Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1529–1552 | Cite as

Nonlinear dynamics of two-dimensional cardiac action potential duration mapping model with memory

  • M. Kesmia
  • S. Boughaba
  • S. JacquirEmail author


The aim of this work is the analysis of the nonlinear dynamics of two-dimensional mapping model of cardiac action potential duration (2D-map APD) with memory derived from one dimensional map (1D-map). Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting cardiac arrhythmias. For a constant rate of stimulation the short action potential during alternans is followed by a longer DI and inversely. It has been suggested that these differences in DI are responsible for the occurrence and maintenance of APD alternans. We focus our attention on the observed bifurcations produced by a change in the stimulation period and a fixed value of a particular parameter in the model. This parameter provides new information about the dynamics of the APD with memory, such as the occurrence of bistabilities not previously described in the literature, as well as the fact that synchronization rhythms occur in different ways and in a new fashion as the stimulation frequency increases. Moreover, we show that this model is flexible enough as to accurately reflect the chaotic dynamics properties of the APD: we have highlighted the fractal structure of the strange attractor of the 2D-map APD, and we have characterized chaos by tools such as the calculation of the Lyapunov exponents, the fractal dimension and the Kolmogorov entropy, with the next objective of refining the study of the nonlinear dynamics of the duration of the action potential and to apply methods of controlling chaos.


Cardiac action potential duration Memory Bifurcation Periodic dynamics Fixed point Strange attractor Fractal structure Chaos 

Mathematics Subject Classification

37D45 37N25 92C50 92B25 93C10 93C55 



We would like to thank Professor Michael R. Guevara (Department of Physiology and Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montreal, Quebec) for his insightful comments and helpful suggestions, which have helped improve the quality and presentation of the paper significantly.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Constantine IConstantineAlgeria
  2. 2.Le2i, FRE CNRS 2005, Arts et MétiersUniversité Bourgogne Franche-ComtéDijonFrance

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