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A MemoryEfficient Finite Element Method for Systems of ReactionDiffusion Equations with NonSmooth Forcing
, 2003
"... The release of calcium ions in a human heart cell is modeled by a system of reactiondi #usion equations, which describe the interaction of the chemical species and the e#ects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposi ..."
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Cited by 17 (12 self)
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The release of calcium ions in a human heart cell is modeled by a system of reactiondi #usion equations, which describe the interaction of the chemical species and the e#ects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposition of many Dirac delta functions in space; such a nonsmooth righthand side leads to divergence for many numerical methods. The calcium ions enter the cell at a large number of regularly spaced points throughout the cell; to resolve those points adequately for a cell with realistic threedimensional dimensions, an extremely fine spatial mesh is needed. A finite element method is developed that addresses the two crucial issues for this and similar applications: Convergence of the method is demonstrated in extension of the classical theory that does not apply to nonsmooth forcing functions like the Dirac delta function; and the memory usage of the method is optimal and thus allows for extremely fine threedimensional meshes with many millions of degrees of freedom, already on a serial computer. Additionally, a coarsegrained parallel implementation of the algorithm allows for the solution on meshes with yet finer resolution than possible in serial.
Stochastic properties of Ca 2+ release of inositol 1,4,5trisphosphate receptor clusters
 Biophys J
, 2002
"... ABSTRACT Intracellular Ca2 release is controlled by inositol 1,4,5trisphosphate (IP3) receptors or ryanodine receptors. These receptors are typically distributed in clusters with several or tens of channels. The random opening and closing of these channels introduces stochasticity into the element ..."
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ABSTRACT Intracellular Ca2 release is controlled by inositol 1,4,5trisphosphate (IP3) receptors or ryanodine receptors. These receptors are typically distributed in clusters with several or tens of channels. The random opening and closing of these channels introduces stochasticity into the elementary calcium release mechanism. Stochastic release events have been experimentally observed in a variety of cell types and have been termed sparks and puffs. We put forward a stochastic version of the LiRinzel model (the deactivation binding process is described by a Markovian scheme) and a computationally more efficient Langevin approach to model the stochastic Ca2 oscillation of single clusters. Statistical properties such as Ca2 puff amplitudes, lifetimes, and interpuff intervals are studied with both models and compared with experimental observations. For clusters with tens of channels, a simply decaying amplitude distribution is typically observed at low IP3 concentration, while a single peak distribution appears at high IP3 concentration.
Solitary waves in a model of dendritic cable with active spines
 SIAM J. Appl. Math
"... We consider a continuum model of dendritic spines with active membrane dynamics uniformly distributed along a passive dendritic cable. Byconsidering a systematic reduction of the HodgkinHuxleydynamics that is valid on all but veryshort timescales we derive 2 dimensional and 1 dimensional systems f ..."
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Cited by 11 (2 self)
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We consider a continuum model of dendritic spines with active membrane dynamics uniformly distributed along a passive dendritic cable. Byconsidering a systematic reduction of the HodgkinHuxleydynamics that is valid on all but veryshort timescales we derive 2 dimensional and 1 dimensional systems for excitable tissue, both of which may be used to model the active processes in spineheads. In the first case the coupling of the spine head dynamics to a passive dendritic cable via a direct electrical connection yields a model that may be regarded as a simplification of the Baer and Rinzel cable theoryof excitable spinynerve tissue [3]. This model is computationallysimple with few free parameters. Importantly, as in the full model, numerical simulation illustrates the possibilityof a traveling wave. We present a systematic numerical investigation of the speed and stabilityof the wave as a function of physiologically important parameters. A further reduction of this model suggests that active spinehead dynamics maybe modeled byan all or none type process which we take to be of the integrateandfire (IF) type. The model is analytically tractable allowing the explicit construction of the shape of traveling waves as well as the calculation of wave speed as a function of system parameters. In general a slow and fast wave are found to coexist. The behavior of the fast wave is found to closelyreproduce the behavior of the wave seen in simulations of the more detailed model. Importantlya linear stabilitytheoryis presented showing that it is the faster of the two solutions that is stable. Beyond a critical value the speed of the stable wave is found to decrease as a function of spine density. Moreover, the speed of this wave is found to decrease as a function of the strength of the electrical resistor coupling the spinehead and the cable, such that beyond some critical value there is propagation failure. Finally we discuss the importance of a model of passive electrical cable coupled to a system of integrateandfire units for physiological studies of branching dendritic tissue with active spines. Key words. cable equation, dendritic spines, integrateandfire AMS subject classifications. 92C20,34A34,76B25
The effect of ion pumps on the speed of travelling waves in the firediffusefire model of Ca2+ release
 Bulletin of Mathematical Biology
, 2001
"... The firediffusefire model provides an idealized model of Ca2+ release within living cells. The effect of calcium pumps, which drive Ca2+ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer e ..."
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Cited by 8 (3 self)
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The firediffusefire model provides an idealized model of Ca2+ release within living cells. The effect of calcium pumps, which drive Ca2+ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer et al. [Keizer, J. E., G. D. Smith, S. Ponce Dawson and J. Pearson (1998). Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595–600.]. For travelling waves, in which release events occur sequentially, we construct the speed of waves in terms of the timescale at which pumps operate. An immediate consequence of this analysis is that the inclusion of calcium pumps leads to multiple solutions. A linear stability analysis determines those solution branches in parameter space which are stable. Numerical continuation is used to provide explicit examples of the bifurcation diagrams of the speed of waves as a function of physiologically significant system parameters. c © 2000 Society for Mathematical Biology 1.
