# Browsing by Author "Schaeffer, David G"

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Item Open Access Item Open Access A Spectral Deferred Correction Method for Solving Cardiac Models(2011) Bowen, Matthew M.Many numerical approaches exist to solving models of electrical activity in the heart. These models consist of a system of stiff nonlinear ordinary differential equations for the voltage and other variables governing channels, with the voltage coupled to a diffusion term. In this work, we propose a new algorithm that uses two common discretization methods, operator splitting and finite elements. Additionally, we incorporate a temporal integration process known as spectral deferred correction. Using these approaches,

we construct a numerical method that can achieve arbitrarily high order in both space and time in order to resolve important features of the models, while gaining accuracy and efficiency over lower order schemes.

Our algorithm employs an operator splitting technique, dividing the reaction-diffusion systems from the models into their constituent parts.

We integrate both the reaction and diffusion pieces via an implicit Euler method. We reduce the temporal and splitting errors by using a spectral deferred correction method, raising the temporal order and accuracy of the scheme with each correction iteration.

Our algorithm also uses continuous piecewise polynomials of high order on rectangular elements as our finite element approximation. This approximation improves the spatial discretization error over the piecewise linear polynomials typically used, especially when the spatial mesh is refined.

As part of these thesis work, we also present numerical simulations using our algorithm of one of the cardiac models mentioned, the Two-Current Model. We demonstrate the efficiency, accuracy and convergence rates of our numerical scheme by using mesh refinement studies and comparison of accuracy versus computational time. We conclude with a discussion of how our algorithm can be applied to more realistic models of cardiac electrical activity.

Item Open Access Bifurcations in the Echebarria-Karma Modulation Equation for Cardiac Alternans in One Dimension(2009) Dai, ShuWhile alternans in a single cardiac cell appears through a simple

period-doubling bifurcation, in extended tissue the exact nature

of the bifurcation is unclear. In particular, the phase of

alternans can exhibit wave-like spatial dependence, either

stationary or traveling, which is known as discordant

alternans. We study these phenomena in simple cardiac models

through a modulation equation proposed by Echebarria-Karma. In

this dissertation, we perform bifurcation analysis for their

modulation equation.

Suppose we have a cardiac fiber of length l, which is

stimulated periodically at its x=0 end. When the pacing period

(basic cycle length) B is below some critical value B

_{c},alternans emerges along the cable. Let a(x,n) be the amplitude

of the alternans along the fiber corresponding to the n-th

stimulus. Echebarria and Karma suppose that a(x,n) varies

slowly in time and it can be regarded as a time-continuous

function a(x,t). They derive a weakly nonlinear modulation

equation for the evolution of a(x,t) under some approximation,

which after nondimensionization is as follows:

&partial

_{t}a = σ a + L a - g a 3,where the linear operator

L a = &partial

_{xx}a - &partial_{x}a -Λ-1 ∫ 0_{x}a(x',t)dx'.In the equation, σ is dimensionless and proportional to

B

_{c}- B, i.e. σ indicates how rapid the pacing is,Λ-1 is related to the conduction velocity (CV) of the

propagation and the nonlinear term -ga3 limits growth after the

onset of linear instability. No flux boundary conditions are

imposed on both ends.

The zero solution of their equation may lose stability, as the

pacing rate is increased. To study the bifurcation, we calculate

the spectrum of operator L. We find that the

bifurcation may be Hopf or steady-state. Which bifurcation occurs

first depends on parameters in the equation, and for one critical

case both modes bifurcate together at a degenerate (codimension 2)

bifurcation.

For parameters close to the degenerate case, we investigate the

competition between modes, both numerically and analytically. We

find that at sufficiently rapid pacing (but assuming a 1:1

response is maintained), steady patterns always emerge as the only

stable solution. However, in the parameter range where Hopf

bifurcation occurs first, the evolution from periodic solution

(just after the bifurcation) to the eventual standing wave

solution occurs through an interesting series of secondary

bifurcations.

