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 β-cells

Electrical bursting oscillations in the β-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 Beta Cells?

The standard model for Ca²⁺ oscillations in insulin-secreting pancreatic beta 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 beta 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 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.

Coordination of pancreatic islet rhythmic activity by delayed negative feedback

Secretion of insulin from the pancreas is pulsatile, driven by intrinsic oscillations within individual islets of Langerhans. The secretions are coordinated among the many islets distributed throughout the pancreas producing a synchronized rhythm in vivo that is essential for maintaining normal glucose levels. One hypothesized mechanism for the coordination of islet activity is negative feedback, whereby sequestration of glucose in response to elevated insulin leads to a reduction in the blood glucose level that is sensed by the islet population. This global signal of glucose then coordinates the individual islets. In this study, we tested how this coordination mechanism is affected by time delays in the negative feedback, using a microfluidic system to monitor Ca2+ levels in a small population of islets and implementing glucose control through a negative feedback system. We found that islet synchronization occurs even with time delays in the feedback of up to 7 min. We also found that a second, slower closed-loop oscillation period is produced during delayed feedback in which islet oscillations are clustered into episodes. The period of this second oscillatory mode increases with the time delay and appears to be a second stable behavior that coexists with the faster synchronized oscillation. The general conclusion is that islet coordination through negative feedback is a viable means of islet coordination that is robust to delays in the timing of the feedback, and could complement other potential coordination mechanisms such as entrainment by pancreatic ganglia.NEW & NOTEWORTHY Insulin secretion from islets of Langerhans is rhythmic, and these rhythms are coordinated to produce oscillatory plasma insulin levels. Using a combination of microfluidics and computational modeling, we demonstrate that coordination can occur through negative feedback of the type provided by the liver, even if that feedback is delayed by several minutes. We also demonstrate that a second, slower, mode of oscillations can occur when feedback is delayed where faster oscillations are grouped into episodes.

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.

Pulsatile Basal Insulin Secretion Is Driven by Glycolytic Oscillations

In fasted and fed states, blood insulin levels are oscillatory. While this phenomenon is well studied at high glucose levels, comparatively little is known about its origin under basal conditions. We propose a possible mechanism for basal insulin oscillations based on oscillations in glycolysis, demonstrated using an established mathematical model. At high glucose, this is superseded by a calcium-dependent mechanism.

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

ATP-sensitive K+ (K(ATP)) channels were first reported in the β-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 β-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 β-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.

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.

Symbiosis of Electrical and Metabolic Oscillations in Pancreatic β-Cells

Insulin is secreted in a pulsatile pattern, with important physiological ramifications. In pancreatic β-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.

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.

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.

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.

Phantom bursting may underlie electrical bursting in single pancreatic β-cells

Insulin is secreted by pancreatic β-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 β-cellsin intact islets, while in other islets the β-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, β-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 β-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 β-cellsmay be due to a phantom bursting mechanism, and by extension, slow bursting in intact islets may also be driven by this mechanism.

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 β-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 β-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 β-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 β-cell adaptability to overstimulation, which is likely an important mechanism for maintaining glucose homeostasis in the face of chronic stimulation.

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.

Where to look and how to look: Combining global sensitivity analysis with fast/slow analysis to study multi-timescale oscillations

Parameterized systems of nonlinear ordinary differential equations, the type of system that is often used in mathematical models for biological systems, can be of sufficient complexity that it can take years to appreciate the full range of behaviors that can be produced. Global sensitivity analysis is one tool that has been developed for determining which parameters have the largest impact on the behavior of the model. Thus, it provides the user with a tool to know where to look in parameter space for important changes in behavior. However, it says nothing about the underlying mechanism mediating a change in behavior. For this, other tools exist. If the system dynamics occur over multiple highly-separated time scales then one useful analysis tool is fast/slow geometric analysis, also known as geometric singular perturbation analysis. This is based on bifurcation analysis of a fast or slow subsystem, and can shed light on the influence that a parameter has on structures of either subsystem, and thus on the system dynamics. Hence, once one knows where to look in parameter space for interesting behavior, this technique describes how to look at the system to extract information about how parameter changes influence the behavior of the system. In this study, we combine the two techniques in the analysis of bursting behavior in a model of insulin-secreting pancreatic β-cells, with the goal of determining the key parameters setting the period of the bursting oscillations, and understanding why they are so influential. This can be viewed as a case study for combining mathematical techniques to build on the strengths of each and thereby achieve a better understanding of what most influences the range of model behaviors and how this influence is brought about.

