Spatially heterogeneous ionic remodelling promotes wavefront instability in simulated atrial tissue

Figures

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  FIGURE 1. Realisations of the 2-dimensional Ising model on a 400×400 grid. White/black regions correspond to remodelled/normal tissue. The left and right plots show instances 10% below and 10% above the critical coupling respectively, and the central plot is an instance at the critical coupling.

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  FIGURE 2. Examples of AP propagation with K=Kc for different values of T . (A) T=300 ms for beat numbers 998–1,000 (from left to right), where the top image in each column shows the AP at the time of stimulus tpace and the bottom shows the state half way between stimuli. (B) T=290 ms for beats 998–1,000. Visible conduction blocks have also been labelled. (C) T=280 ms for beats 995–1,000. Significant conduction blocks have been labelled in this case.

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  FIGURE 3. Difference metrics dn(1) and dn(2) for T=300 ms and 290 m (see Equation 2). (A) T=300 ms. It is evident from the top panel (showing dn(1) ) that the difference between Vm over successive beats decays exponentially for all experiments, indicating periodicity. The bottom panel shows the same for dn(2) . Note that the sharp drops near the beginning reflect transient behaviour due to the initial conditions. (B) T=290 ms. In this case neither K=Kc nor K=1.1Kc displays periodic behaviour, except for a short transient period of period 2 like behaviour (indicated by small values of dn(2) in the bottom panel), but K=0.9Kc does exhibit period 2 behaviour, although the plateau in the decay (indicated by the decreasing slope in the bottom panel) suggests that this behaviour may not be stable.

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  FIGURE 4. Envelope of Vm spatial mean for T=290 ms experiments. The plot shows the behaviour for different values of K , where we are only showing one instance of K=Kc . The inset displays the upper envelop of the K=0.9Kc experiment over the last 200 beats, and shows that there is small scale oscillatory behaviour present.

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  FIGURE 5. (A) Values of dn(2) for homogeneous model (with non-remodelled parameters) compared to the experiment with K=1.1Kc , both paced at T=290 ms. It can be inferred from this that in the homogeneous model, uniform propagation failure occurs regularly before transitioning to periodic propagation failure at around cycle 700. The intermittent drops for K=1.1Kc are much more sporadic in comparison. (B) Average sodium concentration for homogeneous model and K=1.1Kc (in the non-remodelled regions only). In the homogeneous model, the decline is approximately linear until propagation failure finally becomes periodic in which case it accelerates, while the decline is much slower for K=1.1Kc , and appears to plateau towards the end. This suggests that the turbulent behaviour observed for K=1.1Kc may be stable.

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  FIGURE 6. dn(2) shown for (A) T=280 ms. (B) T=270 ms, and (C) T=260 ms. Evidently, for T=270 ms and 280 m, K=0.9Kc and K=1.1Kc exhibit aperiodic, unpredictable behaviour (Note that for K=Kc , the average over 10 instances of heterogeneity is shown, which dampens the fluctuations compared to K=0.9Kc ), and for T=260 ms they both fall into period 2 behaviour. K=1.1Kc always falls into period 2 behaviour, however it displays a long transient of aperiodic behaviour for T=280 ms, which shortens for T=270 ms (and still more for T=260 ms). (D) dn(8) for K=Kc and T=280 ms. The solid orange line shows the average of dn(8) over the 10 instances of heterogeneity as a reference, and the black dashed line shows a single instance that exhibits transient episodes of period 8 behaviour.

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  FIGURE 7. Average [Na]i in proportion of its initial value in remodelled and non-remodelled regions. (A) T=280 ms and K=1.1Kc , along with the value of dn(2) (with the x-axis scaled to match the beat number) shown on the right axis as a reference. (B) An instance of K=Kc and T=270 ms with transient period 8 behaviour (see Figure 6D). The right axis shows dn(8) as a reference.

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  FIGURE 8. T=300 ms with altered initial condition. (A) Envelope of spatial mean of Vm observable for K=0.9Kc and K=Kc , which shows an initial period of chaotic behaviour that eventually settles down to periodic behaviour. (B) Envelope of spatial mean of Vm for K=1.1Kc , which shows an initial period of stable periodic behaviour (corresponding to propagation failure) followed by persistent chaotic behaviour. (C) Average [Na]i for non-remodelled regions (left) and remodelled regions (right).