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Phan NA, Wang Y. Mixed-mode oscillations in a three-timescale coupled Morris-Lecar system. CHAOS (WOODBURY, N.Y.) 2024; 34:053119. [PMID: 38717416 PMCID: PMC11087137 DOI: 10.1063/5.0181308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 04/19/2024] [Indexed: 05/12/2024]
Abstract
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
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Affiliation(s)
- Ngoc Anh Phan
- Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA
| | - Yangyang Wang
- Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA
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2
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Martin M, Pedersen MG. Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels. PLoS Comput Biol 2024; 20:e1011559. [PMID: 38517941 PMCID: PMC10990174 DOI: 10.1371/journal.pcbi.1011559] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2023] [Revised: 04/03/2024] [Accepted: 03/10/2024] [Indexed: 03/24/2024] Open
Abstract
Cyclic AMP controls neuronal ion channel activity. For example hyperpolarization-activated cyclic nucleotide-gated (HCN) and M-type K+ channels are activated by cAMP. These effects have been suggested to be involved in astrocyte control of neuronal activity, for example, by controlling the action potential firing frequency. In cortical neurons, cAMP can induce mixed-mode oscillations (MMOs) consisting of small-amplitude, subthreshold oscillations separating complete action potentials, which lowers the firing frequency greatly. We extend a model of neuronal activity by including HCN and M channels, and show that it can reproduce a series of experimental results under various conditions involving and inferring with cAMP-induced activation of HCN and M channels. In particular, we find that the model can exhibit MMOs as found experimentally, and argue that both HCN and M channels are crucial for reproducing these patterns. To understand how M and HCN channels contribute to produce MMOs, we exploit the fact that the model is a three-time scale dynamical system with one fast, two slow, and two super-slow variables. We show that the MMO mechanism does not rely on the super-slow dynamics of HCN and M channel gating variables, since the model is able to produce MMOs even when HCN and M channel activity is kept constant. In other words, the cAMP-induced increase in the average activity of HCN and M channels allows MMOs to be produced by the slow-fast subsystem alone. We show that the slow-fast subsystem MMOs are due to a folded node singularity, a geometrical structure well known to be involved in the generation of MMOs in slow-fast systems. Besides raising new mathematical questions for multiple-timescale systems, our work is a starting point for future research on how cAMP signalling, for example resulting from interactions between neurons and glial cells, affects neuronal activity via HCN and M channels.
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Affiliation(s)
- Matteo Martin
- Department of Information Engineering, University of Padova, Padova, Italy
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3
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Fazli M, Bertram R. Conversion of spikers to bursters in pituitary cell networks: Is it better to disperse for maximum exposure or circle the wagons? PLoS Comput Biol 2024; 20:e1011811. [PMID: 38289902 PMCID: PMC10826967 DOI: 10.1371/journal.pcbi.1011811] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2023] [Accepted: 01/08/2024] [Indexed: 02/01/2024] Open
Abstract
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.
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Affiliation(s)
- Mehran Fazli
- Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America
- Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America
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Bertram R, Marinelli I, Fletcher PA, Satin LS, Sherman AS. Deconstructing the integrated oscillator model for pancreatic β-cells. Math Biosci 2023; 365:109085. [PMID: 37802364 PMCID: PMC10991200 DOI: 10.1016/j.mbs.2023.109085] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2023] [Revised: 09/12/2023] [Accepted: 09/26/2023] [Indexed: 10/10/2023]
Abstract
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.