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.

Synchronization of pancreatic islets by periodic or non-periodic muscarinic agonist pulse trains

Pulsatile insulin secretion into the portal vein from the many pancreatic islets of Langerhans is critical for efficient glucose homeostasis. The islets are themselves endogenous oscillators, but since they are not physically coupled it is not obvious how their oscillations are synchronized across the pancreas. It has been proposed that synchronization of islets is achieved through periodic activity of intrapancreatic ganglia, and indeed there are data supporting this proposal. Postganglionic nerves are cholinergic, and their product, acetylcholine, can influence islet β-cells through actions on M₃ muscarinic receptors which are coupled to G_q type G-proteins. In addition, the neurons secrete several peptide hormones that act on β-cell receptors. The data supporting synchronization via intrapancreatic ganglia are, however, limited. In particular, it has not been shown that trains of muscarinic pulses are effective at synchronizing islets in vitro. Also, if as has been suggested, there is a ganglionic pacemaker driving islets to a preferred frequency, no neural circuitry for this pacemaker has been identified. In this study, both points are addressed using a microfluidic system that allows for the pulsed application of the muscarinic agonist carbachol. We find that murine islets are entrained and synchronized over a wide range of frequencies when the carbachol pulsing is periodic, adding support to the hypothesis that ganglia can synchronize islets in vivo. We also find that islet synchronization is very effective even if the carbachol pulses are applied at random times. This suggests that a neural pacemaker is not needed; all that is required is that islets receive occasional coordinated input from postganglionic neurons. The endogenous rhythmic activity of the islets then sets the frequency of the islet population rhythm, while the input from ganglia acts only to keep the islet oscillators in phase.

Female zebra finches do not sing yet share neural pathways necessary for singing in males

Adult female zebra finches (Taeniopygia guttata), which do not produce learned songs, have long been thought to possess only vestiges of the forebrain network that supports learned song in males. This view ostensibly explains why females do not sing-many of the neural populations and pathways that make up the male song control network appear rudimentary or even missing in females. For example, classic studies of vocal-premotor cortex (HVC, acronym is name) in male zebra finches identified prominent efferent pathways from HVC to vocal-motor cortex (RA, robust nucleus of the arcopallium) and from HVC to the avian basal ganglia (Area X). In females, by comparison, the efferent targets of HVC were thought to be only partially innervated by HVC axons (RA) or absent (Area X). Here, using a novel visually guided surgical approach to target tracer injections with precision, we mapped the extrinsic connectivity of the adult female HVC. We find that female HVC shows a mostly male-typical pattern of afferent and efferent connectivity, including robust HVC innervation of RA and Area X. As noted by earlier investigators, we find large sex differences in the volume of many regions that control male singing (male > female). However, sex differences in volume were diminished in regions that convey ascending afferent input to HVC. Our findings do not support a vestigial interpretation of the song control network in females. Instead, our findings support the emerging view that the song control network may have an altogether different function in nonsinging females.

Interhemispheric dominance switching in a neural network model for birdsong

Male zebra finches produce a sequence-invariant set of syllables, separated by short inspiratory gaps. These songs are learned from an adult tutor and maintained throughout life, making them a tractable model system for learned, sequentially ordered behaviors, particularly speech production. Moreover, much is known about the cortical, thalamic, and brain stem areas involved in producing this behavior, with the premotor cortical nucleus HVC (proper name) being of primary importance. In a previous study, our group developed a behavioral neural network model for birdsong constrained by the structural connectivity of the song system, the signaling properties of individual neurons and circuits, and circuit-breaking behavioral studies. Here we describe a more computationally tractable model and use it to explain the behavioral effects of unilateral cooling and electrical stimulations of HVC on song production. The model demonstrates that interhemispheric switching of song control is sufficient to explain these results, consistent with the hypotheses proposed when the experiments were initially conducted. Finally, we use the model to make testable predictions that can be used to validate the model framework and explain the effects of other perturbations of the song system, such as unilateral ablations of the primary input and output nuclei of HVC.

