Conversion of spikers to bursters in pituitary cell networks: Is it better to disperse for maximum exposure or circle the wagons?

The endocrine cells of the pituitary gland are electrically active, and in vivo they form small networks where the bidirectional cell-cell coupling is through gap junctions. Numerous studies of dispersed pituitary cells have shown that typical behaviors are tonic spiking and bursting, the latter being more effective at evoking secretion. In this article, we use mathematical modeling to examine the dynamics of small networks of spiking and bursting pituitary cells. We demonstrate that intrinsic bursting cells are capable of converting intrinsic spikers into bursters, and perform a fast/slow analysis to show why this occurs. We then demonstrate the sensitivity of network dynamics to the placement of bursting cells within the network, and demonstrate strategies that are most effective at maximizing secretion from the population of cells. This study provides insights into the in vivo behavior of cells such as the stress-hormone-secreting pituitary corticotrophs that are switched from spiking to bursting by hypothalamic neurohormones. While much is known about the electrical properties of these cells when isolated from the pituitary, how they behave when part of an electrically coupled network has been largely unstudied.

KATP channel activity and slow oscillations in pancreatic beta cells are regulated by mitochondrial ATP production.

Pancreatic beta cells secrete insulin in response to plasma glucose. The ATP-sensitive potassium channel (KATP ) links glucose metabolism to islet electrical activity in these cells by responding to increased cytosolic [ATP]/[ADP]. It was recently proposed that pyruvate kinase (PK) in close proximity to beta cell KATP locally produces the ATP that inhibits KATP activity. This proposal was largely based on the observation that applying phosphoenolpyruvate (PEP) and ADP to the cytoplasmic side of excised inside-out patches inhibited KATP . To test the relative contributions of local vs. mitochondrial ATP production, we recorded KATP activity using mouse beta cells and INS-1 832/13 cells. In contrast to prior reports, we could not replicate inhibition of KATP activity by PEP + ADP. However, when the pH of the PEP solutions was not corrected for the addition of PEP, strong channel inhibition was observed as a result of the well-known action of protons to inhibit KATP . In cell-attached recordings, perifusing either a PK activator or an inhibitor had little or no effect on KATP channel closure by glucose, further suggesting that PK is not an important regulator of KATP . In contrast, addition of mitochondrial inhibitors robustly increased KATP activity. Finally, by measuring the [ATP]/[ADP] responses to imposed calcium oscillations in mouse beta cells, we found that oxidative phosphorylation could raise [ATP]/[ADP] even when ADP was at its nadir during the burst silent phase, in agreement with our mathematical model. These results indicate that ATP produced by mitochondrial oxidative phosphorylation is the primary controller of KATP in pancreatic beta cells. KEY POINTS: Phosphoenolpyruvate (PEP) plus adenosine diphosphate does not inhibit KATP activity in excised patches. PEP solutions only inhibit KATP activity if the pH is unbalanced. Modulating pyruvate kinase has minimal effects on KATP activity. Mitochondrial inhibition, in contrast, robustly potentiates KATP activity in cell-attached patches. Although the ADP level falls during the silent phase of calcium oscillations, mitochondria can still produce enough ATP via oxidative phosphorylation to close KATP . Mitochondrial oxidative phosphorylation is therefore the main source of the ATP that inhibits the KATP activity of pancreatic beta cells.

Deconstructing the integrated oscillator model for pancreatic ß-cells.

Electrical bursting oscillations in the ß-cells of pancreatic islets have been a focus of investigation for more than fifty years. This has been aided by mathematical models, which are descendants of the pioneering Chay-Keizer model. This article describes the key biophysical and mathematical elements of this model, and then describes the path forward from there to the Integrated Oscillator Model (IOM). It is both a history and a deconstruction of the IOM that describes the various elements that have been added to the model over time, and the motivation for adding them. Finally, the article is a celebration of the 40th anniversary of the publication of the Chay-Keizer model.

A mathematical study of the efficacy of possible negative feedback pathways involved in neuronal polarization.

