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

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.

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 Ca(2+) 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 Ca(2+) 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 Ca(2+) 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.

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 CRH-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. ABSTRACT 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 CRH-induced bursting that enhances ACTH secretion. Previous studies revealed HPA 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 increases in spontaneous bursting, that 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 calcium (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; that 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. This article is protected by copyright. All rights reserved.

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 StatementNeurons 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.

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.

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.

Early afterdepolarizations (EADs) are pathological voltage fluctuations that can occur in cardiac cells and are a potent source of potentially fatal arrhythmias. Recent works examining the mechanis...

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.

The growing demand for electric vehicles and portable devices has brought significant attention to the scalability, recyclability, and economics of both traditional and emerging battery technologies. Lithium-ion batteries (LIB) are approaching their theoretical energy density yet remain the most widespread and promising technology for the next 5-10 years, as alternative (e.g., Li-metal) chemistries and solid-state architectures are still in relatively early stages of commercial scale-up. An alternative route to improve LIB performance and therefore increase energy density, is to redesign the cell geometry to incorporate thick electrodes. Batteries with thick planar (2D) electrodes suffer from power limitations and capacity loss due to increased Li-ion diffusion distances and tortuosity. 3D electrode designs compensate for this weakness by providing micro-scale channels within the electrode to enable rapid charge transport. Prototype LIBs using 3D electrodes have been fabricated using a wide variety of laboratory techniques (e.g., lithography, imprinting, direct-ink writing), which have afforded study of diverse geometries (e.g., pillar arrays, lattices, interpenetrating structures) and active materials (e.g., Si, Sn, carbon-based, lithium transition metal oxides). However, these methods do not have requisite scalability or compatibility with industrial LIB processing techniques, which demand: (1) integration with cost-effective metal substrates as current collectors; (2) compatibility with commodity anode/cathode materials and electrolytes; and (3) high-rate continuous production and cell assembly. Moreover, the materials used in 3D electrodes must be mechanically robust, electrically conductive, patterned at the microscale level, and have sufficient porosity to accommodate expansion and contraction upon charging and discharging. We are developing thick 3D “honeycomb” battery electrodes using patterned, vertically aligned carbon nanotubes (VA-CNTs) as current collectors. CNTs are widely known to have high thermal and electrical conductivities, and to be mechanically durable. VA-CNTs (“forests”) form by self-organization of CNTs during chemical vapor deposition (CVD) on substrates using common hydrocarbon sources (e.g., C2H2, C2H4). However, well-established CNT growth techniques utilize rigid non-conductive substrates, such as silicon wafers, necessitating an additional transfer step to a suitable substrate. Recently, we translated these insights from CNT growth on silicon wafer substrates to thin metal foils (Cu) suitable for battery electrode fabrication, using optimized in-furnace moisture conditions, and a thin-film interface that forms high-density nanoparticles upon annealing, and prevents diffusion of Cu that is known to poison CNT growth. The resulting CNT forests can reach up to 500 μm thickness and can be patterned to have regular hole arrays with feature sizes as small as ~5 μm. Thick composite electrodes are then created by coating CNT forests with Si thin films by CVD. The inherent nanoporosity of the CNT forests (>95%) allows for precursor diffusion into the entirety of the forest cross-section. The CVD conditions are tuned to create dense electrodes with tailored Si loading ranging from ~10 to ~90 at.%. Half-cells using monolithic and patterned Si-CNT electrodes (~200 μm tall), Li-metal foil and a liquid electrolyte/separator combination are cycled over a range of currents, demonstrating electronic connection between the deposited Si and Cu foil via the aligned CNTs. At low currents (C/5) these electrodes demonstrate large gravimetric capacities, indicating the lithiation of CNTs beneath the deposited Si. We will also investigate cycling performance at high currents (>5C) and perform post-mortem characterization of the Si-CNT interface to study how the CNTs and their patterning govern the stability of the electrode. Future work will look to optimize electrode performance by manipulating the geometric parameters of the CNT forests and assess the economics of CNT-based electrode manufacturing at commercial scales.

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.