NEW & NOTEWORTHY In this report, we propose a two-hemisphere neural network model for the bilaterally symmetrical song system underlying birdsong in the male zebra finch. This model captures the behavioral effects of unilateral cooling and electrical stimulations of the premotor cortical nucleus HVC during song production, supporting the hypothesis of interhemispheric switching of song control. We use the model to make testable predictions regarding the behavioral effects of other unilateral perturbations to the song system.

Transitions between bursting modes in the integrated oscillator model for pancreatic β-cells

Insulin-secreting β-cells of pancreatic islets of Langerhans produce bursts of electrical impulses, resulting in intracellular Ca²⁺ oscillations and pulsatile insulin secretion. The mechanism for this bursting activity has been the focus of mathematical modeling for more than three decades, and as new data are acquired old models are modified and new models are developed. Comprehensive models must now account for the various modes of bursting observed in islet β-cells, which include fast bursting, slow bursting, and compound bursting. One such model is the Integrated Oscillator Model (IOM), in which β-cell electrical activity, intracellular Ca²⁺, and glucose metabolism interact via numerous feedforward and feedback pathways. These interactions can produce metabolic oscillations with a sawtooth time course or a pulsatile time course, reflecting very different oscillation mechanisms. In this report, we determine conditions favorable to one form of oscillations or the other, and examine the transitions between modes of bursting and the relationship of the transitions to the patterns of metabolic oscillations. Importantly, this work clarifies what can be expected in experimental measurements of β-cell oscillatory activity, and suggests pathways through which oscillations of one type can be converted to oscillations of another type.

Expansion of scroll wave filaments induced by chiral mismatch

In three-dimensional excitable systems, scroll waves are rotating vortex states that consist of smoothly stacked spirals. This stacking occurs along one-dimensional phase singularities called filaments. If the system has a positive filament tension, these curves either straighten or collapse over time. The collapse can be prevented if the filament pins to a nonreactive object or a group of objects, but even in this case, the filament length does not typically grow. Using numerical simulations, we provide examples of filament growth induced by pinning, such as a scroll ring pinning to an inert trefoil knot, and explain the mechanism of this growth. Surprisingly, the corresponding filament loop thus not only persists in time but also steadily extends far from the pinning object.

Closing in on the Mechanisms of Pulsatile Insulin Secretion

Insulin secretion from pancreatic islet β-cells occurs in a pulsatile fashion, with a typical period of ∼5 min. The basis of this pulsatility in mouse islets has been investigated for more than four decades, and the various theories have been described as either qualitative or mathematical models. In many cases the models differ in their mechanisms for rhythmogenesis, as well as other less important details. In this Perspective, we describe two main classes of models: those in which oscillations in the intracellular Ca²⁺ concentration drive oscillations in metabolism, and those in which intrinsic metabolic oscillations drive oscillations in Ca²⁺ concentration and electrical activity. We then discuss nine canonical experimental findings that provide key insights into the mechanism of islet oscillations and list the models that can account for each finding. Finally, we describe a new model that integrates features from multiple earlier models and is thus called the Integrated Oscillator Model. In this model, intracellular Ca²⁺ acts on the glycolytic pathway in the generation of oscillations, and it is thus a hybrid of the two main classes of models. It alone among models proposed to date can explain all nine key experimental findings, and it serves as a good starting point for future studies of pulsatile insulin secretion from human islets.

The Effects of GABAergic Polarity Changes on Episodic Neural Network Activity in Developing Neural Systems

Early in development, neural systems have primarily excitatory coupling, where even GABAergic synapses are excitatory. Many of these systems exhibit spontaneous episodes of activity that have been characterized through both experimental and computational studies. As development progress the neural system goes through many changes, including synaptic remodeling, intrinsic plasticity in the ion channel expression, and a transformation of GABAergic synapses from excitatory to inhibitory. What effect each of these, and other, changes have on the network behavior is hard to know from experimental studies since they all happen in parallel. One advantage of a computational approach is that one has the ability to study developmental changes in isolation. Here, we examine the effects of GABAergic synapse polarity change on the spontaneous activity of both a mean field and a neural network model that has both glutamatergic and GABAergic coupling, representative of a developing neural network. We find some intuitive behavioral changes as the GABAergic neurons go from excitatory to inhibitory, shared by both models, such as a decrease in the duration of episodes. We also find some paradoxical changes in the activity that are only present in the neural network model. In particular, we find that during early development the inter-episode durations become longer on average, while later in development they become shorter. In addressing this unexpected finding, we uncover a priming effect that is particularly important for a small subset of neurons, called the “intermediate neurons.” We characterize these neurons and demonstrate why they are crucial to episode initiation, and why the paradoxical behavioral change result from priming of these neurons. The study illustrates how even arguably the simplest of developmental changes that occurs in neural systems can present non-intuitive behaviors. It also makes predictions about neural network behavioral changes that occur during development that may be observable even in actual neural systems where these changes are convoluted with changes in synaptic connectivity and intrinsic neural plasticity.

