scholarly journals Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

2020 ◽  
Vol 32 (10) ◽  
pp. 1775-1835
Author(s):  
Shusen Pu ◽  
Peter J. Thomas
Keyword(s):  
Diffusion Tensor ◽  
Planck Equation ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Langevin Equations ◽  
Time Step ◽  
State Transition Graph ◽  
State Dependent ◽  
System 2 ◽  
Matrix Formula

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor [Formula: see text] and suggested a Langevin formulation with noise coefficient matrix [Formula: see text] such that SS[Formula: see text]. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 [Formula: see text] 28 noise coefficient matrix [Formula: see text]. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix [Formula: see text] that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the [Formula: see text] and [Formula: see text] metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies [Formula: see text] with minimal loss of accuracy under both voltage- and current-clamp conditions.

A Kind of Preconditioned IGMRES(m) Algorithm

2012 ◽  
Vol 482-484 ◽  
pp. 413-416
Author(s):  
Chun Xiao Yu
Keyword(s):  
Krylov Subspace ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Residual Method ◽  
Minimal Residual Method ◽  
Algorithm Convergence ◽  
Minimal Residual

Fundamental theories are studied for an Incomplete Generalized Minimal Residual Method(IGMRES(m)) in Krylov subspace. An algebraic equations generated from the IGMRES(m) algorithm is presented. The relationships are deeply researched for the algorithm convergence and the coefficient matrix of the equations. A kind of preconditioned method is proposed to improve the convergence of the IGMRES(m) algorithm. It is proved that the best convergence can be obtained through appropriate matrix decomposition.

BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES

2013 ◽  
Vol 55 (2) ◽  
pp. 93-108 ◽  
Author(s):  
JACK D. HYWOOD ◽  
KERRY A. LANDMAN
Keyword(s):  
Statistical Physics ◽  
Planck Equation ◽  
Step Length ◽  
Physical Object ◽  
Cell Diameter ◽  
Agent Based Models ◽  
Time Step ◽  
Partial Differential ◽  
Agent Based ◽  
Random Walkers

AbstractThere is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

scholarly journals Calcium homeostasis in a local/global whole cell model of permeabilized ventricular myocytes with a Langevin description of stochastic calcium release

2015 ◽  
Vol 308 (5) ◽  
pp. H510-H523 ◽  
Author(s):  
Xiao Wang ◽  
Seth H. Weinberg ◽  
Yan Hao ◽  
Eric A. Sobie ◽  
Gregory D. Smith
Keyword(s):  
Calcium Release ◽  
Ventricular Myocytes ◽  
Cell Model ◽  
Ryanodine Receptors ◽  
Planck Equation ◽  
Linear Increase ◽  
Langevin Equations ◽  
Cell Models ◽  
Whole Cell ◽  
Alternative Approach

Population density approaches to modeling local control of Ca2+-induced Ca2+ release in cardiac myocytes can be used to construct minimal whole cell models that accurately represent heterogeneous local Ca2+ signals. Unfortunately, the computational complexity of such “local/global” whole cell models scales with the number of Ca2+ release unit (CaRU) states, which is a rapidly increasing function of the number of ryanodine receptors (RyRs) per CaRU. Here we present an alternative approach based on a Langevin description of the collective gating of RyRs coupled by local Ca2+ concentration ([Ca2+]). The computational efficiency of this approach no longer depends on the number of RyRs per CaRU. When the RyR model is minimal, Langevin equations may be replaced by a single Fokker-Planck equation, yielding an extremely compact and efficient local/global whole cell model that reproduces and helps interpret recent experiments that investigate Ca2+ homeostasis in permeabilized ventricular myocytes. Our calculations show that elevated myoplasmic [Ca2+] promotes elevated network sarcoplasmic reticulum (SR) [Ca2+] via SR Ca2+-ATPase-mediated Ca2+ uptake. However, elevated myoplasmic [Ca2+] may also activate RyRs and promote stochastic SR Ca2+ release, which can in turn decrease SR [Ca2+]. Increasing myoplasmic [Ca2+] results in an exponential increase in spark-mediated release and a linear increase in nonspark-mediated release, consistent with recent experiments. The model exhibits two steady-state release fluxes for the same network SR [Ca2+] depending on whether myoplasmic [Ca2+] is low or high. In the later case, spontaneous release decreases SR [Ca2+] in a manner that maintains robust Ca2+ sparks.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

scholarly journals THE STOCHASTIC QUANTIZATION METHOD IN PHASE SPACE AND A NEW GAUGE FIXING PROCEDURE

1994 ◽  
Vol 09 (30) ◽  
pp. 2803-2815
Author(s):  
RIUJI MOCHIZUKI
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Equilibrium Solution ◽  
Planck Equation ◽  
Langevin Equations ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Canonical Variables ◽  
Gauge Fixing ◽  
Integral Distribution

We study the stochastic quantization of the system with first class constraints in phase space. Though the Langevin equations of the canonical variables are defined without ordinary gauge fixing procedure, gauge fixing conditions are automatically selected and introduced by imposing stochastic consistency conditions upon the first class constraints. Then the equilibrium solution of the Fokker–Planck equation is identical to the corresponding path-integral distribution.

