Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology

Beata  Jackowska-Zduniak; Urszula Foryś

doi:10.3934/mbe.2016035

Article Contents

2016, Volume 13, Issue 6: 1143-1158. Doi: 10.3934/mbe.2016035

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Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology

Beata Jackowska-Zduniak^1, and
Urszula Foryś^2,

1.
Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences, Nowoursynowska 159, 02-776 Warsaw
2.
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw

Received: September 30, 2015

Revised: April 30, 2016

Published: July 2016

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Abstract

A proposed model consisting of two coupled models (Hodgkin-Huxley and Yanagihara-Noma-Irisawa model) is considered as a description of the heart's action potential. System of ordinary differential equations is used to recreate pathological behaviour in the conducting heart's system such as double fire and the most common tachycardia: atrioventricular nodal reentrant tachycardia (AVNRT). Part of the population has an abnormal accessory pathways: fast and slow (Fujiki, 2008). These pathways in the atrioventricular node (AV node) are anatomical and functional contributions of supraventricular tachycardia. However, the appearance of two pathways in the AV node may be a contribution of arrhythmia, which is caused by coexistent conduction by two pathways (called double fire). The difference in the conduction time between these pathways is the most important factor. This is the reason to introduce three types of couplings and delay to our system in order to reproduce various types of the AVNRT. In our research, introducing the feedback loops and couplings entails the creation of waves which can correspond to the re-entry waves occurring in the AVNRT. Our main aim is to study solutions of the given equations and take into consideration the influence of feedback and delays which occur in these pathological modes. We also present stability analysis for both components, that is Hodgkin-Huxley and Yanagihara-Noma-Irisawa models, as well as for the final double-fire model.

Keywords:

Hodgkin-Huxley model,
ordinary differential equations with delay,
feedback,
couplings,
action potential,
double-fire pathology,
AVNRT.

Mathematics Subject Classification: Primary: 37N25, 92C30; Secondary: 70K20, 70K50.

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