S. E. Geneser, R. M. Kirby, D. Xiu, and F. B. Sachse. Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents. In J Theor Biol, vol. 245(4) , pp. 627-637, 2007
Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.
Based on the Weiner-Hermite polynomial chaos expansion, the stochastic Galerkin method efficiently computes nu- merical solutions for stochastic systems. Unlike such tech- niques as sensitivity analysis, perturbation methods, and second moment-analysis, this method is applicable to a large number of systems while requiring less computational effort than sampling based stochastic methods like Monte Carlo. We utilize the stochastic Galerkin method to assess the impact of stochastic rate coefficients on the predictions of Markovian cardiac ion channel models
Mathematical models of biophysical phenomena have proven useful in the reconstruction of experimental data and prediction of biological behavior. By quantifying the sensitivity of a model to certain parameters, one can place an appropriate amount of emphasis in the accuracy with which those parameters are determined. In addition, investigation of stochastic parameters can lead to a greater understanding of the behavior captured by the model. This can lead to possible model reductions, or point out shortcomings to be addressed. We present polynomial chaos as a computationally efficient alternative to Monte Carlo for assessing the impact of stochastically distributed parameters on the model predictions of several cardiac electrophysiological models.
Electrical activity in biological media can be described in a mathematical way, which is applicable to computer-based simulation. Biophysically mathematical descriptions provide important insights into the electrical and electrophysiological properties of cells, tissues, and organs. Examples of these descriptions are Maxwell's and Poisson's equations for electromagnetic and electric fields. Commonly, numerical techniques are applied to calculate electrical fields, e.g. the finite element method. Finite elements can be classified on the order of the underlying Interpolation. High-order finite elements provide enhanced geometric flexibility and can increase the accuracy of a solution. The aim of this work is the design of a framework for describing and solving high-order finite elements in the SCIRun/BioPSE software system, which allows geometric modeling, simulation, and visualization for solving bioelectric field problems. Currently, only low-order elements are supported. Our design for high-order elements concerns interpolation of geometry and physical fields. The design is illustrated by an implementation of one-dimensional elements with cubic interpolation of geometry and field variables.