We also find that for some extreme range of parameters, the

modulation equation also includes chaotic solutions. Chaotic waves

in recent years has been regarded to be closely related with

dreadful cardiac arrhythmia. Proceeding work illustrated some

chaotic phenomena in two- or three-dimensional space, for instance

spiral and scroll waves. We show the existence of chaotic waves in

one dimension by the Echebarria-Karma modulation equation for

cardiac alternans. This new discovery may provide a different

mechanism accounting for the instabilities in cardiac dynamics.

Item Open Access Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans(2010) Dai, Shu; Schaeffer, David GInstabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria-Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations. (C) 2010 American Institute of Physics. [doi:10.1063/1.3456058]Item Open Access Modeling Oscillations in the cAMP-PKA Network Within Budding Yeast(2011) Gonzales, Kevin EdmondIn our work we develop and analyze an ordinary differential equation

model that describes the cyclic adenosine monophosphate (cAMP) --Protein Kinase A (PKA) pathway in budding yeast. In particular our

model describes the effect of glucose stimulation on the concentration of cAMP in the short term,

and the effect of stress in the long term. We develop this model in

order to understand two specific experimental results, reported by

Ma et al. (1999) and Garmendia-Torres et al. (2007). In order to describe the

surprising results published by Ma et al. (1999) we make a key assumption

that three enzymes within the cAMP-PKA network compete with one

another for activation by PKA. This assumption sets our model apart

from previous models of the cAMP-PKA network.

Our model focuses on two forms of negative feedback that

drive oscillations in the concentration of cAMP. Under high or low

stress conditions (for example, following glucose stimulation) our model reduces to a single negative

feedback loop, resulting in decaying oscillations in the concentration of cAMP towards a unique

equilibrium point. Under intermediate stress levels, a second negative feedback loop also exists, resulting in the possible loss of stability

through a Hopf bifurcation, which leads to sustained oscillations in the concentration of cAMP. Given the novel prediction that

the concentration of cAMP experiences decaying oscillations for a

wide range of parameters, our collaborators in biology, Dr. Magwene's Lab, undertook new experiments in

which they verified decaying cAMP oscillations at low stress levels. In an initial

experiment they also verify the possibility of sustained oscillations at intermediate stress levels as predicted by our model.

Our model of the cAMP-PKA network has both predictive and explanatory

power and will serve as a foundation for future mathematical and

experimental studies of this key signaling network.

Item Open Access Principles that govern competition or co-existence in Rho-GTPase driven polarization.(PLoS computational biology, 2018-04-12) Chiou, Jian-Geng; Ramirez, Samuel A; Elston, Timothy C; Witelski, Thomas P; Schaeffer, David G; Lew, Daniel JRho-GTPases are master regulators of polarity establishment and cell morphology. Positive feedback enables concentration of Rho-GTPases into clusters at the cell cortex, from where they regulate the cytoskeleton. Different cell types reproducibly generate either one (e.g. the front of a migrating cell) or several clusters (e.g. the multiple dendrites of a neuron), but the mechanistic basis for unipolar or multipolar outcomes is unclear. The design principles of Rho-GTPase circuits are captured by two-component reaction-diffusion models based on conserved aspects of Rho-GTPase biochemistry. Some such models display rapid winner-takes-all competition between clusters, yielding a unipolar outcome. Other models allow prolonged co-existence of clusters. We investigate the behavior of a simple class of models and show that while the timescale of competition varies enormously depending on model parameters, a single factor explains a large majority of this variation. The dominant factor concerns the degree to which the maximal active GTPase concentration in a cluster approaches a "saturation point" determined by model parameters. We suggest that both saturation and the effect of saturation on competition reflect fundamental properties of the Rho-GTPase polarity machinery, regardless of the specific feedback mechanism, which predict whether the system will generate unipolar or multipolar outcomes.