Neuronal polarization, a process wherein nascent neurons develop a single long axon and multiple short dendrites, can occur within in vitro cell cultures without environmental cues. This is an apparently random process in which one of several short processes, called neurites, grows to become long, while the others remain short. In this study, we propose a minimum model for neurite growth, which involves bistability and random excitations reflecting actin waves. Positive feedback is needed to produce the bistability, while negative feedback is required to ensure that no more than one neurite wins the winner-takes-all contest. By applying the negative feedback to different aspects of the neurite growth process, we demonstrate that targeting the negative feedback to the excitation amplitude results in the most persistent polarization. Also, we demonstrate that there are optimal ranges of values for the neurite count, and for the excitation rate and amplitude that best maintain the polarization. Finally, we show that a previously published model for neuronal polarization based on competition for limited resources shares key features with our best-performing minimal model: bistability and negative feedback targeted to the size of random excitations.

Ca2+ release or Ca2+ entry, that is the question: what governs Ca2+ oscillations in pancreatic ß cells?

The standard model for Ca2+ oscillations in insulin-secreting pancreatic ß cells centers on Ca2+ entry through voltage-activated Ca2+ channels. These work in combination with ATP-dependent K+ channels, which are the bridge between the metabolic state of the cells and plasma membrane potential. This partnership underlies the ability of the ß cells to secrete insulin appropriately on a minute-to-minute time scale to control whole body plasma glucose. Though this model, developed over more than 40 years through many cycles of experimentation and mathematical modeling, has been very successful, it has been challenged by a hypothesis that calcium-induced calcium release from the endoplasmic reticulum through ryanodine or inositol trisphosphate (IP3) receptors is instead the key driver of islet oscillations. We show here that the alternative model is in fact incompatible with a large body of established experimental data and that the new observations offered in support of it can be better explained by the standard model.

Active Licking Shapes Cortical Taste Coding.

Neurons in the gustatory cortex (GC) represent taste through time-varying changes in their spiking activity. The predominant view is that the neural firing rate represents the sole unit of taste information. It is currently not known whether the phase of spikes relative to lick timing is used by GC neurons for taste encoding. To address this question, we recorded spiking activity from >500 single GC neurons in male and female mice permitted to freely lick to receive four liquid gustatory stimuli and water. We developed a set of data analysis tools to determine the ability of GC neurons to discriminate gustatory information and then to quantify the degree to which this information exists in the spike rate versus the spike timing or phase relative to licks. These tools include machine learning algorithms for classification of spike trains and methods from geometric shape and functional data analysis. Our results show that while GC neurons primarily encode taste information using a rate code, the timing of spikes is also an important factor in taste discrimination. A further finding is that taste discrimination using spike timing is improved when the timing of licks is considered in the analysis. That is, the interlick phase of spiking provides more information than the absolute spike timing itself. Overall, our analysis demonstrates that the ability of GC neurons to distinguish among tastes is best when spike rate and timing is interpreted relative to the timing of licks.SIGNIFICANCE STATEMENT Neurons represent information from the outside world via changes in their number of action potentials (spikes) over time. This study examines how neurons in the mouse gustatory cortex (GC) encode taste information when gustatory stimuli are experienced through the active process of licking. We use electrophysiological recordings and data analysis tools to evaluate the ability of GC neurons to distinguish tastants and then to quantify the degree to which this information exists in the spike rate versus the spike timing relative to licks. We show that the neuron's ability to distinguish between tastes is higher when spike rate and timing are interpreted relative to the timing of licks, indicating that the lick cycle is a key factor for taste processing.

Network Properties of Electrically Coupled Bursting Pituitary Cells.