scholarly journals TIMING OPTIMIZATION OF HEAD AND NECK CT ANGIOGRAPHY VIA THE INVERSE PROBLEM ALGORITHM: IN VIVO SURVEY FOR 1001 PATIENTS IN 2020–2021

2021 ◽  
Author(s):  
CHUN-CHIEH LIANG ◽  
LUNG-FA PAN ◽  
MING-HSIANG CHEN ◽  
JIE DENG ◽  
DENG-HO YANG ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Correlation Coefficient ◽  
Ct Angiography ◽  
Empirical Formula ◽  
Coefficient Matrix ◽  
Biological Data ◽  
Survey Results ◽  
Matrix Formula ◽  
Semi Empirical

This study processed the recent in vivo survey results for over a thousand patients and optimized their neck and head CT angiography triggered timing (CTA-TT) via the inverse problem algorithm, which ensured the maximal ratio of both left and right arterial to upper sinuses (LRA/US). These results are instrumental in examining the ischemic stroke syndromes along the neck and head. These 1001 patients were randomly categorized into test surveyed (802 patients) and verification group (199 patients), then a six factors semi-empirical formula was constructed by the STATISTICA program. The six factors were assigned a patient’s biological data and preset of the CTA facility; namely Age, mean arterial pressure (MAP), heart rate (HR), contrast media dose (CMD), Pre (injected pressure of CMD), and body surface area (BSA). Each factor was normalized into dimensionless values and incorporated into the dataset matrix [Formula: see text] to analyze the coefficient matrix [Formula: see text]. The derived semi-empirical formula closely correlated with experimental data, according to the loss function [Formula: see text], correlation coefficient [Formula: see text], and variance of 0.8965. The formula verification for 199 more patients (verification group) yielded a correlation coefficient [Formula: see text]. Thus, it can be used for the CTA-TT estimation of patients without their preliminary tests, avoiding unnecessary irradiation. The estimated LRA/US was [Formula: see text] for the verification group in this study. A simplified three-factor formula, featuring only age, MAP, and BSA, was also proposed.

An Introduction to Quantum Optics and Quantum Fluctuations

2019 ◽  
Author(s):  
Peter W. Milonni
Keyword(s):  
Quantum Optics ◽  
Spontaneous Emission ◽  
Quantum Electrodynamics ◽  
Planck Equation ◽  
Quantum Fluctuations ◽  
Van Der Waals Interactions ◽  
Vacuum Field ◽  
Langevin Equations ◽  
Uncertainty Relations ◽  
Point Energy

This book is an introduction to quantum optics for students who have studied electromagnetism and quantum mechanics at an advanced undergraduate or graduate level. It provides detailed expositions of theory with emphasis on general physical principles. Foundational topics in classical and quantum electrodynamics, including the semiclassical theory of atom-field interactions, the quantization of the electromagnetic field in dispersive and dissipative media, uncertainty relations, and spontaneous emission, are addressed in the first half of the book. The second half begins with a chapter on the Jaynes-Cummings model, dressed states, and some distinctly quantum-mechanical features of atom-field interactions, and includes discussion of entanglement, the no-cloning theorem, von Neumann’s proof concerning hidden variable theories, Bell’s theorem, and tests of Bell inequalities. The last two chapters focus on quantum fluctuations and fluctuation-dissipation relations, beginning with Brownian motion, the Fokker-Planck equation, and classical and quantum Langevin equations. Detailed calculations are presented for the laser linewidth, spontaneous emission noise, photon statistics of linear amplifiers and attenuators, and other phenomena. Van der Waals interactions, Casimir forces, the Lifshitz theory of molecular forces between macroscopic media, and the many-body theory of such forces based on dyadic Green functions are analyzed from the perspective of Langevin noise, vacuum field fluctuations, and zero-point energy. There are numerous historical sidelights throughout the book, and approximately seventy exercises.

Multiloop state-dependent nonlinear time-varying sliding mode control of unmanned small-scale helicopter

2019 ◽  
Vol 234 (3) ◽  
pp. 585-606 ◽  
Author(s):  
Sinan Ozcan ◽  
Metin U Salamci ◽  
Volkan Nalbantoglu
Keyword(s):  
Sliding Mode Control ◽  
Sliding Mode ◽  
Operating Conditions ◽  
Small Scale ◽  
Time Varying ◽  
Parameter Uncertainties ◽  
Time Step ◽  
Mode Control ◽  
State Dependent ◽  
Improved Performance

Time delays, parameter uncertainties, and disturbances are the fundamental problems that hinder the stability and reduce dramatically the tracking performance of dynamical systems. In this paper, a new state-dependent nonlinear time-varying sliding mode control autopilot structure is proposed to cope with these dynamical and environmental complexities for an unmanned helicopter. The presented technique is based on freezing the nonlinear system equations on each time step and designing a controller using the frozen system model at this time step. The proposed method offers an improved performance in the presence of major disturbances and parameter uncertainties by adapting itself to possible dynamical varieties without a need of trimming the system on different operating conditions. Unlike the existing linear cascade autopilot structure, this study also proposes a nonlinear cascade state-dependent coefficient helicopter autopilot structure consisting of four separate nonlinear sub-systems. The proposed method is tested through the real time and PC-based simulations. To show the performance of the proposed robust method, it is also bench-marked against a linear sliding control control in PC-based simulations.

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