Intrinsic physiology of inhibitory neurons changes over auditory development

During auditory development, changes in membrane properties promote the ability of excitatory neurons in the brain stem to code aspects of sound, including the level and timing of a stimulus. Some of these changes coincide with hearing onset, suggesting that sound-driven neural activity produces developmental plasticity of ion channel expression. While it is known that the coding properties of excitatory neurons are modulated by inhibition in the mature system, it is unknown whether there are also developmental changes in the membrane properties of brain stem inhibitory neurons. We investigated the primary source of inhibition in the avian auditory brain stem, the superior olivary nucleus (SON). The present studies test the hypothesis that, as in excitatory neurons, the membrane properties of these inhibitory neurons change after hearing onset. We examined SON neurons at different stages of auditory development: embryonic days 14-16 (E14-E16), a time at which cochlear ganglion neurons are just beginning to respond to sound; later embryonic stages (E18-E19); and after hatching (P0-P2). We used in vitro whole cell patch electrophysiology to explore physiological changes in SON. Age-related changes were observed at the level of a single spike and in multispiking behavior. In particular, tonic behavior, measured as a neuron's ability to sustain tonic firing over a range of current steps, became more common later in development. Voltage-clamp recordings and biophysical models were employed to examine how age-related increases in ion currents enhance excitability in SON. Our findings suggest that concurrent increases in sodium and potassium currents underlie the emergence of tonic behavior. NEW & NOTEWORTHY This article is the first to examine heterogeneity of neuronal physiology in the inhibitory nucleus of the avian auditory system and demonstrate that tonic firing here emerges over development. By pairing computer simulations with physiological data, we show that increases in both sodium and potassium channels over development are necessary for the emergence of tonic firing.

Upregulation of an inward rectifying K+ channel can rescue slow Ca2+ oscillations in K(ATP) channel deficient pancreatic islets

Plasma insulin oscillations are known to have physiological importance in the regulation of blood glucose. In insulin-secreting β-cells of pancreatic islets, K(ATP) channels play a key role in regulating glucose-dependent insulin secretion. In addition, they convey oscillations in cellular metabolism to the membrane by sensing adenine nucleotides, and are thus instrumental in mediating pulsatile insulin secretion. Blocking K(ATP) channels pharmacologically depolarizes the β-cell plasma membrane and terminates islet oscillations. Surprisingly, when K(ATP) channels are genetically knocked out, oscillations in islet activity persist, and relatively normal blood glucose levels are maintained. Compensation must therefore occur to overcome the loss of K(ATP) channels in K(ATP) knockout mice. In a companion study, we demonstrated a substantial increase in Kir2.1 protein occurs in β-cells lacking K(ATP) because of SUR1 deletion. In this report, we demonstrate that β-cells of SUR1 null islets have an upregulated inward rectifying K⁺ current that helps to compensate for the loss of K(ATP) channels. This current is likely due to the increased expression of Kir2.1 channels. We used mathematical modeling to determine whether an ionic current having the biophysical characteristics of Kir2.1 is capable of rescuing oscillations that are similar in period to those of wild-type islets. By experimentally testing a key model prediction we suggest that Kir2.1 current upregulation is a likely mechanism for rescuing the oscillations seen in islets from mice deficient in K(ATP) channels.

