A Differential Model of Controlled Cardiac Pacemaker Cell

Abstract

A differential model of a cardiac pacemaker cell with only ten state variables is proposed. It is intended for 0D or 3D simulation of the heart under the vagal control of the autonomous nervous system. Three variables are used to describe the membrane (membrane potential and two gate variables of ionic channels), taking into account the dynamics of the main ionic currents (inward sodium, L-type calcium and outward potassium), N a + /Ca 2+ exchangers and N a + /K + pumps. The remaining seven variables are associated with the fluid compartment model that includes Ca 2+ binding by myoplasmic proteins, and the intracellular concentrations of free Calcium, Sodium and Potassium. Despite its moderate number of state variables, this model includes the main processes thought to be important in pacemaking on the cell scale and predicts the experimentally observed ionic concentration of calcium, sodium and potassium, action potential and membrane currents. The control by the calcium of the pacemaking activity is also considered.

Modelación del funcionamiento de un marcapasos y su implementación electrónica

“…Este trabajo parte de la idea de una ecuación diferencial logística como modelo matemático. 3 Actualmente la utilización de ecuaciones diferenciales en los modelos matemáticos tienen un papel sumamente relevante en el desarrollo de nuevas tecnologías, tanto para el diseño como para la construcción de los marcapasos (Djabella y Sorine, 2006). Por ejemplo, es gracias a los modelos en ecuaciones diferenciales que podemos comprender el comportamiento de las células cardiacas (Keener y Sneyd, 1998), lo cual resulta sumamente útil para definir la magnitud del impulso eléctrico que debe emitir el marcapasos.…”

Bifurcation Analysis of the Control Mechanism of Pacemaker Activity in a Mathematical Cardiac Cell Model

“…It is possible that any of these currents, or a combination of them, is responsible for membrane depolarization during diastole. In our simulations using a new reduced order model of the SA node cell (Djabella and Sorine, 2006), we concluded that all the currents directly or indirectly contribute to pacemaker depolarization, with the background calcium current (I b,Ca ) being the most dominant inward current during diastole. Thus, the roles of all individual currents (like pump and exchanger) in pacemaker activity must be considered.…”

“…To do that the nonlinear dynamics of the mathematical model of the pacemaker cell of (Djabella and Sorine, 2006) is studied by a bifurcation analysis, as well as by numerical simulations. We construct bifurcation diagrams by calculating stationary points, periodic orbits, stability of the equilibrium points, Hopf bifurcation points as functions of bifurcation parameters (maximal conductances) to characterize the dynamics properties of each individual ionic components in the SA node pacemaker cell model.…”