Pamela Burrage

Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable–step–size implementations based on various types of control.

Order Conditions of Stochastic Runge--Kutta Methods by B -Series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Numerical solutions of stochastic differential equations — implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

Determination of Somatic and Cancer Stem Cell Self-Renewing Symmetric Division Rate Using Sphere Assays

Representing a renewable source for cell replacement, neural stem cells have received substantial attention in recent years. The neurosphere assay represents a method to detect the presence of neural stem cells, however owing to a deficiency of specific and definitive markers to identify them, their quantification and the rate they expand is still indefinite. Here we propose a mathematical interpretation of the neurosphere assay allowing actual measurement of neural stem cell symmetric division frequency. The algorithm of the modeling demonstrates a direct correlation between the overall cell fold expansion over time measured in the sphere assay and the rate stem cells expand via symmetric division. The model offers a methodology to evaluate specifically the effect of diseases and treatments on neural stem cell activity and function. Not only providing new insights in the evaluation of the kinetic features of neural stem cells, our modeling further contemplates cancer biology as cancer stem-like cells have been suggested to maintain tumor growth as somatic stem cells maintain tissue homeostasis. Indeed, tumor stem cells resistance to therapy makes these cells a necessary target for effective treatment. The neurosphere assay mathematical model presented here allows the assessment of the rate malignant stem-like cells expand via symmetric division and the evaluation of the effects of therapeutics on the self-renewal and proliferative activity of this clinically relevant population that drive tumor growth and recurrence.

A Variable Stepsize Implementation for Stochastic Differential Equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge--Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge--Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Stochastic Simulation for Spatial Modelling of Dynamic Processes in a Living Cell

One of the fundamental motivations underlying computational cell biology is to gain insight into the complicated dynamical processes taking place, for example, on the plasma membrane or in the cytosol of a cell. These processes are often so complicated that purely temporal mathematical models cannot adequately capture the complex chemical kinetics and transport processes of, for example, proteins or vesicles. On the other hand, spatial models such as Monte Carlo approaches can have very large computational overheads. This chapter gives an overview of the state of the art in the development of stochastic simulation techniques for the spatial modelling of dynamic processes in a living cell.

Populations of models, experimental designs and coverage of parameter space by Latin Hypercube and Orthogonal sampling

In this paper we have used simulations to make a conjecture about the coverage of a t-dimensional subspace of a d-dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k,n,d,t) = 1 - e^(-k/n^(t-1)). We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t-dimensional subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses.

Unlocking data sets by calibrating populations of models to data density: A study in atrial electrophysiology

We describe a statistically informed calibration of in silico populations to explore variability in complex systems. The understanding of complex physical or biological systems nearly always requires a characterization of the variability that underpins these processes. In addition, the data used to calibrate these models may also often exhibit considerable variability. A recent approach to deal with these issues has been to calibrate populations of models (POMs), multiple copies of a single mathematical model but with different parameter values, in response to experimental data. To date, this calibration has been largely limited to selecting models that produce outputs that fall within the ranges of the data set, ignoring any trends that might be present in the data. We present here a novel and general methodology for calibrating POMs to the distributions of a set of measured values in a data set. We demonstrate our technique using a data set from a cardiac electrophysiology study based on the differences in atrial action potential readings between patients exhibiting sinus rhythm (SR) or chronic atrial fibrillation (cAF) and the Courtemanche-Ramirez-Nattel model for human atrial action potentials. Not only does our approach accurately capture the variability inherent in the experimental population, but we also demonstrate how the POMs that it produces may be used to extract additional information from the data used for calibration, including improved identification of the differences underlying stratified data. We also show how our approach allows different hypotheses regarding the variability in complex systems to be quantitatively compared.

Sampling methods for exploring between-subject variability in cardiac electrophysiology experiments

Between-subject and within-subject variability is ubiquitous in biology and physiology, and understanding and dealing with this is one of the biggest challenges in medicine. At the same time, it is difficult to investigate this variability by experiments alone. A recent modelling and simulation approach, known as population of models (POM), allows this exploration to take place by building a mathematical model consisting of multiple parameter sets calibrated against experimental data. However, finding such sets within a high-dimensional parameter space of complex electrophysiological models is computationally challenging. By placing the POM approach within a statistical framework, we develop a novel and efficient algorithm based on sequential Monte Carlo (SMC). We compare the SMC approach with Latin hypercube sampling (LHS), a method commonly adopted in the literature for obtaining the POM, in terms of efficiency and output variability in the presence of a drug block through an in-depth investigation via the Beeler–Reuter cardiac electrophysiological model. We show improved efficiency for SMC that produces similar responses to LHS when making out-of-sample predictions in the presence of a simulated drug block. Finally, we show the performance of our approach on a complex atrial electrophysiological model, namely the Courtemanche–Ramirez–Nattel model.

Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.