The endocrine cells of the anterior pituitary gland are electrically active when stimulated or, in some cases, when not inhibited. The activity pattern thought to be most effective in releasing hormones is bursting, which consists of depolarization with small spikes that are much longer than single spikes. Although a majority of the research on cellular activity patterns has been performed on dispersed cells, the environment in situ is characterized by networks of coupled cells of the same type, at least in the case of somatotrophs and lactotrophs. This produces some degree of synchronization of their activity, which can be greatly increased by hormones and changes in the physiological state. In this computational study, we examine how electrical coupling among model cells influences synchronization of bursting oscillations among the population. We focus primarily on weak electrical coupling, since strong coupling leads to complete synchronization that is not characteristic of pituitary cell networks. We first look at small networks to point out several unexpected behaviors of the coupled system, and then consider a larger random scale-free network to determine what features of the structural network formed through gap junctional coupling among cells produce a high degree of functional coupling, i.e., clusters of synchronized cells. We employ several network centrality measures, and find that cells that are closely related in terms of their closeness centrality are most likely to be synchronized. We also find that structural hubs (cells with extensive coupling to other cells) are typically not functional hubs (cells synchronized with many other cells). Overall, in the case of weak electrical coupling, it is hard to predict the functional network that arises from a structural network, or to use a functional network as a means for determining the structural network that gives rise to it.

Oscillations in K(ATP) conductance drive slow calcium oscillations in pancreatic ß-cells.

ATP-sensitive K+ (K(ATP)) channels were first reported in the ß-cells of pancreatic islets in 1984, and it was soon established that they are the primary means by which the blood glucose level is transduced to cellular electrical activity and consequently insulin secretion. However, the role that the K(ATP) channels play in driving the bursting electrical activity of islet ß-cells, which drives pulsatile insulin secretion, remains unclear. One difficulty is that bursting is abolished when several different ion channel types are blocked pharmacologically or genetically, making it challenging to distinguish causation from correlation. Here, we demonstrate a means for determining whether activity-dependent oscillations in K(ATP) conductance play the primary role in driving electrical bursting in ß-cells. We use mathematical models to predict that if K(ATP) is the driver, then contrary to intuition, the mean, peak, and nadir levels of ATP/ADP should be invariant to changes in glucose within the concentration range that supports bursting. We test this in islets using Perceval-HR to image oscillations in ATP/ADP. We find that mean, peak, and nadir levels are indeed approximately invariant, supporting the hypothesis that oscillations in K(ATP) conductance are the main drivers of the slow bursting oscillations typically seen at stimulatory glucose levels in mouse islets. In conclusion, we provide, for the first time to our knowledge, causal evidence for the role of K(ATP) channels not only as the primary target for glucose regulation but also for their role in driving bursting electrical activity and pulsatile insulin secretion.

Slow oscillations persist in pancreatic beta cells lacking phosphofructokinase M.

Pulsatile insulin secretion by pancreatic beta cells is necessary for tight glucose control in the body. Glycolytic oscillations have been proposed as the mechanism for generating the electrical oscillations underlying pulsatile insulin secretion. The glycolytic enzyme 6-phosphofructokinase-1 (PFK) synthesizes fructose-1,6-bisphosphate (FBP) from fructose-6-phosphate. It has been proposed that the slow electrical and Ca2+ oscillations (periods of 3-5 min) observed in islets result from allosteric feedback activation of PFKM by FBP. Pancreatic beta cells express three PFK isozymes: PFKL, PFKM, and PFKP. A prior study of mice that were engineered to lack PFKM using a gene-trap strategy to delete Pfkm produced a mosaic reduction in global Pfkm expression, but the islets isolated from the mice still exhibited slow Ca2+ oscillations. However, these islets still expressed residual PFKM protein. Thus, to more fully test the hypothesis that beta cell PFKM is responsible for slow islet oscillations, we made a beta-cell-specific knockout mouse that completely lacked PFKM. While PFKM deletion resulted in subtle metabolic changes in vivo, islets that were isolated from these mice continued to exhibit slow oscillations in electrical activity, beta cell Ca2+ concentrations, and glycolysis, as measured using PKAR, an FBP reporter/biosensor. Furthermore, simulations obtained with a mathematical model of beta cell activity shows that slow oscillations can persist despite PFKM loss provided that one of the other PFK isoforms, such as PFKP, is present, even if its level of expression is unchanged. Thus, while we believe that PFKM may be the main regulator of slow oscillations in wild-type islets, PFKP can provide functional redundancy. Our model also suggests that PFKM likely dominates, in vivo, because it outcompetes PFKP with its higher FBP affinity and lower ATP affinity. We thus propose that isoform redundancy may rescue key physiological processes of the beta cell in the absence of certain critical genes.

