Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents

Slow negative feedback enhances robustness of square-wave bursting

Square-wave bursting is an activity pattern common to a variety of neuronal and endocrine cell models that has been linked to central pattern generation for respiration and other physiological functions. Many of the reduced mathematical models that exhibit square-wave bursting yield transitions to an alternative pseudo-plateau bursting pattern with small parameter changes. This susceptibility to activity change could represent a problematic feature in settings where the release events triggered by spike production are necessary for function. In this work, we analyze how model bursting and other activity patterns vary with changes in a timescale associated with the conductance of a fast inward current. Specifically, using numerical simulations and dynamical systems methods, such as fast-slow decomposition and bifurcation and phase-plane analysis, we demonstrate and explain how the presence of a slow negative feedback associated with a gradual reduction of a fast inward current in these models helps to maintain the presence of spikes within the active phases of bursts. Therefore, although such a negative feedback is not necessary for burst production, we find that its presence generates a robustness that may be important for function.

Dopamine receptor agonists mechanism of actions on glucose lowering and their connections with prolactin actions

Robust experiment evidence suggests that prolactin can enhance beta-cell proliferation and increase insulin secretion and sensitivity. Apart from acting as an endocrine hormone, it also function as an adipokine and act on adipocytes to modulate adipogenesis, lipid metabolism and inflammation. Several cross-sectional epidemiologic studies consistently showed that circulating prolactin levels positive correlated with increased insulin sensitivity, lower glucose and lipid levels, and lower prevalence of T2D and metabolic syndrome. Bromocriptine, a dopamine receptor agonist used to treat prolactinoma, is approved by Food and Drug Administration for treatment in type 2 diabetes mellitus since 2009. Prolactin lowering suppress insulin secretion and decrease insulin sensitivity, therefore dopamine receptor agonists which act at the pituitary to lower serum prolactin levels are expected to impair glucose tolerance. Making it more complicating, studies exploring the glucose-lowering mechanism of bromocriptine and cabergoline have resulted in contradictory results; while some demonstrated actions independently on prolactin status, others showed glucose lowering partly explained by prolactin level. Previous studies showed that a moderate increase in central intraventricular prolactin levels stimulates hypothalamic dopamine with a decreased serum prolactin level and improved glucose metabolism. Additionally, sharp wave-ripples from the hippocampus modulates peripheral glucose level within 10 minutes, providing evidence for a mechanistic link between hypothalamus and blood glucose control. Central insulin in the mesolimbic system have been shown to suppress dopamine levels thus comprising a feedback control loop. Central dopamine and prolactin levels plays a key role in the glucose homeostasis control, and their dysregulation could lead to the pathognomonic central insulin resistance depicted in the “ominous octet”. This review aims to provide an in-depth discussion on the glucose-lowering mechanism of dopamine receptor agonists and on the diverse prolactin and dopamine actions on metabolism targets.

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Ion channel noise shapes the electrical activity of endocrine cells

Endocrine cells in the pituitary gland typically display either spiking or bursting electrical activity, which is related to the level of hormone secretion. Recent work, which combines mathematical modelling with dynamic clamp experiments, suggests the difference is due to the presence or absence of a few large-conductance potassium channels. Since endocrine cells only contain a handful of these channels, it is likely that stochastic effects play an important role in the pattern of electrical activity. Here, for the first time, we explicitly determine the effect of such noise by studying a mathematical model that includes the realistic noisy opening and closing of ion channels. This allows us to investigate how noise affects the electrical activity, examine the origin of spiking and bursting, and determine which channel types are responsible for the greatest noise. Further, for the first time, we address the role of cell size in endocrine cell electrical activity, finding that larger cells typically display more bursting, while the smallest cells almost always only exhibit spiking behaviour.

