# American Institute of Mathematical Sciences

March  2018, 23(2): 975-989. doi: 10.3934/dcdsb.2018051

## New approach of controlling cardiac alternans

 1 Département de Mathématiques, Université Mentouri I, Constantine, Algérie 2 Le2i FRE 2005, CNRS, Arts et Métiers, Univ. Bourgogne Franche-Comté, Dijon, France

* Corresponding author: sjacquir@u-bourgogne.fr

Received  February 2017 Revised  September 2017 Published  December 2017

The alternans of the cardiac action potential duration is a pathological rhythm. It is considered to be relating to the onset of ventricular fibrillation and sudden cardiac death. It is well known that, the predictive control is among the control methods that use the chaos to stabilize the unstable fixed point. Firstly, we show that alternans (or period-2 orbit) can be suppressed temporally by the predictive control of the periodic state of the system. Secondly, we determine an estimation of the size of a restricted attraction's basin of the unstable equilibrium point representing the unstable regular rhythm stabilized by the control. This result allows the application of predictive control after one beat of alternans. In particular, using predictive control of periodic dynamics, we can delay the onset of bifurcations and direct a trajectory to a desired target stationary state. Examples of the numerical results showing the stabilization of the unstable normal rhythm are given.

Citation: Mounira Kesmia, Soraya Boughaba, Sabir Jacquir. New approach of controlling cardiac alternans. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 975-989. doi: 10.3934/dcdsb.2018051
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##### References:
The time evolution of transmembrane action potential using cable simulations (Beeler-Reuter model) for a periodically paced cell: (a) the response $1:1$, (b) the presence of alternans $2:2$, (c) the response $2:1$, (d) the irregular response. APD means Action Potential Duration, DI means Diastolic Interval.
Bifurcation diagram from [29]
At $t_s = 302$ $ms$ the alternans without control evolution over 20000 iterations starting from initial condition $APD_1 = 240$ $ms.$
Initiation of predictive control of the alternans $2:2$, (a) after $838$ beats of alternans, (b) after only one beat of alternans. After control is initiated in figure (a) or (b), $APD_{n}$ alternates around $APD^{\ast}$ as the asymptotic stability of $1:1$ rhythm is performed.
Bifurcation diagram ($APD_i$ vs. $t_s$) without control for $t_s = 200-400$ $ms$. At each $t_s$, the map (1) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $t_s$ was $0.1$ $ms$, $APD_1 = 198$ $ms$.
Bifurcation diagram ($APD_i$ vs. $t_s$) with predictive control for $t_s = 200-400$ $ms$. At each $t_s$, the controlled map (3) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $t_s$ was $0.1$ $ms$. $APD_1 = 198$ $ms$, $\varepsilon = 7.7$ $ms$, $K = -0.2.$
Initiation of predictive control of periodic rhythm $6:2$, (a) after 2 beats, (b) after 1 beat. After control is initiated in Fig(a) or (b), $APD_{n}$ alternates about $APD^{\ast}$ as the asymptotic stability of $6:1$ rhythm is performed.
The suppression of noisy alternans with the predictive control.
Bifurcation diagram ($APD_i$ vs. $\varepsilon$) with predictive control for $\varepsilon = 0-15$ $ms$. At each $\varepsilon$, the controlled map (3) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $\varepsilon$ was $0.1$ $ms$, $t_s = 302$ $ms$, $APD_1 = 240$ $ms$, $K = -0.1$.
Example of parameter values $K$, $\varepsilon$ and $APD_{1}$ for the stabilisation of the unstable equilibrium point (or $1:1$ unstable rhythm).
 K $\varepsilon$ $APD_{1}$ -0.1 9 ms 240 ms -0.1 7.7 ms 198 ms
 K $\varepsilon$ $APD_{1}$ -0.1 9 ms 240 ms -0.1 7.7 ms 198 ms
Example of parameter values $K$, $\varepsilon$ and $APD_{1}$ for for controlling the periodic rhythm $6:2$.
 K $\varepsilon$ $APD_{1}$ -0.4 1.96 ms 240 ms -0.4 1.96 ms 202 ms
 K $\varepsilon$ $APD_{1}$ -0.4 1.96 ms 240 ms -0.4 1.96 ms 202 ms
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