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Affiliation(s)
- Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, United States.
| | - Isabella Marinelli
- Centre for Systems Modeling and Quantitative Biomedicine, University of Birmingham, United Kingdom
| | - Patrick A Fletcher
- Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, United States
| | - Leslie S Satin
- Department of Pharmacology and Brehm Center for Diabetes Research, University of Michigan Medical School, Ann Arbor, MI, United States
| | - Arthur S Sherman
- Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, United States
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Duncan PJ, Romanò N, Nair SV, Murray JF, Le Tissier P, Shipston MJ. Sex differences in pituitary corticotroph excitability. Front Physiol 2023; 14:1205162. [PMID: 37534368 PMCID: PMC10391550 DOI: 10.3389/fphys.2023.1205162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 07/05/2023] [Indexed: 08/04/2023] Open
Abstract
Stress-related illness represents a major burden on health and society. Sex differences in stress-related disorders are well documented, with women having twice the lifetime rate of depression compared to men and most anxiety disorders. Anterior pituitary corticotrophs are central components of the hypothalamic-pituitary-adrenal (HPA) axis, receiving input from hypothalamic neuropeptides corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP), while regulating glucocorticoid output from the adrenal cortex. The dynamic control of electrical excitability by CRH/AVP and glucocorticoids is critical for corticotroph function; however, whether corticotrophs contribute to sexually differential responses of the HPA axis, which might underlie differences in stress-related disorders, is very poorly understood. Using perforated patch clamp electrophysiology in corticotrophs from mice expressing green fluorescent protein under the control of the Pomc promoter, we characterized basal and secretagogue-evoked excitability. Both male and female corticotrophs show predominantly single-spike action potentials under basal conditions; however, males predominantly display spikes with small-amplitude (<20 mV) afterhyperpolarizations (B-type), whereas females displayed a mixture of B-type spikes and spikes with a large-amplitude (>25 mV) afterhyperpolarization (A-type). In response to CRH, or CRH/AVP, male cells almost exclusively transition to a predominantly pseudo-plateau bursting, whereas only female B-type cells display bursting in response to CRH±AVP. Treatment of male or female corticotrophs with 1 nM estradiol (E2) for 24-72 h has no effect on the proportion of cells with A- or B-type spikes in either sex. However, E2 results in the cessation of CRH-induced bursting in both male and female corticotrophs, which can be partially reversed by adding a BK current using a dynamic clamp. RNA-seq analysis of purified corticotrophs reveals extensive differential gene expression at the transcriptional level, including more than 71 mRNAs encoding ion channel subunits. Interestingly, there is a two-fold lower level (p < 0.01) of BK channel pore-forming subunit (Kcnma1) expression in females compared to males, which may partially explain the decrease in CRH-induced bursting. This study identified sex differences at the level of the anterior pituitary corticotroph ion channel landscape and control of both spontaneous and CRH-evoked excitability. Determining the mechanisms of sex differences of corticotroph and HPA activity at the cellular level could be an important step for better understanding, diagnosing, and treating stress-related disorders.
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John SR, Krauskopf B, Osinga HM, Rubin JE. Slow negative feedback enhances robustness of square-wave bursting. J Comput Neurosci 2023; 51:239-261. [PMID: 37067661 PMCID: PMC10181982 DOI: 10.1007/s10827-023-00846-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2022] [Revised: 01/17/2023] [Accepted: 02/15/2023] [Indexed: 04/18/2023]
Abstract
Square-wave bursting is an activity pattern common to a variety of neuronal and endocrine cell models that has been linked to central pattern generation for respiration and other physiological functions. Many of the reduced mathematical models that exhibit square-wave bursting yield transitions to an alternative pseudo-plateau bursting pattern with small parameter changes. This susceptibility to activity change could represent a problematic feature in settings where the release events triggered by spike production are necessary for function. In this work, we analyze how model bursting and other activity patterns vary with changes in a timescale associated with the conductance of a fast inward current. Specifically, using numerical simulations and dynamical systems methods, such as fast-slow decomposition and bifurcation and phase-plane analysis, we demonstrate and explain how the presence of a slow negative feedback associated with a gradual reduction of a fast inward current in these models helps to maintain the presence of spikes within the active phases of bursts. Therefore, although such a negative feedback is not necessary for burst production, we find that its presence generates a robustness that may be important for function.
