the same mechanism revealed by experiments in vitro may also operate in vivo. This model provides one plausible explanation for the apparently controversial conclusions based on experiments on the effects of the ultra-short feedback loop of GnRH on its own release in vivo.

Selection of in-phase or out-of-phase synchronization in a model based on global coupling of cells undergoing metabolic oscillations

On the basis of experimental observations, it has been suggested that glycolytic oscillations underlie the pulsatile secretion of insulin by pancreatic beta cells, with a periodicity of about 13 min. If beta cells within an islet are synchronized through gap junctions, the question arises as to how beta cells located in different islets of Langerhans synchronize to produce oscillations in plasma levels of insulin. We address this question by means of a minimal model that incorporates the secretion of insulin by cells undergoing glycolytic oscillations. Global coupling and synchronization result from the inhibition exerted by insulin on the production of glucose, which serves as the substrate for metabolic oscillations. Glycolytic oscillations are described by a simple two-variable model centered on the product-activated reaction catalyzed by the allosteric enzyme phosphofructokinase. We obtain bifurcation diagrams for the cases in which insulin secretion is controlled solely by the product or by the substrate of the metabolic oscillator. Remarkably, we find that the oscillating cells in these conditions synchronize, respectively, in phase or out of phase. Numerical simulations show that in-phase and out-of-phase synchronization can sometimes coexist when insulin release is controlled by both the substrate and the product of the metabolic oscillator. The results provide an example of a system in which the selection of in-phase or out-of-phase synchronization is governed by the nature of the coupling between the intracellular oscillations and the secretion of the biochemical signal through which the oscillating cells are globally coupled.

A mathematical model for the actions of activin, inhibin, and follistatin on pituitary gonadotrophs

The timed secretion of the luteinizing hormone (LH) and follicle stimulating hormone (FSH) from pituitary gonadotrophs during the estrous cycle is crucial for normal reproductive functioning. The release of LH and FSH is stimulated by gonadotropin releasing hormone (GnRH) secreted by hypothalamic GnRH neurons. It is controlled by the frequency of the GnRH signal that varies during the estrous cycle. Curiously, the secretion of LH and FSH is differentially regulated by the frequency of GnRH pulses. LH secretion increases as the frequency increases within a physiological range, and FSH secretion shows a biphasic response, with a peak at a lower frequency. There is considerable experimental evidence that one key factor in these differential responses is the autocrine/paracrine actions of the pituitary polypeptides activin and follistatin. Based on these data, we develop a mathematical model that incorporates the dynamics of these polypeptides. We show that a model that incorporates the actions of activin and follistatin is sufficient to generate the differential responses of LH and FSH secretion to changes in the frequency of GnRH pulses. In addition, it shows that the actions of these polypeptides, along with the ovarian polypeptide inhibin and the estrogen-mediated variations in the frequency of GnRH pulses, are sufficient to account for the time courses of LH and FSH plasma levels during the rat estrous cycle. That is, a single peak of LH on the afternoon of proestrus and a double peak of FSH on proestrus and early estrus. We also use the model to identify which regulation pathways are indispensable for the differential regulation of LH and FSH and their time courses during the estrous cycle. We conclude that the actions of activin, inhibin, and follistatin are consistent with LH/FSH secretion patterns, and likely complement other factors in the production of the characteristic secretion patterns in female rats.

Emergent synchronous bursting of oxytocin neuronal network

When young suckle, they are rewarded intermittently with a let-down of milk that results from reflex secretion of the hormone oxytocin; without oxytocin, newly born young will die unless they are fostered. Oxytocin is made by magnocellular hypothalamic neurons, and is secreted from their nerve endings in the pituitary in response to action potentials (spikes) that are generated in the cell bodies and which are propagated down their axons to the nerve endings. Normally, oxytocin cells discharge asynchronously at 1–3 spikes/s, but during suckling, every 5 min or so, each discharges a brief, intense burst of spikes that release a pulse of oxytocin into the circulation. This reflex was the first, and is perhaps the best, example of a physiological role for peptide-mediated communication within the brain: it is coordinated by the release of oxytocin from the dendrites of oxytocin cells; it can be facilitated by injection of tiny amounts of oxytocin into the hypothalamus, and it can be blocked by injection of tiny amounts of oxytocin antagonist. Here we show how synchronized bursting can arise in a neuronal network model that incorporates basic observations of the physiology of oxytocin cells. In our model, bursting is an emergent behaviour of a complex system, involving both positive and negative feedbacks, between many sparsely connected cells. The oxytocin cells are regulated by independent afferent inputs, but they interact by local release of oxytocin and endocannabinoids. Oxytocin released from the dendrites of these cells has a positive-feedback effect, while endocannabinoids have an inhibitory effect by suppressing the afferent input to the cells.

When young suckle, they are rewarded intermittently with a let-down of milk that results from reflex secretion of the hormone oxytocin. Oxytocin is a neuropeptide made by specialised neurons in the hypothalamus, and is secreted from nerve endings in the pituitary gland. During suckling, every 5 min or so, each of these neurons discharges a brief, intense burst of action potentials; these are propagated down the axons, and release a pulse of oxytocin into the circulation. Here, we have built a computational model to understand how these bursts arise and how they are synchronized. In our model, bursting is an emergent behaviour of a complex system, involving both positive and negative feedbacks, between many, sparsely connected cells. The oxytocin cells are regulated by independent afferent inputs, but they interact by local release of oxytocin and endocannabinoids. Oxytocin released from the dendrites of these cells has a positive-feedback effect, while endocannabinoids have an inhibitory effect by suppressing the afferent input to the cells. Many neurons make peptides that act as messengers within the brain, and many of these are also released from dendrites, so this model may reflect a common pattern-generating mechanism in the brain.

Mathematical modelling in neuroendocrinology

In neuroendocrinology, mathematical modelling is about formalising our understanding of the behaviour of the complex biological systems with which we deal. Formulating our explanations mathematically ensures their logical consistency, and makes them open to structured analysis; it is a stringent test of their intellectual coherence. In addition, however, modellers are seeking to extend our understanding in new ways, by seeking novel, simple explanations for complex behaviour. Here we discuss some styles of modelling as they have been applied to neuroendocrine systems, and discuss some of their strengths and limitations.

Facilitation of quantal release induced by a D1-like receptor on bovine chromaffin cells

Dopaminergic receptors are found on bovine adrenal chromaffin cells and have been implicated in the facilitation of an inward calcium current [Artalejo et al., (1990) Nature 348, 239-242] that could enhance release. However, previous studies using incubations of long duration (minutes) with dopaminergic receptor antagonists have found instead an inhibition of catecholamine release. In this work we used brief (subsecond) chemical depolarizing stimuli to reexamine the role of dopaminergic receptors on exocytosis from bovine adrenal chromaffin cells. Responses to consecutive depolarizing stimuli were compared using amperometry to monitor vesicular release events and intracellular fura-2 to examine Ca2+ dynamics within individual cells. Restoration of intracellular Ca2+ levels to their initial values following exposure to 60 mM K+ was found to be prolonged unless the exposure was brief (0.5 s) and the cells were maintained at 37 degrees C. However, with these optimum conditions, a second stimulation evoked more exocytotic events than the first. This effect was blocked by SCH-23390, a D1 antagonist, in a dose dependent fashion, but not by raclopride, a D2 antagonist. The D1 agonist, SKF-38393, enhanced the number of exocytotic events as did prior exposure of the cell to epinephrine. Taken together, the data indicate that released catecholamines can enhance their own release by interaction with a D1-like receptor on bovine adrenal chromaffin cells.