scholarly journals Geometric Analysis of Mixed-mode Oscillations in a Model of Electrical Activity in Human Beta-cells

2021 ◽  
Author(s):  
Simone Battaglin ◽  
Morten Gram Pedersen
Keyword(s):  
Hopf Bifurcation ◽  
Electrical Activity ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Original Model ◽  
Biophysical Model ◽  
Reduced Model ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations

Abstract Human pancreatic beta-cells may exhibit complex mixed-mode oscillatory electrical activity, which underlies insulin secretion. A recent biophysical model of human beta-cell electrophysiology can simulate such bursting behavior, but a mathematical understanding of the model's dynamics is still lacking. Here we exploit time-scale separation to simplify the original model to a simpler three-dimensional model that retains the behavior of the original model and allows us to apply geometric singular perturbation theory to investigate the origin of mixed-mode oscillations. Changing a parameter modeling the maximal conductance of a potassium current, we nd that the reduced model possesses a singular Hopf bifurcation that results in small-amplitude oscillations, which go through a period-doubling sequence and chaos until the birth of a large-scale return mechanism and bursting dynamics. The theory of folded node singularities provide insight into the bursting dynamics further away from the singular Hopf bifurcation and the eventual transition to simple spiking activity. Numerical simulations confirm that the insight obtained from the analysis of the reduced model can be lifted back to the original model.

scholarly journals Geometric analysis of mixed-mode oscillations in a model of electrical activity in human beta-cells

2021 ◽  
Author(s):  
Simone Battaglin ◽  
Morten Gram Pedersen
Keyword(s):  
Hopf Bifurcation ◽  
Electrical Activity ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Original Model ◽  
Biophysical Model ◽  
Reduced Model ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations

AbstractHuman pancreatic beta-cells may exhibit complex mixed-mode oscillatory electrical activity, which underlies insulin secretion. A recent biophysical model of human beta-cell electrophysiology can simulate such bursting behavior, but a mathematical understanding of the model’s dynamics is still lacking. Here we exploit time-scale separation to simplify the original model to a simpler three-dimensional model that retains the behavior of the original model and allows us to apply geometric singular perturbation theory to investigate the origin of mixed-mode oscillations. Changing a parameter modeling the maximal conductance of a potassium current, we find that the reduced model possesses a singular Hopf bifurcation that results in small-amplitude oscillations, which go through a period-doubling sequence and chaos until the birth of a large-scale return mechanism and bursting dynamics. The theory of folded node singularities provide insight into the bursting dynamics further away from the singular Hopf bifurcation and the eventual transition to simple spiking activity. Numerical simulations confirm that the insight obtained from the analysis of the reduced model can be lifted back to the original model.

Symmetric Fold/Super-Hopf Bursting, Chaos and Mixed-Mode Oscillations in Pernarowski Model of Pancreatic Beta-Cells

2016 ◽  
Vol 26 (09) ◽  
pp. 1630022 ◽  
Author(s):  
Haniyeh Fallah
Keyword(s):  
Blood Glucose ◽  
Blood Glucose Level ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Current System ◽  
Period Doubling ◽  
Slow Manifold ◽  
Mixed Mode Oscillations ◽  
Route To Chaos

Pancreatic beta-cells produce insulin to regularize the blood glucose level. Bursting is important in beta cells due to its relation to the release of insulin. Pernarowski model is a simple polynomial model of beta-cell activities indicating bursting oscillations in these cells. This paper presents bursting behaviors of symmetric type in this model. In addition, it is shown that the current system exhibits the phenomenon of period doubling cascades of canards which is a route to chaos. Canards are also observed symmetrically near folds of slow manifold which results in a chaotic transition between [Formula: see text] and [Formula: see text] spikes symmetric bursting. Furthermore, mixed-mode oscillations (MMOs) and combination of symmetric bursting together with MMOs are illustrated during the transition between symmetric bursting and continuous spiking.

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Circuit World ◽  
2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yue Yu ◽  
Cong Zhang ◽  
Zhenyu Chen ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Large Amplitude ◽  
Mixed Mode ◽  
Periodic Perturbation ◽  
Van Der Pol ◽  
Content Type ◽  
Stable Focus ◽  
Mixed Mode Oscillations ◽  
Transition Mechanism

Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

scholarly journals Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model

2015 ◽  
Vol 22 (2) ◽  
pp. 71-79 ◽  
Author(s):  
Morten Brøns ◽  
Mathieu Desroches ◽  
Martin Krupa
Keyword(s):  
Hopf Bifurcation ◽  
Mixed Mode ◽  
Forest Pest ◽  
Mixed Mode Oscillations

Mixed-mode oscillations for slow-fast perturbed systems

2021 ◽  
Author(s):  
Yaru Liu ◽  
Shenquan Liu ◽  
Bo Lu ◽  
Jürgen Kurths
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Mixed Mode ◽  
Dynamic Properties ◽  
Differential Algebraic Equations ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations ◽  
Average Analysis ◽  
Perturbed Systems

Abstract This article concerns the dynamics of mixed-mode oscillations (MMOs) emerging from the calcium-based inner hair cells (IHCs) model in the auditory cortex. The paper captures the MMOs generation mechanism based on the geometric singular perturbation theory (GSPT) after exploiting the average analysis for reducing the full model. Our analysis also finds that the critical manifold and folded surface are central to the mechanism of the existence of MMOs at the folded saddle for the perturbed system. The system parameters, such like the maximal calcium channels conductance, controls the firing patterns, and many new oscillations occur for the IHCs model. Tentatively, we conduct dynamic analysis combined with dynamic method based on GSPT by giving slow-fast analysis for the singular perturbed models and bifurcation analysis. In particular, we explore the two-slow-two-fast and three-slow-one-fast IHCs perturbed systems with layer and reduced problems so that differential-algebraic equations are obtained. This paper reveals the underlying dynamic properties of perturbed systems under singular perturbation theory.

Contributions of modeling to understanding stimulus-secretion coupling in pancreatic beta-cells

1996 ◽  
Vol 271 (2) ◽  
pp. E362-E372 ◽  
Author(s):  
A. Sherman
Keyword(s):  
Electrical Activity ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Electrical Coupling ◽  
Calcium Handling ◽  
Isolated Cells ◽  
Ionic Fluxes ◽  
Pancreatic Islets Of Langerhans ◽  
Stimulus Secretion Coupling ◽  
Model Independent

Mechanisms of ionic control of insulin secretion in beta-cells of the pancreatic islets of Langerhans are reviewed. The focus is on aspects that have been treated by mathematical models, especially those related to bursting electrical activity. The study of these mechanisms is difficult because of the need to consider ionic fluxes, calcium handling, metabolism, and electrical coupling with other cells in the islet. The data come either from islets, where experimental maneuvers tend to have multiple effects, or from isolated cells, which have degraded electrical activity and secretory sensitivity. Modeling aids in the process by integrating data on individual components such as channels and calcium handling and testing hypotheses for coherence and quantitative plausibility. The study of a variety of models has led to some general mathematical results that have yielded qualitative model-independent insights.

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