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Affiliation(s)
- Sushmita Rose John
- Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, 15260, PA, USA
| | - Bernd Krauskopf
- Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand
| | - Hinke M Osinga
- Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand.
| | - Jonathan E Rubin
- Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, 15260, PA, USA.
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Pedersen MG, Brøns M, Sørensen MP. Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking. CHAOS (WOODBURY, N.Y.) 2022; 32:013121. [PMID: 35105132 DOI: 10.1063/5.0072497] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2021] [Accepted: 01/03/2022] [Indexed: 06/14/2023]
Abstract
Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.
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Affiliation(s)
- Morten Gram Pedersen
- Department of Information Engineering and Department of Mathematics "Tullio-Levi Civita," University of Padova, I-35131 Padova, Italy
| | - Morten Brøns
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
| | - Mads Peter Sørensen
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
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Fazli M, Bertram R. Network Properties of Electrically Coupled Bursting Pituitary Cells. Front Endocrinol (Lausanne) 2022; 13:936160. [PMID: 35872987 PMCID: PMC9299381 DOI: 10.3389/fendo.2022.936160] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 05/04/2022] [Accepted: 06/01/2022] [Indexed: 12/03/2022] Open
Abstract
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.
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Affiliation(s)
- Mehran Fazli
- Department of Mathematics, Florida State University, Tallahassee, FL, United States
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, FL, United States
- Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, United States
- *Correspondence: Richard Bertram,
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Duncan PJ, Fazli M, Romanò N, Le Tissier P, Bertram R, Shipston MJ. Chronic stress facilitates bursting electrical activity in pituitary corticotrophs. J Physiol 2021; 600:313-332. [PMID: 34855218 DOI: 10.1113/jp282367] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2021] [Accepted: 11/18/2021] [Indexed: 11/08/2022] Open
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 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.
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Affiliation(s)
- Peter J Duncan
- Centre for Discovery Brain Sciences, Edinburgh Medical School: Biomedical Sciences, University of Edinburgh, Edinburgh, UK
| | - Mehran Fazli
- Department of Mathematics, Florida State University, Tallahassee, FL, USA
| | - Nicola Romanò
- Centre for Discovery Brain Sciences, Edinburgh Medical School: Biomedical Sciences, University of Edinburgh, Edinburgh, UK
| | - Paul Le Tissier
- Centre for Discovery Brain Sciences, Edinburgh Medical School: Biomedical Sciences, University of Edinburgh, Edinburgh, UK
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, FL, USA.,Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, USA
| | - Michael J Shipston
- Centre for Discovery Brain Sciences, Edinburgh Medical School: Biomedical Sciences, University of Edinburgh, Edinburgh, UK
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Dynamics of ramping bursts in a respiratory neuron model. J Comput Neurosci 2021; 50:161-180. [PMID: 34704174 DOI: 10.1007/s10827-021-00800-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2021] [Revised: 09/24/2021] [Accepted: 09/29/2021] [Indexed: 10/20/2022]
Abstract
Intensive computational and theoretical work has led to the development of multiple mathematical models for bursting in respiratory neurons in the pre-Bötzinger Complex (pre-BötC) of the mammalian brainstem. Nonetheless, these previous models have not captured the pre-inspiratory ramping aspects of these neurons' activity patterns, in which relatively slow tonic spiking gradually progresses to faster spiking and a full-blown burst, with a corresponding gradual development of an underlying plateau potential. In this work, we show that the incorporation of the dynamics of the extracellular potassium ion concentration into an existing model for pre-BötC neuron bursting, along with some parameter adjustments, suffices to induce this ramping behavior. Using fast-slow decomposition, we show that this activity can be considered as a form of parabolic bursting, but with burst termination at a homoclinic bifurcation rather than as a SNIC bifurcation. We also investigate the parameter-dependence of these solutions and show that the proposed model yields a greater dynamic range of burst frequencies, durations, and duty cycles than those produced by other models in the literature.
