[Paper Review] A Map-Based Model of the Cardiac Action Potential

Nikolai F. Rulkov|ArXiv.org|Aug 8, 2007

Functional Brain Connectivity Studies3 citations

TL;DR

This paper proposes a computationally efficient map-based model of the cardiac action potential using a two-dimensional discrete-time map to replicate the key phases of ventricular cell dynamics—rapid depolarization, plateau, and repolarization. By coupling a fast spiking map with a slower recovery variable modulated by a control parameter, the model enables large-scale simulations with significantly larger time steps than traditional ODE models, achieving stable wave propagation and complex dynamics like frequency entrainment and wavebreak in coupled networks.

ABSTRACT

A discrete time model that is capable of replicating the basic features of cardiac cell action potentials is suggested. The paper shows how the map-based approaches can be used to design highly efficient computational models (algorithms) that enable large-scale simulations and analysis of discrete network models of cardiac activity.

Motivation & Objective

• To develop a computationally efficient model of the cardiac action potential that avoids the numerical constraints of traditional ODE-based models.
• To apply map-based dynamics to replicate the five-phase action potential of non-pacemaker cardiac cells, including rapid depolarization and plateau phases.
• To enable large-scale simulations of electrically coupled cardiac networks by using discrete-time maps with large integration steps.
• To demonstrate emergent network behaviors such as wave propagation, frequency entrainment, and wavebreak in coupled systems.

Proposed method

• The model uses a two-dimensional discrete-time map: $ x_{n+1} = P(x_n, y_n) $, $ y_{n+1} = Q(x_n, y_n) $, where $ x_n $ represents fast depolarization and $ y_n $ represents recovery and repolarization.
• The $ x $-dynamics are governed by a nonlinear function $ f(x_n, x_{n-1}, u) $ that produces spike-like behavior when $ \varepsilon(y_n) = 0 $, and is deactivated via $ \varepsilon(y_n) \to 1 $ to simulate Na+ channel inactivation.
• The $ y $-dynamics are modeled by a one-dimensional map $ Q(x_n, y_n) $ with a control parameter $ \mu $, enabling plateau and repolarization phases.
• A coupling mechanism via gap junctions is introduced through $ y_{n+1} = Q(x_n, y_n) + \mu_{\text{gap}} \sum_{j \in J} I_{n,j,i}^{\text{gap}} $, allowing wave propagation in networks.
• The model uses a linear combination $ V_n = a x_n + b y_n $ to map the state variables to the membrane potential.
• The model is validated in one-dimensional rings of 500 coupled cells, showing stable wave propagation, frequency ratio switching (1:2 to 1:1), and wavebreak due to damaged cells.

Experimental results

Research questions

• RQ1 Can a map-based discrete-time model replicate the key phases of the cardiac action potential with high computational efficiency?
• RQ2 How can a map-based approach enable large time steps in simulations of cardiac electrical activity without sacrificing dynamical fidelity?
• RQ3 What emergent network behaviors, such as wave propagation and frequency entrainment, can be captured in coupled map-based cardiac models?
• RQ4 How do localized cell abnormalities (e.g., shortened action potential duration) lead to wavebreak and new wave sources in a coupled network?

Key findings

• The model successfully replicates the five-phase action potential: resting, rapid depolarization (phase 0), initial repolarization (phase 1), plateau (phase 2), and repolarization (phase 3).
• The use of a modulated map allows for large integration time steps, overcoming the stiffness issues common in ODE-based cardiac models.
• In a ring of 500 coupled cells, the model exhibits stable 1:1 and 1:2 frequency wave patterns, with switching between them induced by a single external trigger pulse.
• A group of 30 cells with shortened action potential duration (from $ y_s = 1.3 $ to $ y_s = 1.2 $) acts as a new wave source, causing persistent doublet wave patterns after perturbation.
• The model captures memory effects in wave dynamics, where the timing of an external stimulus determines whether the system remains in a 1:2 or switches to a 1:1 frequency regime.
• The gap junction coupling model with $ \mu_{\text{gap}} = 0.0001 $ and $ g_{\text{gap}} = 0.004 $ enables realistic wave propagation and recovery dynamics in the network.