Pulsatile insulin secretion is important for the proper regulation of blood glucose, and disruption of this pulsatility is a hallmark of type II diabetes. An ion channel was discovered more than three decades ago that conveys the metabolic state of insulin-secreting β-cells to the plasma membrane because it is blocked by ATP and opened by ADP, and thereby controls the activity of these electrically-excitable cells on a rapid time scale according to the prevailing blood glucose level. In addition to setting the appropriate level of insulin secretion, K(ATP) channels play a key role in generating the oscillations in cellular activity that underlie insulin pulsatility. It is therefore surprising that oscillations in activity persist in islets in which the K(ATP) channels are genetically knocked out. In this combined modeling and experimental study, we demonstrate that the role played by K(ATP) current in wild-type β-cells can be taken over by an inward-rectifying K⁺ current which, we show here, is upregulated in β-cells from SUR1 knockout mice. This result helps to resolve a mystery in the field that has remained elusive for more than a decade, since the first studies showing oscillations in SUR1^-/- islets.

Calcium Oscillation Frequency-Sensitive Gene Regulation and Homeostatic Compensation in Pancreatic β-Cells

Pancreatic islet [Formula: see text]-cells are electrically excitable cells that secrete insulin in an oscillatory fashion when the blood glucose concentration is at a stimulatory level. Insulin oscillations are the result of cytosolic [Formula: see text] oscillations that accompany bursting electrical activity of [Formula: see text]-cells and are physiologically important. ATP-sensitive [Formula: see text] channels (K(ATP) channels) play the key role in setting the overall activity of the cell and in driving bursting, by coupling cell metabolism to the membrane potential. In humans, when there is a defect in K(ATP) channel function, [Formula: see text]-cells fail to respond appropriately to changes in the blood glucose level, and electrical and [Formula: see text] oscillations are lost. However, mice compensate for K(ATP) channel defects in islet [Formula: see text]-cells by employing alternative mechanisms to maintain electrical and [Formula: see text] oscillations. In a recent study, we showed that in mice islets in which K(ATP) channels are genetically knocked out another [Formula: see text] current, provided by inward-rectifying [Formula: see text] channels, is increased. With mathematical modeling, we demonstrated that a sufficient upregulation in these channels can account for the paradoxical electrical bursting and [Formula: see text] oscillations observed in these [Formula: see text]-cells. However, the question of determining the correct level of upregulation that is necessary for this compensation remained unanswered, and this question motivates the current study. [Formula: see text] is a well-known regulator of gene expression, and several examples have been shown of genes that are sensitive to the frequency of the [Formula: see text] signal. In this mathematical modeling study, we demonstrate that a [Formula: see text] oscillation frequency-sensitive gene transcription network can adjust the gene expression level of a compensating [Formula: see text] channel so as to rescue electrical bursting and [Formula: see text] oscillations in a model [Formula: see text]-cell in which the key K(ATP) current is removed. This is done without the prescription of a target [Formula: see text] level, but evolves naturally as a consequence of the feedback between the [Formula: see text]-dependent enzymes and the cell's electrical activity. More generally, the study indicates how [Formula: see text] can provide the link between gene expression and cellular electrical activity that promotes wild-type behavior in a cell following gene knockout.

A distributed neural network model for the distinct roles of medial and lateral HVC in zebra finch song production

Male zebra finches produce a song consisting of a canonical sequence of syllables, learned from a tutor and repeated throughout its adult life. Much of the neural circuitry responsible for this behavior is located in the cortical premotor region HVC (acronym is name). In a recent study from our laboratory, we found that partial bilateral ablation of the medial portion of HVC has effects on the song that are qualitatively different from those of bilateral ablation of the lateral portion. In this report we describe a neural network organization that can explain these data, and in so doing suggests key roles for other brain nuclei in the production of song. We also suggest that syllables and the gaps between them are each coded separately by neural chains within HVC, and that the timing mechanisms for syllables and gaps are distinct. The design principles underlying this model assign distinct roles for medial and lateral HVC circuitry that explain the data on medial and lateral ablations. In addition, despite the fact that the neural coding of song sequence is distributed among several brain nuclei in our model, it accounts for data showing that cooling of HVC stretches syllables uniformly and to a greater extent than gaps. Finally, the model made unanticipated predictions about details of the effects of medial and lateral HVC ablations that were then confirmed by reanalysis of these previously acquired behavioral data.NEW & NOTEWORTHY Zebra finch song consists of a string of syllables repeated in a nearly invariant sequence. We propose a neural network organization that can explain recent data indicating that the medial and lateral portions of the premotor cortical nucleus HVC have different roles in zebra finch song production. Our model explains these data, as well as data on the effects on song of cooling HVC, and makes predictions that we test in the singing bird.