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Abstract
Excitable biological cells, such as cardiac muscle cells, can exhibit complex patterns of oscillations such as spiking and bursting. Moreover, it is well known that an enhancement in calcium currents may yield certain kind of cardiac arrhythmia, so-called early afterdepolarisations (EADs). The presence of EADs strongly correlates with the onset of dangerous cardiac arrhythmia. In this paper we study mathematically and numerically the dynamics of a cardiac muscle cell with respect to the calcium current by investigating a simplistic system of differential equations. For the study of this phenomena, we use bifurcation theory, numerical bifurcation analysis, geometric singular perturbation theory and computational methods to investigate a nonlinear multiple time scales system. It will turn out that EADs related to an enhanced calcium current are canard–induced and that we have to combine these theories to derive a better understanding of the dynamics behind EADs. Moreover, a suitable time scale separation argument determines the important and sensitive system parameters which are related to the occurrence of EADs.
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12
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Fletcher PA, Sherman A, Stojilkovic SS. Common and diverse elements of ion channels and receptors underlying electrical activity in endocrine pituitary cells. Mol Cell Endocrinol 2018; 463:23-36. [PMID: 28652171 PMCID: PMC5742314 DOI: 10.1016/j.mce.2017.06.022] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 04/24/2017] [Revised: 06/20/2017] [Accepted: 06/22/2017] [Indexed: 12/30/2022]
Abstract
The pituitary gland contains six types of endocrine cells defined by hormones they secrete: corticotrophs, melanotrophs, gonadotrophs, thyrotrophs, somatotrophs, and lactotrophs. All these cell types are electrically excitable, and voltage-gated calcium influx is the major trigger for their hormone secretion. Along with hormone intracellular content, G-protein-coupled receptor and ion channel expression can also be considered as defining cell type identity. While many aspects of the developmental and activity dependent regulation of hormone and G-protein-coupled receptor expression have been elucidated, much less is known about the regulation of the ion channels needed for excitation-secretion coupling in these cells. We compare the spontaneous and receptor-controlled patterns of electrical signaling among endocrine pituitary cell types, including insights gained from mathematical modeling. We argue that a common set of ionic currents unites these cells, while differential expression of another subset of ionic currents could underlie cell type-specific patterns. We demonstrate these ideas using a generic mathematical model, showing that it reproduces many observed features of pituitary electrical signaling. Mapping these observations to the developmental lineage suggests possible modes of regulation that may give rise to mature pituitary cell types.
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Affiliation(s)
- Patrick A Fletcher
- Laboratory of Biological Modeling, National Institute of Diabetes, Digestive and Kidney Diseases, NIH, Bethesda, MD, USA.
| | - Arthur Sherman
- Laboratory of Biological Modeling, National Institute of Diabetes, Digestive and Kidney Diseases, NIH, Bethesda, MD, USA
| | - Stanko S Stojilkovic
- Section on Cellular Signaling, Eunice Kennedy Shriver National Institute of Child Health and Human Development, NIH, Bethesda, MD, USA
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13
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Shipston MJ. Control of anterior pituitary cell excitability by calcium-activated potassium channels. Mol Cell Endocrinol 2018; 463:37-48. [PMID: 28596131 DOI: 10.1016/j.mce.2017.06.003] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 04/14/2017] [Revised: 06/02/2017] [Accepted: 06/02/2017] [Indexed: 01/01/2023]
Abstract
In anterior pituitary endocrine cells, large (BK), small (SK) and intermediate (IK) conductance calcium activated potassium channels are key determinants in shaping cellular excitability in a cell type- and context-specific manner. Indeed, these channels are targeted by multiple signaling pathways that stimulate or inhibit cellular excitability. BK channels can, paradoxically, both promote electrical bursting as well as terminate bursting and spiking dependent upon intrinsic BK channel properties and proximity to voltage gated calcium channels in somatotrophs, lactotrophs and corticotrophs. In contrast, SK channels are predominantly activated by calcium released from intracellular IP3-sensitive calcium stores and mediate membrane hyperpolarization in cells including gonadotrophs and corticotrophs. IK channels are predominantly expressed in corticotrophs where they limit membrane excitability. A major challenge for the future is to determine the cell-type specific molecular composition of calcium-activated potassium channels and how they control anterior pituitary hormone secretion as well as other calcium-dependent processes.
