Resetting Behavior in a Model of Bursting in Secretory Pituitary Cells: Distinguishing Plateaus from Pseudo-Plateaus

Stern, Julie; Osinga, Hinke M.; LeBeau, Andrew; Sherman, Arthur

doi:10.1007/s11538-007-9241-x

Cited by 45 publications

(52 citation statements)

References 21 publications

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“…Note that there are other Hopf bifurcations on Z, but these have been found (numerically) to occur on the repelling branch Z r of Z, and so we will concentrate only on those Hopfs that are O(ε) close to the fold surface L. The criticality of the fast subsystem Hopf typically differentiates between plateau and pseudoplateau bursting [47,39,56,54]. In our model system, Z H has always been found (numerically) to be subcritical so that the associated bursts are of pseudo-plateau type.…”

Section: Geometric Singular Perturbation Analysismentioning

confidence: 99%

“…Historically, the analysis of bursting in slow/fast systems was pioneered by [40], and several treatments of pseudoplateau bursting followed suit [38,39,47,56]. In this traditional 3-fast/1-slow approach, the small oscillations are born from a slow passage through a dynamic Hopf bifurcation [36,37,3] and the MMOs are hysteresis loops that alternately jump at a fold and a subcritical Hopf (fold/sub-Hopf bursts) [41,24].…”

Section: Delayed Hopf Bifurcation and Tourbillonmentioning

confidence: 99%

“…Characteristics of the burst pattern such as frequency and duration determine how much calcium enters the cell, which in turn determines the level of hormone secretion [59]. One particular type of bursting that has been the focus of recent modeling efforts is pseudo-plateau bursting, which features small amplitude oscillations or spikes in the active phase superimposed on relaxation type oscillations [51,47,39,56]. Pseudo-plateau bursts are distinguished from plateau bursts, which feature large amplitude fast spiking in the active phase [40,4,32,57].…”

Section: Introductionmentioning

confidence: 99%

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Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Vo¹,

Bertram²,

Wechselberger³

2013

SIAM J. Appl. Dyn. Syst.

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Abstract. Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found in neural bursting models. In a recent model for the electrical activity and calcium signaling in a pituitary lactotroph, two types of pseudo-plateau bursts were discovered: one in which the calcium drives the bursts and another in which the calcium simply follows them. Multiple methods from dynamical systems theory have been used to understand the bursting. The classic 2-timescale approach treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems. A more novel and successful 2-timescale approach divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics. Both methods can be effective analytic tools, but there has been little justification for one approach over the other. In this work, we use the lactotroph model to demonstrate that the two analysis techniques are different unfoldings of a 3-timescale system. We show that elementary applications of geometric singular perturbation theory in the 2-timescale and 3-timescale methods provide us with substantial predictive power. We use that predictive power to explain the transient and long-term dynamics of the pituitary lactotroph model.

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Section: Geometric Singular Perturbation Analysismentioning

confidence: 99%

Section: Delayed Hopf Bifurcation and Tourbillonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Vo¹,

Bertram²,

Wechselberger³

2013

SIAM J. Appl. Dyn. Syst.

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“…Both mathematical models are described by a system of nonlinear Ordinary Differential Equations (ODEs). In addition, bursting dynamics has been studied in a mathematical model (Stern model) that was developed by modification of a pituitary corticotroph cell model [4]. An important issue in the analysis of mathematical models that show bursting dynamics is clarification of the dependence of the dynamic states on a system parameter such as long-lasting external stimulation.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Pituitary Cell Model: Dependence on Long-Lasting External Stimulation and Potassium Conductance Kinetics

Shirahata¹

2016

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Stern et al. have developed a mathematical model describing pseudo-plateau bursting of pituitary cells. This model is formulated based on the Hodgkin-Huxley scheme and described by a system of nonlinear ordinary differential equations. In the present study, computer simulation analysis of this model was performed to evaluate the correlation between the dynamic states of the model and two system parameters: long-lasting external stimulation (Iapp) and the time constant of delayed-rectifier potassium conductance activation (τn). Computer simulation results revealed that the model showed four different dynamic states: a hyperpolarized steady state, a depolarized steady state, a repetitive spiking state, and a bursting state. An increase in Iapp changed the dynamic states from the hyperpolarized steady state to bursting state to depolarized steady state when τn was fixed at smaller values, whereas it changed the dynamic states from the hyperpolarized steady state to bursting state to repetitive spiking state when τn was fixed at larger values. An increase in τn 1) did not change the dynamic states when Iapp was fixed at a very small value, 2) changed the dynamic states from the depolarized steady state to repetitive spiking state when Iapp was fixed at a very large value, and 3) changed the dynamic states from the depolarized steady state to bursting state to repetitive spiking state when Iapp was fixed at an intermediate value.

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“…This approach has been very successful for understanding bursting in pancreatic islets 25 and neurons. [2][3][4]21 It has also been useful in understanding various aspects of bursting in pituitary cells such as resetting properties, 26 how fast subsystem manifolds affect burst termination, 23 and how parameter changes convert the system from one burst type to the other. 24 In the alternate one-fast/two-slow analysis, one associates a variable with an intermediate time scale with the slow subsystem rather than the fast subsystem.…”

Section: Introductionmentioning

confidence: 99%

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

Teka

Tabak

Bertram

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the twofast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow. Bursting electrical oscillations are common in nerve cells and endocrine cells. These consist of episodes of electrical activity followed by periods of quiescence. Since each electrical impulse is itself an oscillation, this is an example of a multi-time scale oscillation, which has been analyzed successfully using a decomposition of the system of equations into fast and slow subsystems. In this article, we compare two alternate fast/slow analysis techniques for the study of the type of bursting oscillations that typically occur in pituitary lactotrophs and somatotrophs. We show the relationships between the key elements of both types of analysis.

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Resetting Behavior in a Model of Bursting in Secretory Pituitary Cells: Distinguishing Plateaus from Pseudo-Plateaus

Cited by 45 publications

References 21 publications

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Dynamics of a Pituitary Cell Model: Dependence on Long-Lasting External Stimulation and Potassium Conductance Kinetics

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

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