From Local to Global Modeling for Characterizing Calcium Dynamics and Their Effects on Electrical Activity and Exocytosis in Excitable Cells

Electrical activity in neurons and other excitable cells is a result of complex interactions between the system of ion channels, involving both global coupling (e.g., via voltage or bulk cytosolic Ca²⁺ concentration) of the channels, and local coupling in ion channel complexes (e.g., via local Ca²⁺ concentration surrounding Ca²⁺ channels (CaVs), the so-called Ca²⁺ nanodomains). We recently devised a model of large-conductance BK_Ca potassium currents, and hence BK_Ca–CaV complexes controlled locally by CaVs via Ca²⁺ nanodomains. We showed how different CaV types and BK_Ca–CaV stoichiometries affect whole-cell electrical behavior. Ca²⁺ nanodomains are also important for triggering exocytosis of hormone-containing granules, and in this regard, we implemented a strategy to characterize the local interactions between granules and CaVs. In this study, we coupled electrical and exocytosis models respecting the local effects via Ca²⁺ nanodomains. By simulating scenarios with BK_Ca–CaV complexes with different stoichiometries in pituitary cells, we achieved two main electrophysiological responses (continuous spiking or bursting) and investigated their effects on the downstream exocytosis process. By varying the number and distance of CaVs coupled with the granules, we found that bursting promotes exocytosis with faster rates than spiking. However, by normalizing to Ca²⁺ influx, we found that bursting is only slightly more efficient than spiking when CaVs are far away from granules, whereas no difference in efficiency between bursting and spiking is observed with close granule-CaV coupling.

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.

Forebrain Dopamine System Regulates Inner Ear Auditory Sensitivity to Socially Relevant Acoustic Signals

Dopamine is integral to attentional and motivational processes, but studies are largely restricted to the central nervous system. In mammals [1, 2] and fishes [3, 4], central dopaminergic neurons project to the inner ear and could modulate acoustic signals at the earliest stages of processing. Studies in rodents show dopamine inhibits cochlear afferent neurons and protects against noise-induced acoustic injury [5-10]. However, other functions for inner ear dopamine have not been investigated, and the effect of dopamine on peripheral auditory processing in non-mammalians remains unknown [11, 12]. Insights could be gained by studies conducted in the context of intraspecific acoustic communication. We present evidence from a vocal fish linking reproductive-state-dependent changes in auditory sensitivity with seasonal changes in the dopaminergic efferent system in the saccule, their primary organ of hearing. Plainfin midshipman (Porichthys notatus) migrate from deep-water winter habitats to the intertidal zone in the summer to breed. Nesting males produce nocturnal vocalizations to attract females [13]. Both sexes undergo seasonal enhancement of hearing sensitivity at the level of the hair cell [14-16], increasing the likelihood of detecting conspecific signals [17, 18]. Importantly, reproductive females concurrently have reduced dopaminergic input to the saccule [19]. Here, we show that dopamine decreases saccule auditory sensitivity via a D2-like receptor. Saccule D2a receptor expression is reduced in the summer and correlates with sensitivity within and across seasons. We propose that reproductive-state-dependent changes to the dopaminergic efferent system provide a release of inhibition in the saccule, enhancing peripheral encoding of social-acoustic signals.

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.

Concise Whole-Cell Modeling of BKCa-CaV Activity Controlled by Local Coupling and Stoichiometry

Large-conductance Ca²⁺-dependent K⁺ (BK_Ca) channels are important regulators of electrical activity. These channels colocalize and form ion channel complexes with voltage-dependent Ca²⁺ (CaV) channels. Recent stochastic simulations of the BK_Ca-CaV complex with 1:1 stoichiometry have given important insight into the local control of BK_Ca channels by fluctuating nanodomains of Ca²⁺. However, such Monte Carlo simulations are computationally expensive, and are therefore not suitable for large-scale simulations of cellular electrical activity. In this work we extend the stochastic model to more realistic BK_Ca-CaV complexes with 1:n stoichiometry, and analyze the single-complex model with Markov chain theory. From the description of a single BK_Ca-CaV complex, using arguments based on timescale analysis, we derive a concise model of whole-cell BK_Ca currents, which can readily be analyzed and inserted into models of cellular electrical activity. We illustrate the usefulness of our results by inserting our BK_Ca description into previously published whole-cell models, and perform simulations of electrical activity in various cell types, which show that BK_Ca-CaV stoichiometry can affect whole-cell behavior substantially. Our work provides a simple formulation for the whole-cell BK_Ca current that respects local interactions in BK_Ca-CaV complexes, and indicates how local-global coupling of ion channels may affect cell behavior.