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Affiliation(s)
- Michael J Shipston
- Centre for Integrative Physiology, Edinburgh Medical School: Biomedical Sciences, University of Edinburgh, Edinburgh, EH89XD, UK.
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14
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Bilinsky LM, Baer SM. Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects. Bull Math Biol 2017; 80:130-150. [PMID: 29150760 DOI: 10.1007/s11538-017-0366-2] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2017] [Accepted: 11/06/2017] [Indexed: 11/27/2022]
Abstract
It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction-diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction-diffusion system is Baer and Rinzel's continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon "complete accommodation."
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Affiliation(s)
- L M Bilinsky
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA.
| | - S M Baer
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA
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15
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V-Ghaffari B, Kouhnavard M, Elbasiouny SM. Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions. PLoS One 2017; 12:e0178244. [PMID: 28591171 PMCID: PMC5462370 DOI: 10.1371/journal.pone.0178244] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2016] [Accepted: 05/10/2017] [Indexed: 11/19/2022] Open
Abstract
Subthreshold oscillations in combination with large-amplitude oscillations generate mixed-mode oscillations (MMOs), which mediate various spatial and temporal cognition and memory processes and behavioral motor tasks. Although many studies have shown that canard theory is a reliable method to investigate the properties underlying the MMOs phenomena, the relationship between the results obtained by applying canard theory and conductance-based models of neurons and their electrophysiological mechanisms are still not well understood. The goal of this study was to apply canard theory to the conductance-based model of pyramidal neurons in layer V of the Entorhinal Cortex to investigate the properties of MMOs under antiepileptic drug conditions (i.e., when persistent sodium current is inhibited). We investigated not only the mathematical properties of MMOs in these neurons, but also the electrophysiological mechanisms that shape spike clustering. Our results show that pyramidal neurons can display two types of MMOs and the magnitude of the slow potassium current determines whether MMOs of type I or type II would emerge. Our results also indicate that slow potassium currents with large time constant have significant impact on generating the MMOs, as opposed to fast inward currents. Our results provide complete characterization of the subthreshold activities in MMOs in pyramidal neurons and provide explanation to experimental studies that showed MMOs of type I or type II in pyramidal neurons under antiepileptic drug conditions.
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Affiliation(s)
- Babak V-Ghaffari
- Department of Neuroscience, Cell Biology and Physiology, Boonshoft School of Medicine and College of Science & Mathematics, Wright State University, Dayton, Ohio, United States of America
- * E-mail: (SME); (BV)
| | - M. Kouhnavard
- Malaysia-Japan Int. Inst. of Tech, University Technology Malaysia, Kuala Lumpur, Malaysia
| | - Sherif M. Elbasiouny
- Department of Neuroscience, Cell Biology and Physiology, Boonshoft School of Medicine and College of Science & Mathematics, Wright State University, Dayton, Ohio, United States of America
- Department of Biomedical, Industrial and Human Factors Engineering, College of Engineering & Computer Science, Wright State University, Dayton, Ohio, United States of America
- * E-mail: (SME); (BV)
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Duncan PJ, Tabak J, Ruth P, Bertram R, Shipston MJ. Glucocorticoids Inhibit CRH/AVP-Evoked Bursting Activity of Male Murine Anterior Pituitary Corticotrophs. Endocrinology 2016; 157:3108-21. [PMID: 27254001 PMCID: PMC4967125 DOI: 10.1210/en.2016-1115] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
Abstract
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.