Fast-slow analysis as a technique for understanding the neuronal response to current ramps.

The standard protocol for studying the spiking properties of single neurons is the application of current steps while monitoring the voltage response. Although this is informative, the jump in applied current is artificial. A more physiological input is where the applied current is ramped up, reflecting chemosensory input. Unsurprisingly, neurons can respond differently to the two protocols, since ion channel activation and inactivation are affected differently. Understanding the effects of current ramps, and changes in their slopes, is facilitated by mathematical models. However, techniques for analyzing current ramps are under-developed. In this article, we demonstrate how current ramps can be analyzed in single neuron models. The primary issue is the presence of gating variables that activate on slow time scales and are therefore far from equilibrium throughout the ramp. The use of an appropriate fast-slow analysis technique allows one to fully understand the neural response to ramps of different slopes. This study is motivated by data from olfactory bulb dopamine neurons, where both fast ramp (tens of milliseconds) and slow ramp (tens of seconds) protocols are used to understand the spiking profiles of the cells. The slow ramps generate experimental bifurcation diagrams with the applied current as a bifurcation parameter, thereby establishing asymptotic spiking activity patterns. The faster ramps elicit purely transient behavior that is of relevance to most physiological inputs, which are short in duration. The two protocols together provide a broader understanding of the neuron's spiking profile and the role that slowly activating ion channels can play.

Chronic stress facilitates bursting electrical activity in pituitary corticotrophs.

Coordination of an appropriate stress response is dependent upon anterior pituitary corticotroph excitability in response to hypothalamic secretagogues and glucocorticoid negative feedback. A key determinant of corticotroph excitability is large conductance calcium- and voltage-activated (BK) potassium channels that are critical for promoting corticotrophin-releasing hormone (CRH)-induced bursting that enhances adrenocorticotrophic hormone secretion. Previous studies revealed hypothalamic-pituitary-adrenal axis hyperexcitability following chronic stress (CS) is partly a function of increased corticotroph output. Thus, we hypothesise that chronic stress promotes corticotroph excitability through a BK-dependent mechanism. Corticotrophs from CS mice displayed significant increase in spontaneous bursting, which was suppressed by the BK blocker paxilline. Mathematical modelling reveals that the time constant of BK channel activation, plus properties and proportion of BK channels functionally coupled to L-type Ca2+ channels determines bursting activity. Surprisingly, CS corticotrophs (but not unstressed) display CRH-induced bursting even when the majority of BK channels are inhibited by paxilline, which modelling suggests is a consequence of the stochastic behaviour of a small number of BK channels coupled to L-type Ca2+ channels. Our data reveal that changes in the stochastic behaviour of a small number of BK channels can finely tune corticotroph excitability through stress-induced changes in BK channel properties. Importantly, regulation of BK channel function is highly context dependent allowing dynamic control of corticotroph excitability over a large range of time domains and physiological challenges in health and disease. This is likely to occur in other BK-expressing endocrine cells, with important implications for the physiological processes they regulate and the potential for therapy. KEY POINTS: Chronic stress (CS) is predicted to modify the electrical excitability of anterior pituitary corticotrophs. Electrophysiological recordings from isolated corticotrophs from CS male mice display spontaneous electrical bursting behaviour compared to the tonic spiking behaviour of unstressed corticotrophs. The increased spontaneous bursting from CS corticotrophs is BK-dependent and mathematical modelling reveals that the time constant of activation, properties and proportion of BK channels functionally coupled to L-type calcium channels determines the promotion of bursting activity. CS (but not unstressed) corticotrophs display corticotrophin-releasing hormone-induced bursting even when the majority of BK channels are pharmacologically inhibited, which can be explained by the stochastic behaviour of a small number of BK channels with distinct properties. Corticotroph excitability can be finely tuned by the stochastic behaviour of a small number of BK channels dependent on their properties and functional co-localisation with L-type calcium channels to control corticotroph excitability over diverse time domains and physiological challenges.