Stabilization of collapsing scroll waves in systems with random heterogeneities

In three-dimensional reaction-diffusion systems, excitation waves may form and rotate around a one-dimensional phase singularity called the filament. If the filament forms a closed curve, it will shrink over time and eventually collapse. However, filaments may pin to non-reactive objects present in the medium, reducing their rate of collapse or even allowing them to persist indefinitely. We use numerical simulations to study how different arrangements of non-reactive spheres affect the dynamics of circular filaments. As the filament contracts, it gets closer to and eventually touches and pins to objects in its path. This causes two possible behaviors. The filament can detach from the spheres in its path, slowing down the rate of contraction, or it can remain pinned to a collection of spheres. In general, more or larger spheres increase the chance that the filament remains pinned, but there are exceptions. It is possible for a small number of small spheres to support the filament and possible for the filament to pass through a large number of large spheres. Our work yields insights into the pinning of scroll waves in excitable tissue such as cardiac muscle, where scar tissue acts in a way similar to the non-reactive domains.

Melt growth and properties of bulk BaSnO3 single crystals

We present the first-time growth of bulk BaSnO₃ single crystals from the melt by direct solidification, their basic electrical and optical properties as well as their structural quality. Our measurement of the melting point (MP) of BaSnO₃ amounts to 1855 °C  ±  25 K. At this temperature an intensive decomposition and non-stoichiometric evaporation takes place as the partial pressure of SnO(g) is about 90 times higher than that of BaO(g). X ray powder diffraction identified only the BaSnO₃ perovskite phase, while narrow rocking curves having a full width at half maximum of 26 arcsec and etch pit densities below 10⁶ cm^-2 confirm a high degree of structural perfection of the single crystals. In this respect they surpass the structural properties of those single crystals that were reported in the literature. The electrical conductivity of nominally undoped crystals depends on the growth conditions and ranges from insulating to medium n-type conductivity. After post-growth annealing in an oxidizing atmosphere undoped crystals are generally insulating. Doping the crystals with lanthanum during growth results in a high n-type conductivity. For a La doping concentration of 0.123 wt.% we measured an electron concentration of 3.3  ×  10¹⁹ cm^-3 and an electron mobility of 219 cm² V^-1 s^-1. Based on optical absorption measurements we determined an energy of 3.17  ±  0.04 eV at 5 K and of 2.99  ±  0.04 eV at 297 K for the indirect band gap of BaSnO₃.

Ca2+ Effects on ATP Production and Consumption Have Regulatory Roles on Oscillatory Islet Activity

Pancreatic islets respond to elevated blood glucose by secreting pulses of insulin that parallel oscillations in β-cell metabolism, intracellular Ca(2+) concentration, and bursting electrical activity. The mechanisms that maintain an oscillatory response are not fully understood, yet several models have been proposed. Only some can account for experiments supporting that metabolism is intrinsically oscillatory in β-cells. The dual oscillator model (DOM) implicates glycolysis as the source of oscillatory metabolism. In the companion article, we use recently developed biosensors to confirm that glycolysis is oscillatory and further elucidate the coordination of metabolic and electrical signals in the insulin secretory pathway. In this report, we modify the DOM by incorporating an established link between metabolism and intracellular Ca(2+) to reconcile model predictions with experimental observations from the companion article. With modification, we maintain the distinguishing feature of the DOM, oscillatory glycolysis, but introduce the ability of Ca(2+) influx to reshape glycolytic oscillations by promoting glycolytic efflux. We use the modified model to explain measurements from the companion article and from previously published experiments with islets.

