Arrhythmogenic effects of ultra-long and bistable cardiac action potentials

Figures

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  Fig 1. Ultra-long action potentials in cardiomyoctes isolated from rabbit hearts and paced at 1 Hz. (A) Example of a normal train of action potentials. (B) Examples of two ultra-long action potentials that remain depolarized for approximately 2.5 s and 1.5 s respectively. (C) Example of an ultra-long action potential that exceeds 15 s.

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  Fig 2. Comparison of monostable versus bistable dynamics in both the isolated cell and the homogeneous spatial medium. (A) Phase plane of the isolated cell in the monostable regime (b = 1). The nullclines are shown in green. The single fixed point (open circle) is stable. It represents the resting membrane potential. The system can be perturbed from rest by a brief injection current (I = 2 for 15 ms) that initiates a trajectory (black line) which makes a large excursion in phase space before returning to rest. That excursion corresponds to the action potential. (B) Time plot of the action potential in the monostable regime. Solid line for b = 1. Dotted lines for b = 1.5 and b = 1.75. (C) Snapshot of the corresponding traveling wave in a homogeneous sheet of 100 × 100 monostable cells with coupling parameter c = 1. Arrows indicate the direction of travel. Color indicates the membrane potential. (D) Phase plane of the isolated cell in the bistable regime (b = 2). Perturbing the resting state (open circle) induces a transition to the up-state (filled circle). (E) Time plot of that transition. (F) The corresponding traveling front in the spatial medium.

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  Fig 3. Bifurcations in the steady-states of the single cell. The resting state (V = −1.25) is stable (solid line) for all values of parameter b > 0. A pair of unstable fixed points (dashed lines) emerge via a saddle-node (SN) bifurcation at b = 1.64. The lower branch is the saddle and the upper branch is the node. The node becomes stable via a Hopf bifurcation (HB) at b = 1.76. The stable up-state co-exists with the stable resting state for b ≥ 1.76.

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  Fig 4. Effect of mixing monostable and bistable cells in the same tissue. Here we compare wave propagation in two simulations of 100 × 300 heterogeneous media where each cell was randomly assigned either b = 1 (monostable) or b = 3 (bistable) according to the distributions in panels A and C. A rightward traveling wave was initiated in the resting medium by briefly stimulating the cells at the left hand boundary. Snapshots of the respective simulations (t = 6000 ms) are shown in panels B and D. The former supports a functional propagating wave whereas the latter leaves self-sustained ectopic activity in its wake. The coupling strength (c = 0.6) is identical in both cases. Panel E shows successive snapshots of the 50x50 region marked with a white square. The ectopic activity in that region grew from a small cluster of cells that failed to repolarize (white circle). The time series of the cell at the center of the white circle (x = 93, y = 87) is shown in panel F (black). The light gray trace in panel F is that of a nearby cell (x = 93, y = 77). See S1 Video for an animated version of this figure.

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  Fig 5. Stability analysis of the reduced model. (A) Schematic of the model. Left: A single bistable cell (Vo) is coupled to n identical cells which are all clamped at rest (Vr). The coupling strength is c2. Right: The resting cells are lumped together to form the reduced model where the equivalent coupling is nc2. (B) Stability map for the reduced model with n = 4 neighbors. The shaded region indicates those configurations of b and c where the cell operates in the bistable regime. Points P = {3, 0.3}, Q = {3, 0.22} and R = {5, 0.3} are described in the text. (C) Stability map for the case of n = 2 neighbors. The stability boundaries for n = 0, n = 1, n = 3 and n = 4 neighbors are included for comparison.

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  Fig 6. Rogue activity emerges preferentially at tissue boundaries. (A–C) Snapshots of three simulations of a 40 × 40 medium in which 10% of the cells were configured to points P, Q or R in the stability map of the reduced model (Fig 5). The snapshots were taken at t = 1000 ms post-onset of a spatially uniform stimulus. The central annulus (black) represents a tissue boundary where cells are absent. The edges of the simulation domain used periodic boundary conditions to eliminate artificial boundary effects. The color scale indicates the membrane potential of the cells. (D–F) Cumulative results for 1000 trials where color is the percentage of cells that remained depolarized in the long term. Histograms (bottom) show the averaged radial profiles. See S2 Video for an animated version of this figure.

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  Fig 7. Effect of physiologically realistic distributions of cell heterogeneity. (A–C) Log-normal (μ, σ) distributions for μ = 0.9, μ = 1.0 and μ = 1.1 where the second moment is fixed at σ = 1. The shaded region denotes the area under the curve where b>1.77 exceeds the critical value for bistability in the single cell. The black triangle marks the arithmetic mean of the distribution. (D–F) Cumulative results for 1000 stimulation trials using the same protocol as Fig 6. Histograms (bottom) show the averaged radial profiles.

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  Fig 8. Ectopic activity induced at the tissue boundary by a passing wave. The 100 × 100 medium is briefly stimulated at the left-hand boundary and the ensuing wave propagates to the right. The black region represents the tissue boundary. The white circle in panel G marks the genesis of ectopic activity by cells on the tissue boundary that fail to repolarize. The medium is densely heterogeneous with parameter b of each cell drawn randomly from a log-normal distribution (μ = 1, σ = 1). The coupling coefficient is c = 0.7. See S3 Video for an animated version of this figure.

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  Fig 9. Pathways for the progression of disease. Healthy tissue corresponds to cells with low intrinsic bistability and high coupling (top-left). Disease can progress by increasing bistability or decreasing coupling strength or both. See S4 Video for an animated version of this figure.

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  Fig 10. Ultra-long and bistable action potentials in the Qu-Chung [16] model. (A) Repolarization failure induced by slowing the activation rate (1/γ) of the delayed rectifier potassium current. (B) Repolarization failure induced by negatively shifting the voltage-dependence of L-type calcium current inactivation by Δ mV. (C) Monostable and bistable action potential configurations used in subsequent simulations. The values γ = 4 and Δ = −15 were chosen to produce a robust bistable action potential that does not repolarize in the isolated cell. All other parameters are indentical to [16] except for IK1,max = 0.3.

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  Fig 11. Effect of mixing monostable and bistable cells in a biophysical model. The medium comprises 20 × 60 ventricular cells with action potentials described by the Qu-Chung model [16]. Cells on the left-hand boundary were stimulated briefly (80 pA for 0.5 ms) to induce a rightward traveling wave. Color indicates the membrane potential of the cells. Arrows indicate the direction of propagation. (A) Case of a medium with 70% monostable cells (γ = 1 and Δ = 0 mV) and 30% bistable cells (γ = 4 and Δ = −15 mV). (B) Case of 60% monostable and 40% bistable cells. The coupling strength (c = 0.15) is identical in both cases. Neumann (no-flux) conditions were applied at the boundaries. See S5 Video for an animated version of this figure.