Bifurcation analysis of a two-compartment hippocampal pyramidal cell model

The Pinsky-Rinzel model is a non-smooth 2-compartmental CA3 pyramidal cell model that has been used widely within the field of neuroscience. Here we propose a modified (smooth) system that captures the qualitative behaviour of the original model, while allowing the use of available, numerical continuation methods to perform full-system bifurcation and fast-slow analysis. We study the bifurcation structure of the full system as a function of the applied current and the maximal calcium conductance. We identify the bifurcations that shape the transitions between resting, bursting and spiking behaviours, and which lead to the disappearance of bursting when the calcium conductance is reduced. Insights gained from this analysis, are then used to firstly illustrate how the irregular spiking activity found between bursting and stable spiking states, can be influenced by phase differences in the calcium and dendritic voltage, which lead to corresponding changes in the calcium-sensitive potassium current. Furthermore, we use fast-slow analysis to investigate the mechanisms of bursting and show that bursting in the model is dependent on the intermediately slow variable, calcium, while the other slow variable, the activation gate of the afterhyperpolarisation current, does not contribute to setting the intraburst dynamics but participates in setting the interburst interval. Finally, we discuss how some of the described bifurcations affect spiking behaviour, during sharp-wave ripples, in a larger network of Pinsky-Rinzel cells.

From global to local: exploring the relationship between parameters and behaviors in models of electrical excitability

Models of electrical activity in excitable cells involve nonlinear interactions between many ionic currents. Changing parameters in these models can produce a variety of activity patterns with sometimes unexpected effects. Further more, introducing new currents will have different effects depending on the initial parameter set. In this study we combined global sampling of parameter space and local analysis of representative parameter sets in a pituitary cell model to understand the effects of adding K (+) conductances, which mediate some effects of hormone action on these cells. Global sampling ensured that the effects of introducing K (+) conductances were captured across a wide variety of contexts of model parameters. For each type of K (+) conductance we determined the types of behavioral transition that it evoked. Some transitions were counterintuitive, and may have been missed without the use of global sampling. In general, the wide range of transitions that occurred when the same current was applied to the model cell at different locations in parameter space highlight the challenge of making accurate model predictions in light of cell-to-cell heterogeneity. Finally, we used bifurcation analysis and fast/slow analysis to investigate why specific transitions occur in representative individual models. This approach relies on the use of a graphics processing unit (GPU) to quickly map parameter space to model behavior and identify parameter sets for further analysis. Acceleration with modern low-cost GPUs is particularly well suited to exploring the moderate-sized (5-20) parameter spaces of excitable cell and signaling models.

TIDAL WAVES: Network mechanisms in the neuroendocrine control of prolactin release

Neuroendocrine tuberoinfundibular dopamine (TIDA) neurons tonically inhibit pituitary release of the hormone, prolactin. Through the powerful actions of prolactin in promoting lactation and maternal behaviour while suppressing sexual drive and fertility, TIDA neurons play a key role in reproduction. We summarize insights from recent in vitro studies into the membrane properties and network behaviour of TIDA neurons including the observations that TIDA neurons exhibit a robust oscillation that is synchronized between cells and depends on intact gap junction communication. Comparisons are made with phasic firing patterns in other neuronal populations. Modulators involved in the control of lactation - including serotonin, thyrotropin-releasing hormone and prolactin itself - have been shown to change the electrical behaviour of TIDA cells. We propose that TIDA discharge mode may play a central role in tuning the amount of dopamine delivered to the pituitary and hence circulating prolactin concentrations in different reproductive states and pathological conditions.

