Reduction of a cardiac pacemaker cell model using singular perturbation theory

2007 ◽  
Author(s):  
Karima Djabella ◽  
Michel Sorine
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Cell Model ◽  
Cardiac Pacemaker ◽  
Singular Perturbation Theory ◽  
Pacemaker Cell

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

scholarly journals Synergism of coupled subsarcolemmal Ca2+ clocks and sarcolemmal voltage clocks confers robust and flexible pacemaker function in a novel pacemaker cell model

2009 ◽  
Vol 296 (3) ◽  
pp. H594-H615 ◽  
Author(s):  
Victor A. Maltsev ◽  
Edward G. Lakatta
Keyword(s):  
Cell Model ◽  
Numerical Models ◽  
Experimental Studies ◽  
Cardiac Pacemaker ◽  
Critical Dimension ◽  
Pacemaker Cell ◽  
Pumping Rate ◽  
Rate Modulation ◽  
Wide Scale ◽  
Fail Safe

Recent experimental studies have demonstrated that sinoatrial node cells (SANC) generate spontaneous, rhythmic, local subsarcolemmal Ca2+ releases (Ca2+ clock), which occur during late diastolic depolarization (DD) and interact with the classic sarcolemmal voltage oscillator (membrane clock) by activating Na+-Ca2+ exchanger current ( INCX). This and other interactions between clocks, however, are not captured by existing essentially membrane-delimited cardiac pacemaker cell numerical models. Using wide-scale parametric analysis of classic formulations of membrane clock and Ca2+ cycling, we have constructed and initially explored a prototype rabbit SANC model featuring both clocks. Our coupled oscillator system exhibits greater robustness and flexibility than membrane clock operating alone. Rhythmic spontaneous Ca2+ releases of sarcoplasmic reticulum (SR)-based Ca2+ clock ignite rhythmic action potentials via late DD INCX over much broader ranges of membrane clock parameters [e.g., L-type Ca2+ current ( ICaL) and/or hyperpolarization-activated (“funny”) current ( If) conductances]. The system Ca2+ clock includes SR and sarcolemmal Ca2+ fluxes, which optimize cell Ca2+ balance to increase amplitudes of both SR Ca2+ release and late DD INCX as SR Ca2+ pumping rate increases, resulting in a broad pacemaker rate modulation (1.8–4.6 Hz). In contrast, the rate modulation range via membrane clock parameters is substantially smaller when Ca2+ clock is unchanged or lacking. When Ca2+ clock is disabled, the system parametric space for fail-safe SANC operation considerably shrinks: without rhythmic late DD INCX ignition signals membrane clock substantially slows, becomes dysrhythmic, or halts. In conclusion, the Ca2+ clock is a new critical dimension in SANC function. A synergism of the coupled function of Ca2+ and membrane clocks confers fail-safe SANC operation at greatly varying rates.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

On voltage stability from the viewpoint of singular perturbation theory

1994 ◽  
Vol 16 (6) ◽  
pp. 409-417 ◽  
Author(s):  
N. Yorino ◽  
H. Sasaki ◽  
Y. Masuda ◽  
Y. Tamura ◽  
M. Kitagawa ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Voltage Stability ◽  
Singular Perturbation Theory
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