Phase Analysis of Metabolic Oscillations and Membrane Potential in Pancreatic Islet β-Cells

Metabolism in islet β-cells displays oscillations that can trigger pulses of electrical activity and insulin secretion. There has been a decades-long debate among islet biologists about whether metabolic oscillations are intrinsic or occur in response to oscillations in intracellular Ca(2+) that result from bursting electrical activity. In this article, the dynamics of oscillatory metabolism were investigated using five different optical reporters. Reporter activity was measured simultaneously with membrane potential bursting to determine the phase relationships between the metabolic oscillations and electrical activity. Our experimental findings suggest that Ca(2+) entry into β-cells stimulates the rate of mitochondrial metabolism, accounting for the depletion of glycolytic intermediates during each oscillatory burst. We also performed Ca(2+) clamp tests in which we clamped membrane potential with the KATP channel-opener diazoxide and KCl to fix Ca(2+) at an elevated level. These tests confirm that metabolic oscillations do not require Ca(2+) oscillations, but show that Ca(2+) plays a larger role in shaping metabolic oscillations than previously suspected. A dynamical picture of the mechanisms of oscillations emerged that requires the restructuring of contemporary mathematical β-cell models, including our own dual oscillator model. In the companion article, we modified our model to account for these new data.

Glucose Oscillations Can Activate an Endogenous Oscillator in Pancreatic Islets

Pancreatic islets manage elevations in blood glucose level by secreting insulin into the bloodstream in a pulsatile manner. Pulsatile insulin secretion is governed by islet oscillations such as bursting electrical activity and periodic Ca²⁺ entry in β-cells. In this report, we demonstrate that although islet oscillations are lost by fixing a glucose stimulus at a high concentration, they may be recovered by subsequently converting the glucose stimulus to a sinusoidal wave. We predict with mathematical modeling that the sinusoidal glucose signal’s ability to recover islet oscillations depends on its amplitude and period, and we confirm our predictions by conducting experiments with islets using a microfluidics platform. Our results suggest a mechanism whereby oscillatory blood glucose levels recruit non-oscillating islets to enhance pulsatile insulin output from the pancreas. Our results also provide support for the main hypothesis of the Dual Oscillator Model, that a glycolytic oscillator endogenous to islet β-cells drives pulsatile insulin secretion.

A global shift throughout the last century toward excessive nutrient intake relative to energy expenditure has fueled a dramatic increase in the incidence of diabetes in humans. The epidemic is primarily of type 2 diabetes, a disease characterized by the inability of the body to effectively control blood glucose levels. Insulin plays a key role in regulating blood glucose levels by restraining endogenous glucose output from the liver and promoting blood glucose uptake by tissue throughout the body. It is secreted in pulses by islets of Langerhans, endocrine cell aggregates dispersed throughout the pancreas. Loss of insulin pulsatility is an early event in the development of type 2 diabetes. Here, we demonstrate, with a combined modeling and experimental approach, that the loss of pulsatile insulin release that results from elevated glucose may be recovered by an oscillatory glucose stimulus. Our results have potential implications for enhancing insulin pulsatility and therefore mitigating the development of type 2 diabetes.

Dual Detection System for Simultaneous Measurement of Intracellular Fluorescent Markers and Cellular Secretion

Glucose-stimulated insulin secretion from pancreatic β-cells within islets of Langerhans plays a critical role in maintaining glucose homeostasis. Although this process is essential for maintaining euglycemia, the underlying intracellular mechanisms that control it are still unclear. To allow simultaneous correlation between intracellular signal transduction events and extracellular secretion, an analytical system was developed that integrates fluorescence imaging of intracellular probes with high-speed automated insulin immunoassays. As a demonstration of the system, intracellular [Ca²⁺] ([Ca²⁺]_i) was measured by imaging Fura-2 fluorescence simultaneously with insulin secretion from islets exposed to elevated glucose levels. Both [Ca²⁺]_i and insulin were oscillatory during application of 10 mM glucose with temporal and quantitative profiles similar to what has been observed elsewhere. In previous work, sinusoidal glucose levels have been used to test the entrainment of islets while monitoring either [Ca²⁺]_i or insulin levels; using this newly developed system, we show unambiguously that oscillations of both [Ca²⁺]_i and insulin release are entrained to oscillatory glucose levels and that the temporal correlation of these are maintained throughout the experiment. It is expected that the developed analytical system can be expanded to investigate a number of other intracellular messengers in islets or other stimulus-secretion pathways in different cells.