Models of electrical activity: calibration and prediction testing on the same cell

Mathematical models are increasingly important in biology, and testability is becoming a critical issue. One limitation is that one model simulation tests a parameter set representing one instance of the biological counterpart, whereas biological systems are heterogeneous in their properties and behavior, and a model often is fitted to represent an ideal average. This is also true for models of a cell's electrical activity; even within a narrowly defined population there can be considerable variation in electrophysiological phenotype. Here, we describe a computational experimental approach for parameterizing a model of the electrical activity of a cell in real time. We combine the inexpensive parallel computational power of a programmable graphics processing unit with the flexibility of the dynamic clamp method. The approach involves 1), recording a cell's electrical activity, 2), parameterizing a model to the recording, 3), generating predictions, and 4), testing the predictions on the same cell used for the calibration. We demonstrate the experimental feasibility of our approach using a cell line (GH4C1). These cells are electrically active, and they display tonic spiking or bursting. We use our approach to predict parameter changes that can convert one pattern to the other.

The relationship between two fast/slow analysis techniques for bursting oscillations

Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow.

Fast-activating voltage- and calcium-dependent potassium (BK) conductance promotes bursting in pituitary cells: a dynamic clamp study

The electrical activity pattern of endocrine pituitary cells regulates their basal secretion level. Rat somatotrophs and lactotrophs exhibit spontaneous bursting and have high basal levels of hormone secretion, while gonadotrophs exhibit spontaneous spiking and have low basal hormone secretion. It has been proposed that the difference in electrical activity between bursting somatotrophs and spiking gonadotrophs is due to the presence of large conductance potassium (BK) channels on somatotrophs but not on gonadotrophs. This is one example where the role of an ion channel type may be clearly established. We demonstrate here that BK channels indeed promote bursting activity in pituitary cells. Blocking BK channels in bursting lacto-somatotroph GH4C1 cells changes their firing activity to spiking, while further adding an artificial BK conductance via dynamic clamp restores bursting. Importantly, this burst-promoting effect requires a relatively fast BK activation/deactivation, as predicted by computational models. We also show that adding a fast-activating BK conductance to spiking gonadotrophs converts the activity of these cells to bursting. Together, our results suggest that differences in BK channel expression may underlie the differences in electrical activity and basal hormone secretion levels among pituitary cell types and that the rapid rate of BK channel activation is key to its role in burst promotion.

CROSS-CURRENTS BETWEEN BIOLOGY AND MATHEMATICS: THE CODIMENSION OF PSEUDO-PLATEAU BURSTING

A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.

The dynamics underlying pseudo-plateau bursting in a pituitary cell model

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.

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

Non-technical summary

Our ability to respond to stress is critically dependent upon the release of the stress hormone adrenocorticotrophic hormone (ACTH) from corticotroph cells of the anterior pituitary gland. ACTH release is controlled by the electrical properties of corticotrophs that are determined by the movement of ions through channel pores in the plasma membrane. We show that a calcium-activated potassium ion channel called SK4 is expressed in corticotrophs and regulates ACTH release. We provide evidence of how SK4 channels control corticotroph function, which is essential for understanding homeostasis and for treating stress-related disorders.

Abstract

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.