Song learning in zebra finches (Taeniopygia guttata) requires exposure to the song of a tutor, resulting in an auditory memory. This memory is the foundation for later sensorimotor learning, resulting in the production of a copy of the tutor's song. The cortical premotor nucleus HVC (proper name) is necessary for auditory and sensorimotor learning as well as the eventual production of adult song. We recently discovered that the intrinsic physiology of HVC neurons changes across stages of song learning, but are those changes the result of learning or are they experience-independent developmental changes? To test the role of auditory experience in driving intrinsic changes, patch-clamp experiments were performed comparing HVC neurons in juvenile birds with varying amounts of tutor exposure. The intrinsic physiology of HVC neurons changed as a function of tutor exposure. Counterintuitively, tutor deprivation resulted in juvenile HVC neurons showing an adult-like phenotype not present in tutor-exposed juveniles. Biophysical models were developed to predict which ion channels were modulated by experience. The models indicate that tutor exposure transiently suppressed theIhand T-type Ca2+currents in HVC neurons that target the basal ganglia, whereas tutor exposure increased the resting membrane potential and decreased the spike amplitude in HVC neurons that drive singing. Our findings suggest that intrinsic plasticity may be part of the mechanism for auditory learning in the HVC. More broadly, models of learning and memory should consider intrinsic plasticity as a possible mechanism by which the nervous system encodes the lasting effects of experience.SIGNIFICANCE STATEMENTIt is well established that learning involves plasticity of the synapses between neurons. However, the activity of a neural circuit can also be dramatically altered by changes in the intrinsic properties (ion channels) of the component neurons. The present experiments show experience-dependent changes in the intrinsic physiology of neurons in the cortical premotor nucleus HVC (proper name) in juvenile zebra finches (Taeniopygia guttata) during auditory learning of a tutor's song. Tutor deprivation does not “arrest” development of intrinsic properties, but rather results in neurons with a premature adult-like physiological phenotype. It is possible that auditory learning involves a form of nonsynaptic plasticity and that experience-dependent suppression of specific ion channels may work in concert with synaptic plasticity to promote vocal learning.

Insulin-secreting pancreatic β-cells are electrically excitable cells that are unusual because their electrical activity is influenced directly by metabolism via ATP-sensitive K+ channels. At the same time, changes in the intracellular Ca2+concentration that result from the cell's electrical activity influence metabolism in several ways. Thus, there is bidirectional coupling between the electrical dynamics and the metabolic dynamics in β-cells. A mathematical model has been previously developed, called the Integrated. Oscillator Model (IOM), to highlight the bidirectional coupling involved in the oscillation mechanism. In this study, we show how this coupling can produce oscillations in β-cell activity. These oscillations have period similar to that of insulin secretion pulses observed in rats, mice, dogs, and humans, which has been shown to facilitate the action of the liver in maintaining glucose homeostasis. In a companion paper we show that the IOM can produce oscillations using two distinct mechanisms, depending on the values of electrical and metabolic parameters. In the present article, we use fast-slow analysis to understand the mechanisms underlying each of these oscillations. In particular, we show why a key variable in the glycolytic pathway generates a pulsatile time course in one type of oscillation, while it generates a sawtooth time course in the other type. The significance of these patterns is that the time course is a reflection of whether an intrinsic glycolytic oscillator is active, or whether the oscillations are a direct consequence of Ca2+ feedback onto glycolysis.

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.

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., Author summary EADs are pathological voltage fluctuations that can occur during the plateau or repolarizing phase of cardiac action potentials. The EADs of single cells, when embedded in a network of cardiac tissue, can lead to deadly cardiac arrhythmia. Because of this, many experimental and theoretical investigations have been conducted to uncover the biophysical and dynamical origins of EAD genesis. A recurring finding is that suitable changes in the properties of the inward L-type calcium current are sufficient for EAD production. A particularly important property of the L-type calcium current, with respect to EAD production, is the size of its window region. In this work, we use a novel geometric approach to analyze the role of the window region in cellular electrical dynamics using a low-dimensional ventricular action potential model. We illustrate the mechanism underlying window region-induced EADs, and demonstrate how the number of EADs produced can be predicted, using dynamical systems techniques together with canard theory. These techniques allow us to explain precisely why the model reproduces myriad experimental observations while also allowing us to make the testable predictions that either advancing the activation rate or slowing the inactivation rate of the L-type calcium current—changes that would reasonably be expected to increase its active duration and the likelihood of EADs—should, instead, reduce its active duration and the likelihood of EADs.

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.