Influence of dynorphin on estradiol- and cervical stimulation-induced prolactin surges in ovariectomized rats

Prolactin is an anterior pituitary hormone necessary for fertility, pregnancy maintenance, lactation, and aspects of maternal behavior. In rodents, there is a surge of prolactin on the afternoon of proestrus, and a semi-circadian pattern of prolactin surges during early pregnancy, with a diurnal and nocturnal surge every day. Both of these patterns can be replicated in ovariectomized rats. A prior study demonstrated that central antagonism of κ-opioid receptors, the target of dynorphin, largely abolished the nocturnal prolactin surge in pregnant rats. We build on this to determine whether dynorphin, perhaps from the arcuate population that co-express kisspeptin, neurokinin B, and dynorphin (KNDy neurons), also contributes to the estradiol- or cervical stimulation-induced surges in ovariectomized rats. Ovariectomized rats were treated with either estradiol or cervical stimulation to induce prolactin surge(s). Blood samples were taken around the expected surge time to determine the effect of either acute κ-opioid receptor antagonism or previous chemical ablation of the KNDy population on prolactin levels. Dynorphin antagonism does significantly disrupt the nocturnal prolactin surge, but it does not contribute to the estradiol-induced surge. Chemical ablation of KNDy neurons had opposite effects; ablation of 40 % of the KNDy neurons had no impact on the nocturnal prolactin surge, while a somewhat larger ablation significantly reduced the size of the estradiol-induced surge. We conclude that dynorphin is likely a controlling factor for the nocturnal surge induced by cervical stimulation, and that other KNDy neuron products must play a role in the estradiol-induced surge.

Glucocorticoids Inhibit CRH/AVP-Evoked Bursting Activity of Male Murine Anterior Pituitary Corticotrophs

Corticotroph cells from the anterior pituitary are an integral component of the hypothalamic-pituitary-adrenal (HPA) axis, which governs the neuroendocrine response to stress. Corticotrophs are electrically excitable and fire spontaneous single-spike action potentials and also display secretagogue-induced bursting behavior. The HPA axis function is dependent on effective negative feedback in which elevated plasma glucocorticoids result in inhibition at the level of both the pituitary and the hypothalamus. In this study, we have used an electrophysiological approach coupled with mathematical modeling to investigate the regulation of spontaneous and CRH/arginine vasopressin-induced activity of corticotrophs by glucocorticoids. We reveal that pretreatment of corticotrophs with 100 nM corticosterone (CORT; 90 and 150 min) reduces spontaneous activity and prevents a transition from spiking to bursting after CRH/arginine vasopressin stimulation. In addition, previous studies have identified a role for large-conductance calcium- and voltage-activated potassium (BK) channels in the generation of secretagogue-induced bursting in corticotrophs. Using the dynamic clamp technique, we demonstrated that CRH-induced bursting can be switched to spiking by subtracting a fast BK current, whereas the addition of a fast BK current can induce bursting in CORT-treated cells. In addition, recordings from BK knockout mice (BK(-/-)) revealed that CORT can also inhibit excitability through BK-independent mechanisms to control spike frequency. Thus, we have established that glucocorticoids can modulate multiple properties of corticotroph electrical excitability through both BK-dependent and BK-independent mechanisms.

Multi-timescale systems and fast-slow analysis

Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.

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.

Is bursting more effective than spiking in evoking pituitary hormone secretion? A spatiotemporal simulation study of calcium and granule dynamics

Endocrine cells of the pituitary gland secrete a number of hormones, and the amount of hormone released by a cell is controlled in large part by the cell's electrical activity and subsequent Ca(2+) influx. Typical electrical behaviors of pituitary cells include continuous spiking and so-called pseudo-plateau bursting. It has been shown that the amplitude of Ca(2+) fluctuations is greater in bursting cells, leading to the hypothesis that bursting cells release more hormone than spiking cells. In this work, we apply computer simulations to test this hypothesis. We use experimental recordings of electrical activity as input to mathematical models of Ca(2+) channel activity, buffered Ca(2+) diffusion, and Ca(2+)-driven exocytosis. To compare the efficacy of spiking and bursting on the same cell, we pharmacologically block the large-conductance potassium (BK) current from a bursting cell or add a BK current to a spiking cell via dynamic clamp. We find that bursting is generally at least as effective as spiking at evoking hormone release and is often considerably more effective, even when normalizing to Ca(2+) influx. Our hybrid experimental/modeling approach confirms that adding a BK-type K(+) current, which is typically associated with decreased cell activity and reduced secretion, can actually produce an increase in hormone secretion, as suggested earlier.