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Affiliation(s)
- Peter J Duncan
- Centre for Integrative Physiology (P.J.D., M.J.S.), College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh EH8 9XD, United Kingdom; Biomedical Neuroscience Research Group (J.T.), University of Exeter Medical School, Exeter EX4 4PL, United Kingdom; Division of Pharmacology, Toxicology, and Clinical Pharmacy (P.R.), Institute for Pharmacy, University of Tübingen, D-72076 Tübingen, Germany; and Department of Mathematics and Programs in Neuroscience and Molecular Biophysics (R.B.), Florida State University, Tallahassee, Florida 32306
| | - Joël Tabak
- Centre for Integrative Physiology (P.J.D., M.J.S.), College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh EH8 9XD, United Kingdom; Biomedical Neuroscience Research Group (J.T.), University of Exeter Medical School, Exeter EX4 4PL, United Kingdom; Division of Pharmacology, Toxicology, and Clinical Pharmacy (P.R.), Institute for Pharmacy, University of Tübingen, D-72076 Tübingen, Germany; and Department of Mathematics and Programs in Neuroscience and Molecular Biophysics (R.B.), Florida State University, Tallahassee, Florida 32306
| | - Peter Ruth
- Centre for Integrative Physiology (P.J.D., M.J.S.), College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh EH8 9XD, United Kingdom; Biomedical Neuroscience Research Group (J.T.), University of Exeter Medical School, Exeter EX4 4PL, United Kingdom; Division of Pharmacology, Toxicology, and Clinical Pharmacy (P.R.), Institute for Pharmacy, University of Tübingen, D-72076 Tübingen, Germany; and Department of Mathematics and Programs in Neuroscience and Molecular Biophysics (R.B.), Florida State University, Tallahassee, Florida 32306
| | - Richard Bertram
- Centre for Integrative Physiology (P.J.D., M.J.S.), College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh EH8 9XD, United Kingdom; Biomedical Neuroscience Research Group (J.T.), University of Exeter Medical School, Exeter EX4 4PL, United Kingdom; Division of Pharmacology, Toxicology, and Clinical Pharmacy (P.R.), Institute for Pharmacy, University of Tübingen, D-72076 Tübingen, Germany; and Department of Mathematics and Programs in Neuroscience and Molecular Biophysics (R.B.), Florida State University, Tallahassee, Florida 32306
| | - Michael J Shipston
- Centre for Integrative Physiology (P.J.D., M.J.S.), College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh EH8 9XD, United Kingdom; Biomedical Neuroscience Research Group (J.T.), University of Exeter Medical School, Exeter EX4 4PL, United Kingdom; Division of Pharmacology, Toxicology, and Clinical Pharmacy (P.R.), Institute for Pharmacy, University of Tübingen, D-72076 Tübingen, Germany; and Department of Mathematics and Programs in Neuroscience and Molecular Biophysics (R.B.), Florida State University, Tallahassee, Florida 32306
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17
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Bertram R, Rubin JE. Multi-timescale systems and fast-slow analysis. Math Biosci 2016; 287:105-121. [PMID: 27424950 DOI: 10.1016/j.mbs.2016.07.003] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2016] [Revised: 07/01/2016] [Accepted: 07/10/2016] [Indexed: 11/28/2022]
Abstract
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.
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Affiliation(s)
- Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University, Florida State University, Tallahassee, FL, United States.
| | - Jonathan E Rubin
- Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, United States
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Fletcher P, Bertram R, Tabak J. From global to local: exploring the relationship between parameters and behaviors in models of electrical excitability. J Comput Neurosci 2016; 40:331-45. [PMID: 27033230 DOI: 10.1007/s10827-016-0600-1] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2015] [Revised: 03/02/2016] [Accepted: 03/07/2016] [Indexed: 01/25/2023]
Abstract
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.