A closed-loop multi-scale model for intrinsic frequency-dependent regulation of axonal growth.

This article develops a closed-loop multi-scale model for axon length regulation based on a frequency-dependent negative feedback mechanism. It builds on earlier models by linking molecular motor dynamics to signaling delays that then determine signal oscillation period. The signal oscillation is treated as a front end for a signaling pathway that modulates axonal length. This model is used to demonstrate the feasibility of such a mechanism and is tested against two previously published reports in which experimental manipulations were performed that resulted in axon growth. The model captures these observations and yields an expression for equilibrium axonal length. One major prediction of the model is that increasing motor density in the body of an axon results in axonal growth-this idea has not yet been explored experimentally.

Symbiosis of Electrical and Metabolic Oscillations in Pancreatic ß-Cells.

Insulin is secreted in a pulsatile pattern, with important physiological ramifications. In pancreatic ß-cells, which are the cells that synthesize insulin, insulin exocytosis is elicited by pulses of elevated intracellular Ca2+ initiated by bursts of electrical activity. In parallel with these electrical and Ca2+ oscillations are oscillations in metabolism, and the periods of all of these oscillatory processes are similar. A key question that remains unresolved is whether the electrical oscillations are responsible for the metabolic oscillations via the effects of Ca2+, or whether the metabolic oscillations are responsible for the electrical oscillations due to the effects of ATP on ATP-sensitive ion channels? Mathematical modeling is a useful tool for addressing this and related questions as modeling can aid in the design of well-focused experiments that can test the predictions of particular models and subsequently be used to improve the models in an iterative fashion. In this article, we discuss a recent mathematical model, the Integrated Oscillator Model (IOM), that was the product of many years of development. We use the model to demonstrate that the relationship between calcium and metabolism in beta cells is symbiotic: in some contexts, the electrical oscillations drive the metabolic oscillations, while in other contexts it is the opposite. We provide new insights regarding these results and illustrate that what might at first appear to be contradictory data are actually compatible when viewed holistically with the IOM.

Fast-slow analysis of a stochastic mechanism for electrical bursting.

Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neurotransmitters and hormones and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations or, in the case of so-called "pseudo-plateau bursting," canards that are induced by a folded node singularity. In this article, we show an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a much longer response in which single action potentials are converted into bursts of action potentials. Without this stochastic element, the system is incapable of bursting. This mechanism can describe stochastic bursting in pituitary corticotrophs, which are small cells that exhibit a great deal of noise as well as other pituitary cells, such as lactotrophs and somatotrophs that exhibit noisy bursts of electrical activity.

Canard analysis reveals why a large Ca2+ window current promotes early afterdepolarizations in cardiac myocytes.

The pumping of blood through the heart is due to a wave of muscle contractions that are in turn due to a wave of electrical activity initiated at the sinoatrial node. At the cellular level, this wave of electrical activity corresponds to the sequential excitation of electrically coupled cardiac cells. Under some conditions, the normally-long action potentials of cardiac cells are extended even further by small oscillations called early afterdepolarizations (EADs) that can occur either during the plateau phase or repolarizing phase of the action potential. Hence, cellular EADs have been implicated as a driver of potentially lethal cardiac arrhythmias. One of the major determinants of cellular EAD production and repolarization failure is the size of the overlap region between Ca2+ channel activation and inactivation, called the window region. In this article, we interpret the role of the window region in terms of the fast-slow structure of a low-dimensional model for ventricular action potential generation. We demonstrate that the effects of manipulation of the size of the window region can be understood from the point of view of canard theory. We use canard theory to explain why enlarging the size of the window region elicits EADs and why shrinking the window region can eliminate them. We also use the canard mechanism to explain why some manipulations in the size of the window region have a stronger influence on cellular electrical behavior than others. This dynamical viewpoint gives predictive power that is beyond that of the biophysical explanation alone while also uncovering a common mechanism for phenomena observed in experiments on both atrial and ventricular cardiac cells.

