Using noise to determine cardiac restitution with memory

Variation in cardiac pacing cycles, as seen, for example, in heart rate variability, has been observed for decades. Contemporarily, various mathematical models have been constructed to investigate the electrical activity of paced cardiac cells. Yet there has not been a study of these cardiac models when there is variation in the pacing cycles such as noise. We present a method that uses the stochasticity of pacing cycles to determine approximate models of the dynamics of cardiac cells, and use these models to detect bifurcations to alternans.

Figure 1

Typical bifurcation diagram for a cardiac mapping model with constant cycle length $\mu$, where period-doubling bifurcation occurs at $\mu ={\mu }_c$. The shaded region indicates the range of APDs when the cycle length has a small random fluctuation around $\mu$.

Figure 2

Illustration of the cycle lengths $T_n$, action potential duration $A_n$, diastolic interval $D_n$, and memory variable $M_n$.

Figure 3

Sequence of interbeat intervals (in seconds) for a healthy individual [20].

Figure 4

(a) Sequence of pacing cycles, (b) sequence of corresponding APDs, and (c) histogram of all APDs.

Figure 5

Comparison of the bifurcation diagrams between the theoretical map (A1) (black) and the approximate form (2.8) (red, light grey) for various orders of $p$. The solid curve indicates stability and the dashed curve indicates instability. Note that for all cases the approximate fixed points are very close to the theory and they are not distinguishable in the graph. In addition, for the case $p=5$, the theoretical and approximate diagrams almost coincide.

Figure 6

Comparison of the bifurcation diagrams between the theoretical map (3.1) (black) and the approximate form (3.5) (red, light grey) for various values of $k$ and $p$. The solid line indicates stability and the dashed line indicates instability. We note that in each of the two cases, $(k,p)=(1,4)$ and $(2,4)$, the dynamics of theory and the approximation are very close and their bifurcation diagrams almost coincide.

Figure 7

Approximate the model of Fox et al.: RSE and BIC values of each polynomial $H_k^p$ vs $p$ for $k=0,1,2,3$.

Figure 8

(a) The value of $h_{1,0,0,...}$, the coefficient for $(A_{n-1}-\overline{A})$ vs the order $p$ for different values of $k$, and (b) the value of $h_{1,1,0,...}$, the coefficient for $(A_{n-1}-\overline{A})(T_{n-1}-\overline{T})$ vs the order $p$ for different values of $k$.

Figure 9

The RSE and BIC values of the approximate polynomial $H_k^p$ vs the order $p$ for various $k$, and bifurcation diagrams of $H_k^p$ at optimal choice of $(k,p)$ for different measurement error $\epsilon ={10}^{-4}$ (a, c, e) and $\epsilon =0.1$ (b, d, f), respectively, for Fox et al.'s model. In (e) and (f), the theoretical bifurcation diagram (black) and the approximation (red, light grey) are drawn for comparison, and in (e) the theory and approximation almost coincide.

Figure 10

Bifurcation diagrams of the optimal approximate $H_k^p$ (red, light grey) compared to the exact result (black) for various values of $\epsilon$ and $N$. In some of the graphs the approximate and exact diagrams almost coincide. The corresponding optimal choice of $(k,p)$ is shown in the bottom right corner of each graph.

Figure 11

Probability density function and histogram of the applied pacing cycles, and bifurcation diagram of the optimal approximate $H_k^p$ (red, light grey) compared to the exact curve (black) for three different distributions of pacing cycles, where $N=1000$ and $\epsilon =0.01$. The optimal choice of $(k,p)$ is labeled in each graph of the bifurcation diagram. We note that the approximate and the exact bifurcation diagrams are very close and hard to distinguish in all the graphs.

Figure 12

Bifurcation diagrams of the optimal approximate $H_k^p$ (red, light grey) compared to the exact result (black) for various testing time intervals, for $\epsilon =0.01$ (a, b, c) and $0.1$ (d, e, f), respectively, and in all cases $N=1000$. In some graphs the approximate and exact diagrams almost coincide. The optimal choice of $(k,p)$ is shown in the bottom right corner of each graph.