Spontaneous termination of chaotic spiral wave dynamics in human cardiac ion channel models

Figures

• 

  Fig 1. Exemplary snapshots of the membrane potential of chaotic spiral wave dynamics in two-dimensional simulations. The rescaled membrane potential u is shown for the Bueno-Orovio-Cherry-Fenton model in subplot (a)(domain size of A = 2.62 × 105 mm2). In the case of the Ten Tusscher-Noble-Noble-Panfilov model Vm is shown in subplot (b) (domain size of A = 0.66 × 105 mm2).

• 

  Fig 2. Exemplary temporal evolutions of the number of chaotic initial conditions NCh(t) which show chaotic dynamics at time t. In subplot (a) and (b) NCh(t) is shown for the BOCF model (domain size of A = 0.66 × 105 mm2) and the TNNP model (domain size of A = 0.32 × 105 mm2), respectively. For both models, 200 initial conditions were created for each domain size. The red lines indicate the fits of the data according to Eq (6), where data points below the fit threshold were discarded.

• 

  Fig 3. The exponential scaling of 〈T〉. For both models, the average transient lifetime 〈T〉 is shown for various domain sizes A in subplot (a) and (b), respectively. The red line in both subplots shows a fit of the data according to Eq (7), which indicates the exponential scaling of the average transient lifetime with the system size. The corresponding fit parameters are given in Table 1. The numerical data underlying this graph and the following figures are also provided as Supporting information S1 Data.

• 

  Fig 4. The average number of phase singularities grows with the domain size. The average number of spiral waves 〈Nspiral〉t (averaged over simulation time t) present in a system is shown for various domain sizes A for the BOCF model (a) and the TNNP model (b), respectively. Error bars represent the standard deviation, which is significant since the number of spiral waves usually fluctuates considerably during a typical episode.

• 

  Fig 5. The temporal evolution of the number of spiral waves Nspiral during typical episodes of transient chaos. The number of spiral waves is shown for a typical example in subplot (a) for the BOCF model (domain size of A = 4.96 × 105 mm2) and in subplot (b) for the TNNP model (domain size of A = 1.11 × 105 mm2). In the latter case, the system terminates “spontaneously”, whereas in the case of the BOCF model, the system decays into the single-spiral state (around t ≈ −25 s) and terminates much later (around t = 0 s).

• 

  Fig 6. The relative appearance of particular numbers of spiral waves Nspiral present in the system at the same time. For three different domain sizes each (denoted by different colors) the distribution of the relative appearance of Nspiral is shown for the BOCF model (a) and the TNNP model (b), respectively. In both cases, the distribution shifts towards larger Nspiral with increasing domain sizes. The shape is, however, biased for low number of spiral waves. In particular in the case of the BOCF model, a distinct peak for the single-spiral state for all three domain sizes can be observed.

• 

  Fig 7. The mean number of spiral waves, averaged over 200 initial conditions 〈Nspiral〉IC is shown for both models. 〈Nspiral〉IC is shown as red solid lines (light red area indicates the standard deviation) in subplot (a) for the BOCF model (domain size of A = 4.10 × 105 mm2) and in subplot (b) for the TNNP model (domain size of A = 1.28 × 105 mm2). Initial conditions were normalized in such a way that self-termination occurs at t = 0 s. In the latter case, 〈Nspiral〉IC decreases around t ≈ −10 s, whereas in the case of the BOCF model, 〈Nspiral〉IC decreases much earlier (around t ≈ −40 s).

• 

  Fig 8. Symbolic Markov chain description of the temporal evolution of spiral wave chaos. Each state of the system can be characterized by the number of spiral waves N. When evolving the system over a small but finite amount of time, the number of spiral waves can remain the same (described by the probability PΔN=0), or increase/decrease (described by the probabilities PΔN=+1, PΔN=+2 and PΔN=−1, PΔN=−2, respectively). When time intervals of the evolution are small enough, the change of Nspiral is limited by ΔN = {±1, ±2}.