The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Hinke M. Osinga - One of the best experts on this subject based on the ideXlab platform.
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Saddle Slow Manifolds and Canard Orbits in R 4
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007 ) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε . The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in R 4 $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model
Journal of mathematical neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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saddle slow manifolds and canard orbits in r 4 mathbb r 4 and application to the full hodgkin Huxley Model
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
Cris R. Hasan - One of the best experts on this subject based on the ideXlab platform.
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Saddle Slow Manifolds and Canard Orbits in R 4
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007 ) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε . The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in R 4 $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model
Journal of mathematical neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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saddle slow manifolds and canard orbits in r 4 mathbb r 4 and application to the full hodgkin Huxley Model
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
Xuejuan Zhang - One of the best experts on this subject based on the ideXlab platform.
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numerical simulations of piecewise deterministic markov processes with an application to the stochastic hodgkin Huxley Model
Journal of Chemical Physics, 2016Co-Authors: Shaojie Ding, Min Qian, Hong Qian, Xuejuan ZhangAbstract:The stochastic Hodgkin-Huxley Model is one of the best-known examples of piecewise deterministic Markov processes (PDMPs), in which the electrical potential across a cell membrane, V(t), is coupled with a mesoscopic Markov jump process representing the stochastic opening and closing of ion channels embedded in the membrane. The rates of the channel kinetics, in turn, are voltage-dependent. Due to this interdependence, an accurate and efficient sampling of the time evolution of the hybrid stochastic systems has been challenging. The current exact simulation methods require solving a voltage-dependent hitting time problem for multiple path-dependent intensity functions with random thresholds. This paper proposes a simulation algorithm that approximates an alternative representation of the exact solution by fitting the log-survival function of the inter-jump dwell time, H(t), with a piecewise linear one. The latter uses interpolation points that are chosen according to the time evolution of the H(t), as the numerical solution to the coupled ordinary differential equations of V(t) and H(t). This computational method can be applied to all PDMPs. Pathwise convergence of the approximated sample trajectories to the exact solution is proven, and error estimates are provided. Comparison with a previous algorithm that is based on piecewise constant approximation is also presented.
Andrea Arnold - One of the best experts on this subject based on the ideXlab platform.
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Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
Applied Sciences, 2020Co-Authors: Kayleigh Campbell, Laura Staugler, Andrea ArnoldAbstract:The classic Hodgkin-Huxley Model is widely used for understanding the electrophysiological dynamics of a single neuron. While applying a low-amplitude constant current to the system results in a single voltage spike, it is possible to produce multiple voltage spikes by applying time-varying currents, which may not be experimentally measurable. The aim of this work is to estimate time-varying applied currents of different deterministic forms given noisy voltage data. In particular, we utilize an augmented ensemble Kalman filter with parameter tracking to estimate four different time-varying applied current parameters and associated Hodgkin-Huxley Model states, along with uncertainty bounds in each case. We test the efficiency of the parameter tracking algorithm in this setting by analyzing the effects of changing the standard deviation of the parameter drift and the frequency of data available on the resulting time-varying applied current estimates and related uncertainty.
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Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
arXiv: Quantitative Methods, 2019Co-Authors: Kayleigh Campbell, Laura Staugler, Andrea ArnoldAbstract:The classic Hodgkin-Huxley Model is widely used for understanding the electrophysiological dynamics of a single neuron. While applying a constant current to the system results in a single voltage spike, it is possible to produce more interesting dynamics by applying time-varying currents, which may not be experimentally measurable. The aim of this work is to estimate time-varying applied currents of different deterministic forms given noisy voltage data. In particular, we utilize an augmented ensemble Kalman filter with parameter tracking to estimate four different deterministic applied currents, analyzing how the Model dynamics change in each case. We test the efficiency of the parameter tracking algorithm in this setting by exploring the effects of changing the standard deviation of the parameter drift and the frequency of data available on the resulting time-varying applied current estimates and related uncertainty.
Bernd Krauskopf - One of the best experts on this subject based on the ideXlab platform.
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Saddle Slow Manifolds and Canard Orbits in R 4
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007 ) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε . The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in R 4 $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model
Journal of mathematical neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.
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saddle slow manifolds and canard orbits in r 4 mathbb r 4 and application to the full hodgkin Huxley Model
The Journal of Mathematical Neuroscience, 2018Co-Authors: Cris R. Hasan, Bernd Krauskopf, Hinke M. OsingaAbstract:Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron Models typically involve observable differences in time scales. The Hodgkin–Huxley Model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this Model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio e. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this Model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley Model in organizing MMOs and determining the firing rates of action potentials. For e sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the Model.