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Affiliation(s)
- Patrick Fletcher
- Currently at the Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, 20892, USA
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.
| | - Joel Tabak
- Currently at the University of Exeter Medical School, Biomedical Neuroscience Research Group, EX4 4PS, Exeter, UK
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Duncan PJ, Shipston MJ. BK Channels and the Control of the Pituitary. INTERNATIONAL REVIEW OF NEUROBIOLOGY 2016; 128:343-68. [PMID: 27238268 DOI: 10.1016/bs.irn.2016.03.004] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/21/2023]
Abstract
The pituitary gland provides the important link between the nervous system and the endocrine system and regulates a diverse range of physiological functions. The pituitary is connected to the hypothalamus by the pituitary stalk and is comprised primarily of two lobes. The anterior lobe consists of five hormone-secreting cell types which are electrically excitable and display single-spike action potentials as well as complex bursting patterns. Bursting is of particular interest as it raises intracellular calcium to a greater extent than spiking and is believed to underlie secretagogue-induced hormone secretion. BK channels have been identified as a key regulator of bursting in anterior pituitary cells. Experimental data and mathematical modeling have demonstrated that BK activation during the upstroke of an action potential results in a prolonged depolarization and an increase in intracellular calcium. In contrast, the posterior lobe is primarily composed of axonal projections of magnocellular neurosecretory cells which extend from the supraoptic and paraventricular nuclei of the hypothalamus. In these neuroendocrine cells, BK channel activation results in a decrease in excitability and hormone secretion. The opposite effect of BK channels in the anterior and posterior pituitary highlights the diverse role of BK channels in regulating the activity of excitable cells. Further studies of pituitary cell excitability and the specific role of BK channels would lead to a greater understanding of how pituitary cell excitability is regulated by both hypothalamic secretagogues and negative feedback loops, and could ultimately lead to novel treatments to pituitary-related disorders.
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Affiliation(s)
- P J Duncan
- Centre for Integrative Physiology, College of Medicine & Veterinary Medicine, University of Edinburgh, Edinburgh, United Kingdom.
| | - M J Shipston
- Centre for Integrative Physiology, College of Medicine & Veterinary Medicine, University of Edinburgh, Edinburgh, United Kingdom
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21
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Duncan PJ, Şengül S, Tabak J, Ruth P, Bertram R, Shipston MJ. Large conductance Ca²⁺-activated K⁺ (BK) channels promote secretagogue-induced transition from spiking to bursting in murine anterior pituitary corticotrophs. J Physiol 2015; 593:1197-211. [PMID: 25615909 PMCID: PMC4358680 DOI: 10.1113/jphysiol.2015.284471] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2014] [Accepted: 12/16/2014] [Indexed: 12/14/2022] Open
Abstract
Anterior pituitary corticotroph cells are a central component of the hypothalamic-pituitary-adrenal (HPA) axis essential for the neuroendocrine response to stress. Corticotrophs are excitable cells that receive input from two hypothalamic secretagogues, corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) to control the release of adrenocorticotrophic hormone (ACTH). Although corticotrophs are spontaneously active and increase in excitability in response to CRH and AVP the patterns of electrical excitability and underlying ionic conductances are poorly understood. In this study, we have used electrophysiological, pharmacological and genetic approaches coupled with mathematical modelling to investigate whether CRH and AVP promote distinct patterns of electrical excitability and to interrogate the role of large conductance calcium- and voltage-activated potassium (BK) channels in spontaneous and secretagogue-induced activity. We reveal that BK channels do not play a significant role in the generation of spontaneous activity but are critical for the transition to bursting in response to CRH. In contrast, AVP promotes an increase in single spike frequency, a mechanism independent of BK channels but dependent on background non-selective conductances. Co-stimulation with CRH and AVP results in complex patterns of excitability including increases in both single spike frequency and bursting. The ability of corticotroph excitability to be differentially regulated by hypothalamic secretagogues provides a mechanism for differential control of corticotroph excitability in response to different stressors.
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Affiliation(s)
- Peter J Duncan
- Centre for Integrative Physiology, College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh, EH8 9XD, UK
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