Network dynamics underlie learning and performance of birdsong.

Understanding the sensorimotor control of the endless variety of human speech patterns stands as one of the apex problems in neuroscience. The capacity to learn - through imitation - to rapidly sequence vocal sounds in meaningful patterns is clearly one of the most derived of human behavioral traits. Selection pressure produced an analogous capacity in numerous species of vocal-learning birds, and due to an increasing appreciation for the cognitive and computational flexibility of avian cortex and basal ganglia, a general understanding of the forebrain network that supports the learning and production of birdsong is beginning to emerge. Here, we review recent advances in experimental studies of the zebra finch (Taeniopygia guttata), which offer new insights into the network dynamics that support this surprising analogue of human speech learning and production.

Phantom bursting may underlie electrical bursting in single pancreatic ß-cells.

Insulin is secreted by pancreatic ß-cellsthat are electrically coupled into micro-organs called islets of Langerhans. The secretion is due to the influx of Ca2+ions that accompany electrical impulses, which are clustered into bursts. So-called "medium bursting" occurs in many ß-cellsin intact islets, while in other islets the ß-cellsexhibit "slow bursting", with a much longer period. Each burst brings in Ca2+ that, through exocytosis, results in insulin secretion. When isolated from an islet, ß-cellsbehave very differently. The electrical activity is much noisier, and consists primarily of trains of irregularly-timed spikes, or fast or slow bursting. Medium bursting, so often seen in intact islets, is rarely if ever observed. In this study, we examine what the isolated cell behavior can tell us about the mechanism for bursting in intact islets. A previous mathematical study concluded that the slow bursting observed in isolated ß-cells, and therefore most likely in islets, must be due to intrinsic glycolytic oscillations, since this mechanism for bursting is robust to noise. It was demonstrated that an alternate mechanism, phantom bursting, was very sensitive to noise, and therefore could not account for the slow bursting in single cells. We re-examine these conclusions, motivated by recent experimental and mathematical modeling evidence that slow bursting in intact islets is, at least in many cases, driven by the phantom bursting mechanism and not endogenous glycolytic oscillations. We employ two phantom bursting models, one minimal and the other more biophysical, to determine the sensitivity of medium and slow bursting to electrical current noise. In the minimal model, both forms of bursting are highly sensitive to noise. In the biophysical model, while medium bursting is sensitive to noise, slow bursting is much less sensitive. This suggests that the slow bursting seen in isolated ß-cellsmay be due to a phantom bursting mechanism, and by extension, slow bursting in intact islets may also be driven by this mechanism.

Spiking and Membrane Properties of Rat Olfactory Bulb Dopamine Neurons.

The mammalian olfactory bulb (OB) has a vast population of dopamine (DA) neurons, whose function is to increase odor discrimination through mostly inhibitory synaptic mechanisms. However, it is not well understood whether there is more than one neuronal type of OB DA neuron, how these neurons respond to different stimuli, and the ionic mechanisms behind those responses. In this study, we used a transgenic rat line (hTH-GFP) to identify fluorescent OB DA neurons for recording via whole-cell electrophysiology. These neurons were grouped based on their localization in the glomerular layer ("Top" vs. "Bottom") with these largest and smallest neurons grouped by neuronal area ("Large" vs. "Small," in µm2). We found that some membrane properties could be distinguished based on a neuron's area, but not by its glomerular localization. All OB DA neurons produced a single action potential when receiving a sufficiently depolarizing stimulus, while some could also spike multiple times when receiving weaker stimuli, an activity that was more likely in Large than Small neurons. This single spiking activity is likely driven by the Na+ current, which showed a sensitivity to inactivation by depolarization and a relatively long time constant for the removal of inactivation. These recordings showed that Small neurons were more sensitive to inactivation of Na+ current at membrane potentials of -70 and -60 mV than Large neurons. The hyperpolarization-activated H-current (identified by voltage sags) was more pronounced in Small than Large DA neurons across hyperpolarized membrane potentials. Lastly, to mimic a more physiological stimulus, these neurons received ramp stimuli of various durations and current amplitudes. When stimulated with weaker/shallow ramps, the neurons needed less current to begin and end firing and they produced more action potentials at a slower frequency. These spiking properties were further analyzed between the four groups of neurons, and these analyses support the difference in spiking induced with current step stimuli. Thus, there may be more than one type of OB DA neuron, and these neurons' activities may support a possible role of being high-pass filters in the OB by allowing the transmission of stronger odor signals while inhibiting weaker ones.

