Asymptotic analysis and analytical solutions of a model of cardiac excitation

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317–352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an “Archetypal Model” with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable “caricature” of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the “caricature” right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model

Mitochondrial Reactive Oxygen Species (ROS) and Arrhythmias.

In this chapter we analyze the onset of cardiac arrhythmias from the perspective of mitochondrial redox state and energetic metabolism. Significant perturbations in the mitochondrial redox environment trigger mitochondrial membrane potential (ΔΨm) depolarization that under critical conditions can scale up to the whole heart, thereby producing fatal arrhythmias. Utilizing a combined experimental–theoretical approach, we evaluate the processes dynamics at each level of organization involved (molecular, mitochondrial, cardiomyocyte, whole heart) while highlighting their mechanistic interrelationships to explain the appearance of novel emergent properties. Under metabolically stressful conditions, the mitochondrial network of cardiac cells accumulate high level of reactive oxygen species (ROS) attaining a critical state – referred to as mitochondrial criticality. Under criticality, the slightest perturbation triggers a cell-wide collapse of ΔΨm, visualized as a depolarization wave throughout the cell, which is followed by whole cell sustained mitochondrial oscillations in ΔΨm, NADH, ROS, and glutathione. This macroscopic dynamic behavior escalates from the mitochondrion to the organ level driving the heart into catastrophic arrhythmias.Fil: Kembro, Jackelyn Melissa. University Johns Hopkins; Estados Unidos. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Investigaciones Biológicas y Tecnológicas. Universidad Nacional de Córdoba. Facultad de Ciencias Exactas, Físicas y Naturales. Instituto de Investigaciones Biológicas y Tecnológicas; ArgentinaFil: Cortassa, Sonia del Carme. University Johns Hopkins; Estados Unidos. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Biotecnológicas. Instituto de Investigaciones Biotecnológicas "Dr. Raúl Alfonsín" (sede Chascomús). Universidad Nacional de San Martín. Instituto de Investigaciones Biotecnológicas. Instituto de Investigaciones Biotecnológicas "Dr. Raúl Alfonsín" (sede Chascomús); ArgentinaFil: Aon, Miguel A.. University Johns Hopkins; Estados Unido