Modeling mechanisms of cell secretion

Secretion is a fundamental cellular process involving the regulated release of intracellular products from cells. Physiological functions such as neurotransmission, or the release of hormones and digestive enzymes, are all governed by cell secretion. Anomalies in the processes involved in secretion contribute to the development and progression of diseases such as diabetes and other hormonal disorders. To unravel the mechanisms that govern such diseases, it is essential to understand how hormones, growth factors and neurotransmitters are synthesized and processed, and how their signals are recognized, amplified and transmitted by intracellular signaling pathways in the target cells. Here, we discuss diverse aspects of the detailed mechanisms involved in secretion based on mathematical models. The models range from stochastic ones describing the trafficking of secretory vesicles to deterministic ones investigating the regulation of cellular processes that underlie hormonal secretion. In all cases, the models are closely related to experimental results and suggest theoretical predictions for the secretion mechanisms.

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.

Dynamics of plateau bursting depending on the location of its equilibrium

We present a mathematical analysis of the dynamics that underlies plateau bursting in models of endocrine cells under variation of the location of the (unstable) equilibrium around which these bursting patterns are organised. We focus primarily on the less well-studied case of pseudo-plateau bursting, but also consider the square-wave case. The behaviour of such models is explained using the theory for systems with multiple time scales and it is well known that the underlying so-called fast subsystem organises their dynamics. However, such results are valid only in a sufficiently small neighbourhood of the singular limit that defines the fast subsystem. Hence, the slow variable (intracellular calcium concentration) must be very slow, which is actually not the case for pseudo-plateau bursting. Furthermore, the theoretical predictions are also only valid for parameter values such that the equilibrium is close to a homoclinic bifurcation occuring in the fast subsystem. In the present study, we use numerical explorations to discuss what happens outside this theoretically known neighbourhood of parameter space. In particular, we consider what happens as the equilibrium moves outside a small neighbourhood of the homoclinic bifurcation that occurs in the fast subsystem, and relatively fast speeds are allowed for the slow variable which is controlled by a relatively large value of a parameter ε. The results obtained complement our earlier work [Tsaneva-Atanasova et al. (2010) J Theor Biol264, 1133-1146], which focussed on how the bursting patterns vary with the rate of change ε of the slow variable: we fix ε and move the equilibrium over the full range of the bursting regime. Our findings show that the transitions between different bursting patterns are rather similar for square-wave and pseudo-plateau bursting, provided that the value of ε for the pseudo-plateau-bursting model is chosen so that it is much larger than for the square-wave bursting model. Furthermore, the two families of tonic spiking and plateau bursting, which are generally viewed as two separately generated families, are actually connected into a single family in the two-parameter plane through branches of unstable periodic orbits.

From plateau to pseudo-plateau bursting: making the transition

Bursting electrical activity is ubiquitous in excitable cells such as neurons and many endocrine cells. The technique of fast/slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the design principles for bursting in endocrine cells are just quantitative variations of those for bursting in neurons.

Full system bifurcation analysis of endocrine bursting models

Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. We investigate so-called square-wave and pseudo-plateau bursting patterns found in endocrine cell models that are characterized by a super- or subcritical Hopf bifurcation in the fast subsystem, respectively. By using the polynomial model of Hindmarsh and Rose (Proceedings of the Royal Society of London B 221 (1222) 87-102), which preserves the main properties of the biophysical class of models that we consider, we perform a detailed bifurcation analysis of the full fast-slow system for both bursting patterns. We find that both cases lead to the same possibility of two routes to bursting, that is, the criticality of the Hopf bifurcation is not relevant for characterizing the route to bursting. The actual route depends on the relative location of the full-system's fixed point with respect to a homoclinic bifurcation of the fast subsystem. Our full-system bifurcation analysis reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis.

Mixed mode oscillations as a mechanism for pseudo-plateau bursting

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called "pseudo-plateau bursting". Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic beta-cells.