Chronic stimulation induces adaptive potassium channel activity that restores calcium oscillations in pancreatic islets in vitro.

Insulin pulsatility is important to hepatic response in regulating blood glucose. Growing evidence suggests that insulin-secreting pancreatic ß-cells can adapt to chronic disruptions of pulsatility to rescue this physiologically important behavior. We determined the time scale for adaptation and examined potential ion channels underlying it. We induced the adaptation both by chronic application of the ATP-sensitive K+ [K(ATP)] channel blocker tolbutamide and by application of the depolarizing agent potassium chloride (KCl). Acute application of tolbutamide without pretreatment results in elevated Ca2+ as measured by fura-2AM and the loss of endogenous pulsatility. We show that after chronic exposure to tolbutamide (12-24 h), Ca2+ oscillations occur with subsequent acute tolbutamide application. The same experiment was conducted with potassium chloride (KCl) to directly depolarize the ß-cells. Once again, following chronic exposure to the cell stimulator, the islets produced Ca2+ oscillations when subsequently exposed to tolbutamide. These experiments suggest that it is the chronic stimulation, and not tolbutamide desensitization, that is responsible for the adaptation that rescues oscillatory ß-cell activity. This compensatory response also causes islet glucose sensitivity to shift rightward following chronic tolbutamide treatment. Mathematical modeling shows that a small increase in the number of K(ATP) channels in the membrane is one adaptation mechanism that is compatible with the data. To examine other compensatory mechanisms, pharmacological studies provide support that Kir2.1 and TEA-sensitive channels play some role. Overall, this investigation demonstrates ß-cell adaptability to overstimulation, which is likely an important mechanism for maintaining glucose homeostasis in the face of chronic stimulation.

Why pacing frequency affects the production of early afterdepolarizations in cardiomyocytes: An explanation revealed by slow-fast analysis of a minimal model.

Early afterdepolarizations (EADs) are pathological voltage oscillations in cardiomyocytes that have been observed in response to a number of pharmacological agents and disease conditions. Phase-2 EADs consist of small voltage fluctuations during the plateau of an action potential, typically under conditions in which the action potential is elongated. Although a single-cell behavior, EADs can lead to tissue-level arrhythmias. Much is currently known about the biophysical mechanisms (i.e., the roles of ion channels and intracellular Ca^{2+} stores) for the various forms of EADs, due partially to the development and analysis of mathematical models. This includes the application of slow-fast analysis, which takes advantage of timescale separation inherent in the system to simplify its analysis. We take this further, using a minimal three-dimensional model to demonstrate that phase-2 EADs are canards formed in the neighborhood of a folded node singularity. This allows us to predict the number of EADs that can be produced for a given parameter set, and provides guidance on parameter changes that facilitate or inhibit EAD production. With this approach, we demonstrate why periodic stimulation, as occurs in intact heart, preferentially facilitates EAD production when applied at low frequencies. We also explain the origin of complex alternan dynamics that can occur with intermediate-frequency stimulation, in which varying numbers of EADs are produced with each pulse. These revelations fall out naturally from an understanding of folded node singularities, but are difficult to glean from knowledge of the biophysical mechanism for EADs alone. Therefore, understanding the canard mechanism is a useful complement to understanding of the biophysical mechanism that has been developed over years of experimental and computational investigations.