Analysing cardiac excitation–contraction coupling with mathematical models of local control. Progr Biophys Molec Biol 85
, 2004
"... Cardiac excitation–contraction (E–C) coupling describes the process that links sarcolemmal Ca2+ influx via Ltype Ca2+ channels to Ca2+ release from the sarcoplasmic reticulum via ryanodine receptors (RyRs). This process has proven difficult to study experimentally, and complete descriptions of how ..."
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Cardiac excitation–contraction (E–C) coupling describes the process that links sarcolemmal Ca2+ influx via Ltype Ca2+ channels to Ca2+ release from the sarcoplasmic reticulum via ryanodine receptors (RyRs). This process has proven difficult to study experimentally, and complete descriptions of how the cell couples surface membrane and intracellular signal transduction proteins to achieve both stable and sensitive intracellular calcium release are still lacking. Mathematical models provide a framework to test our understanding of how this is achieved. While no single model is yet capable of describing all features of cardiac E–C coupling, models of increasing complexity are revealing unexpected subtlety in the process. In particular, modelling has established a general failure of ‘commonpool ’ models and has emphasized the requirement for ‘local control ’ so that microscopic subcellular domains can separate local behaviour from the wholecell average (commonpool) behaviour. The microarchitecture of the narrow diadic cleft in which the local control takes place is a key factor in determining local Ca2+ dynamics. There is still considerable uncertainty about the number of Ca2+ ions required to open RyRs within the cleft and various gating models have been proposed, many of which are in reasonable agreement with available experimental data. However, not all models exhibit a realistic voltage dependence of E–C coupling gain.
Stochastic calcium oscillations
 Mathematical Medicine and Biology
, 2006
"... Abstract While the oscillatory release of calcium from intracellular stores is comprised of fundamentally stochastic events, most models of calcium oscillations are deterministic. As a result, the transition to calcium oscillations as parameters, such as IP 3 concentration, are changed, is not desc ..."
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Abstract While the oscillatory release of calcium from intracellular stores is comprised of fundamentally stochastic events, most models of calcium oscillations are deterministic. As a result, the transition to calcium oscillations as parameters, such as IP 3 concentration, are changed, is not described correctly. The fundamental difficulty is that whole cell models of calcium dynamics are based on the assumptions that the calcium concentration is spatially homogeneous, and that there are a sufficiently large number of release sites per unit volume so that the law of large numbers is applicable. For situations where these underlying assumptions are not applicable, a new modeling approach is needed. In this paper we present a model and its analysis of calcium dynamics that incorporates the fundamental stochasticity of release events. The model is based on the assumptions that release events are rapid, while reactivation is slow. The model presented here is comprised of two parts. In the first, a stochastic version of the firediffusefire model is studied in order to understand the sparktowave transition and the probability of sparks resulting in abortive waves vs. whole cell calcium release. In the second, this information about the sparktowave transition is incorporated into a stochastic model (a ChapmanKolmogorov equation) that tracks the number of activated and inactivated calcium release sites as a function of time. By solving this model numerically, information about the timing of whole cell calcium release is obtained. The results of this analysis show a transition to oscillations that agrees well with data and with Monte Carlo simulations.
of Ca 2+ release
"... The effect of ion pumps on the speed of travelling waves in the firediffusefire model ..."
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The effect of ion pumps on the speed of travelling waves in the firediffusefire model
A guide to sparkology: The taxonomy of elementary
, 2007
"... Since the discovery of the Ca 2+ spark as an elementary event of cellular Ca 2+ signaling almost 15 years ago, the family of newly described Ca 2+ signal entities has been ever growing. While scientists working in Ca 2+ signaling may have maintained an overview over the specifics of this nomenclatur ..."
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Since the discovery of the Ca 2+ spark as an elementary event of cellular Ca 2+ signaling almost 15 years ago, the family of newly described Ca 2+ signal entities has been ever growing. While scientists working in Ca 2+ signaling may have maintained an overview over the specifics of this nomenclature, those outside the field often make the complaint that they feel hopelessly lost. With the present review we collect and summarize systematic information on the many Ca 2+ signaling events described in a variety of tissues and cells, and we emphasize why and how each of them has its own importance. Most of these signals are taking place in the cytosol of the respective cells, but several events have been recorded from intracellular organelles as well, where they may serve their own specific functions. Finally, we also try to convey an integrated view as to why cellular microdomain signaling is of fundamental biological importance.
Journal of Mathematical Biology manuscript No.
"... The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP3 ) receptors. Systematic reductions of the kinetic schemes for IP3 dynamics have proved especially fruitful in understanding the transition from exci ..."
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The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP3 ) receptors. Systematic reductions of the kinetic schemes for IP3 dynamics have proved especially fruitful in understanding the transition from excitable to oscillatory behaviour. With the inclusion of di#usive transport of calcium ions the model also supports wave propagation. The analysis of waves, even in reduced models, is typically only possible with the use of numerical bifurcation techniques. In this paper we review the travelling wave properties of the biophysical De Young Keizer model and show that much of its behaviour can be reproduced by a much simpler FireDi#useFire (FDF) type model. The FDF model includes both a refractory process and an IP3 dependent threshold. Parameters of the FDF model are constrained using a comprehensive numerical bifurcation analysis of solitary pulses and periodic waves in the De Young Keizer model. The linear stability of numerically constructed solution branches is calculated using pseudospectral techniques. The combination of numerical bifurcation and stability analysis also allows us to highlight the mechanisms that give rise to propagation failure. Moreover, a kinematic theory of wave propagation, based around numerically computed dispersion curves is used to predict waves which connect periodic orbits. Direct numerical simulations of the De Young Keizer model confirm this prediction. Corresponding travelling wave solutions of the FDF model are obtained analytically and are shown to be in good qualitative agreement with those of the De Young Keizer model. Moreover, the FDF model may be naturally extended to include the discrete nature of calcium stores within a cell, withou...