Correlation analysis a tool for comparing relaxation-type models to experimental data

We describe a new technique for comparing mathematical models to the biological systems that are described. This technique is appropriate for systems that produce relaxation oscillations or bursting oscillations, and takes advantage of noise that is inherent to all biological systems. Both types of oscillations are composed of active phases of activity followed by silent phases, repeating periodically. The presence of noise adds variability to the durations of the different phases. The central idea of the technique is that the active phase duration may be correlated with either/both the previous or next silent phase duration, and the resulting correlation pattern provides information about the dynamic structure of the system. Correlation patterns can easily be determined by making scatter plots and applying correlation analysis to the cluster of data points. This could be done both with experimental data and with model simulation data. If the model correlation pattern is in general agreement with the experimental data, then this adds support for the validity of the model. Otherwise, the model must be corrected. While this tool is only one test of many required to validate a mathematical model, it is easy to implement and is noninvasive.

Predicting the activity phase of a follower neuron with A-current in an inhibitory network

The transient potassium A-current is present in most neurons and plays an important role in determining the timing of action potentials. We examine the role of the A-current in the activity phase of a follower neuron in a rhythmic feed-forward inhibitory network with a reduced three-variable model and conduct experiments to verify the usefulness of our model. Using geometric analysis of dynamical systems, we explore the factors that determine the onset of activity in a follower neuron following release from inhibition. We first analyze the behavior of the follower neuron in a single cycle and find that the phase plane structure of the model can be used to predict the potential behaviors of the follower neuron following release from inhibition. We show that, depending on the relative scales of the inactivation time constant of the A-current and the time constant of the recovery variable, the follower neuron may or may not reach its active state following inhibition. Our simple model is used to derive a recursive set of equations to predict the contribution of the A-current parameters in determining the activity phase of a follower neuron as a function of the duration and frequency of the inhibitory input it receives. These equations can be used to demonstrate the dependence of activity phase on the period and duty cycle of the periodic inhibition, as seen by comparing the predictions of the model with the activity of the pyloric constrictor (PY) neurons in the crustacean pyloric network.

A-type K(+) current can act as a trigger for bursting in the absence of a slow variable

Models of bursting in single cells typically include two subsystems with different timescales. Variations in one or more slow variables switch the system between a silent and a spiking state. We have developed a model for bursting in the pituitary lactotroph that does not include any slow variable. The model incorporates fast, noninactivating calcium and potassium currents (the spike-generating mechanism), as well as the fast, inactivating A-type potassium current (I(A)). I(A) is active only briefly at the beginning of a burst, but this brief impulse of I(A) acts as a burst trigger, injecting the spike trajectory close to an unstable steady state. The spiraling of the trajectory away from the steady state produces a period of low-amplitude spiking typical of lactotrophs. Increasing the conductance of A-type potassium current brings the trajectory closer to the unstable steady state, increasing burst duration. However, this also increases interburst interval, and for larger conductance values, all activity stops. To our knowledge, this is the first example of a physiologically based, single-compartmental model of bursting with no slow subsystem.

Phantom bursting is highly sensitive to noise and unlikely to account for slow bursting in beta-cells: considerations in favor of metabolically driven oscillations

Pancreatic beta-cells show bursting electrical activity with a wide range of burst periods ranging from a few seconds, often seen in isolated cells, over tens of seconds (medium bursting), usually observed in intact islets, to several minutes. The phantom burster model [Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic beta-cells. Biophys. J. 79, 2880-2892] provided a framework, which covered this span, and gave an explanation of how to obtain medium bursting combining two processes operating on different time scales. However, single cells are subjected to stochastic fluctuations in plasma membrane currents, which are likely to disturb the bursting mechanism and transform medium bursters into spikers or very fast bursters. We present a polynomial, minimal, phantom burster model and show that noise modifies the plateau fraction and lowers the burst period dramatically in phantom bursters. It is therefore unlikely that slow bursting in single cells is driven by the slow phantom bursting mechanism, but could instead be driven by oscillations in glycolysis, which we show are stable to random ion channel fluctuations. Moreover, so-called compound bursting can be converted to apparent slow bursting by noise, which could explain why compound bursting and mixed Ca(2+) oscillations are seen mainly in intact islets.