Mixed-mode oscillations and bifurcation analysis in a pituitary model

Zhan, Feibiao; Liu, Shenquan; Zhang, Xiaohan; Wang, Jing; Lu, Bo

doi:10.1007/s11071-018-4395-7

Cited by 27 publications

(9 citation statements)

References 38 publications

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“…They introduced the definition of MMOs and showed MMOs in a neuron model as well as the theoretical conclusions of the interaction among canards. Since the relationship between canards and MMOs was first discovered in the neuron model [1,17], the potential connection between the two has been revealed and proven [33][34][35]46,66,122,123]. The bursting mode of neurons is strongly connected with the canards of folded singularities.…”

Section: Introductionmentioning

confidence: 96%

“…The bursting mode of neurons is strongly connected with the canards of folded singularities. For example, the canard of folded nodes can cause MMOs [33,34,123] and the folded saddle is closely related to the parabolic burster and the firing-threshold manifold [39,47]. The torus canard is related to the transition from single spiking to bursting [52,54,55] and so on.…”

Section: Introductionmentioning

confidence: 99%

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Canard Mechanism and Rhythm Dynamics of Neuron Models

et al. 2023

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Canards are a type of transient dynamics that occur in singularly perturbed systems, and they are specific types of solutions with varied dynamic behaviours at the boundary region. This paper introduces the emergence and development of canard phenomena in a neuron model. The singular perturbation system of a general neuron model is investigated, and the link between the transient transition from a neuron model to a canard is summarised. First, the relationship between the folded saddle-type canard and the parabolic burster, as well as the firing-threshold manifold, is established. Moreover, the association between the mixed-mode oscillation and the folded node type is unique. Furthermore, the connection between the mixed-mode oscillation and the limit-cycle canard (singular Hopf bifurcation) is stated. In addition, the link between the torus canard and the transition from tonic spiking to bursting is illustrated. Finally, the specific manifestations of these canard phenomena in the neuron model are demonstrated, such as the singular Hopf bifurcation, the folded-node canard, the torus canard, and the “blue sky catastrophe”. The summary and outlook of this paper point to the realistic possibility of canards, which have not yet been discovered in the neuron model.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Canard Mechanism and Rhythm Dynamics of Neuron Models

et al. 2023

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“…In this class of models, geometrical and mathematical approaches are used to explain temporal dynamics, and the mechanisms generating MMOs are increasingly well understood [8]. Using these mathematical tools, it has been possible to understand and explain MMOs observed in many cellular quantities, e.g., complex patterns of electrical activity in neurons [9][10][11], pituitary cells [12][13][14], human beta cells [15], and cardiomyocytes [16,17], as well as mixed-mode calcium oscillations [18], complex dynamics appearing from cell-to-cell interaction [19][20][21], etc.…”

Section: Introductionmentioning

confidence: 99%

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels

Martin,

Pedersen

2024

PLoS Comput Biol

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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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Section: Introductionmentioning

confidence: 99%

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels

Martin,

Pedersen

2023

Preprint

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Cyclic AMP controls neuronal ion channel activity. For example hyperpolarizationactivated 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 multiple-time scale dynamical system. We show that the MMO mechanism does not rely on the very 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 fast subsystem alone. We show that the 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.Author summaryNeurons use the frequency of electrical signals called action potentials to encode information, and various messenger systems interact with ion channels to control this so-called firing frequency. Recent experimental recordings show that the intracellular messenger cAMP can induce mixed-mode oscillations (MMOs) consisting of small-amplitude, subthreshold oscillations separating action potentials, which lowers the firing frequency greatly. We extend a recent mathematical model of neuronal electrical activity to investigate how MMOs occur from interactions between ion channels regulated by cAMP. Our simulations reproduce a range of experimental results, including cAMP-induced MMOs. We explain the model dynamics using modern geometrical methods that exploit the different timescales in the model. Our analyses show that the slow dynamics of cAMP-regulated HCN and M ion channels is not crucial for creating MMOs, but rather that the cAMP-induced increase in their average activity is important. Our analyses suggest that both HCN and M channels are crucial for MMOs and controlling the firing frequency, which has implications for our understanding of how astrocytes control neuronal information processing. Moreover, our study raises new mathematical questions related to how slow dynamical variables modify MMOs.

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Mixed-mode oscillations and bifurcation analysis in a pituitary model

Cited by 27 publications

References 38 publications

Canard Mechanism and Rhythm Dynamics of Neuron Models

Canard Mechanism and Rhythm Dynamics of Neuron Models

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels

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