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Spatiotemporal dynamics of calcium-driven cardiac alternans

Per Sebastian Skardal, Alain Karma, and Juan G. Restrepo
Phys. Rev. E 89, 052707 – Published 14 May 2014
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  • INTRODUCTION
  • DERIVATION OF THE AMPLITUDE EQUATIONS
  • NUMERICAL SURVEY AND PHASE SPACE
  • LINEAR STABILITY ANALYSIS: ONSET OF…
  • STRONGLY NONLINEAR REGIME: DYNAMICS OF…
  • IONIC MODEL, RESTITUTION CURVES, AND…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We focus on the common case where the bidirectional coupling of intracellular calcium concentration and membrane voltage dynamics produces calcium and voltage alternans that are temporally in phase. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We identify universal dynamical features of SDA pattern formation and evolution in the presence of those jumps. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of “unidirectional pinning”: node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sublinearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations due to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.

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    • Received 26 March 2014

    DOI:https://doi.org/10.1103/PhysRevE.89.052707

    ©2014 American Physical Society

    Authors & Affiliations

    Per Sebastian Skardal1,2,*, Alain Karma3, and Juan G. Restrepo2

    • 1Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
    • 2Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA
    • 3Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA

    • *skardals@gmail.com

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    Issue

    Vol. 89, Iss. 5 — May 2014

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    • Figure 1
      Figure 1

      (a) Phase space of Eqs. (19) and (20) describing three solution regimes. As r increases (left to right) these regimes are no alternans (blue), smooth wave patterns (yellow), and discontinuous patterns (red), each separated by bifurcations r1(Λ) and r2(Λ). For Λ=15 (b) the maximal alternans amplitude c∞, jump amplitude |c+−c−|, and (c) velocity for a range of r values. Other parameter values are α,γ=0.3, β=0, ξ=1, and w=0 with L=30 and Δx=0.005.

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    • Figure 2
      Figure 2

      Examples of solutions of Eqs. (19) and (20) for (a) r=0.85 and (b) r=1.15. Calcium cn(x) and voltage an(x) profiles are plotted as blue dots and red curves, respectively. Other parameter values are Λ=15, α,γ=0.3, β=0, ξ=1, and w=0 with L=20 and Δx=0.04. With the choice ξ=1, x and Λ are in units of ξ (see text).

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    • Figure 3
      Figure 3

      Example solutions of the cable equation with the Shiferaw-Fox model for (a) BCL=340 ms and (b) BCL=330 ms. The amplitude of calcium alternans cn(x) and voltage alternans an(x) are plotted as blue dots and red curves, respectively.

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    • Figure 4
      Figure 4

      The critical value r1(Λ) describing the onset of alternans computed directly from numerical simulations of Eqs. (19) and (20) (blue circles) compared to our theoretical prediction given by Eq. (29) (dashed red). Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=100 and Δx=0.05.

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    • Figure 5
      Figure 5

      (a) The spatial wavelength λs and (b) velocity v of smooth solutions near the onset of alternans computed directly from numerical simulations of Eqs. (19) and (20) (blue circles) compared to our theoretical predictions given by Eqs. (30) and (31) (dashed red). Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=100 and Δx=0.05.

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    • Figure 6
      Figure 6

      Illustration of jumping points c− and c+ and the jump amplitude |c+−c−| at a node x0 for a discontinuous calcium profile c(x) using parameters r=1.15, Λ=15, α,γ=0.3, β=0, ξ=1, and w=0 with L=20 and Δx=0.005.

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    • Figure 7
      Figure 7

      The velocity and jump amplitude |c+−c−| of steady-state solutions, plotted in blue circles and red dots, respectively, over a range of r values for fixed Λ=15. The theoretical prediction for the jump amplitude of normal jumps is plotted in dashed black and the bifurcation value r1(Λ) describing the transition from smooth to discontinuous solutions is denoted by the vertical green dot-dashed line. Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=30 and Δx=0.005.

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    • Figure 8
      Figure 8

      The length scale l of phase reversals defined by Eq. (35) over a range of r values given Λ=15. Note that as r approaches r2(Λ) (denoted by the vertical dot-dashed green line) from below, l approaches zero. Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=30 and Δx=0.005. Inset: Comparison to the analytical expression for the flat CV limit.

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    • Figure 9
      Figure 9

      (a) Jumping points values |c−| and |c+| and (b) asymmetry of nodes Δ as Λ is increased from a steady-state profile with normal jumps with r=1.2 and Λ=10, plotted in solid blue and dashed red. Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=30 and Δx=0.005.

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    • Figure 10
      Figure 10

      First node location x1 and Λ, plotted in blue circles and dashed red, as Λ is “zig-zagged” starting from a steady-state profile with normal jumps with r=1.2 and Λ=16. Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=20 and Δx=0.02.

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    • Figure 11
      Figure 11

      (a) Two paths plotted in dashed blue and dot-dashed red connecting (r1,Λ1)=(1.16,30) and (r2,Λ2)=(1.26,14). (b) Zoomed-in view of the first node of the initial profile c0(x) at (r1,Λ) and resulting profiles c1(x) and c2(x) after moving along paths 1 and 2 plotted in dashed blue and dot-dashed red. [(c) and (d)] Asymmetry Δ (blue circles) and first node location x1 (red crosses) along paths 1 and 2. Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=20 and Δx=0.02.

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    • Figure 12
      Figure 12

      Potential well V(c) for a normal jump with r=1.2 whose equilibria give the jumping points c− (blue circle) and c+ (red cross).

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    • Figure 13
      Figure 13

      Potential well V(c) for non-normal jump with r=1.2 after A(x0) is (a) decreased and (b) increased (solid lines). The original potential is shown in dashed lines. Equilibria represent the jumping points c− (blue circle) and c+ (red cross).

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    • Figure 14
      Figure 14

      (a) First node location x1 and (b) spatial wavelength λs of solutions in the discontinuous regime for r=1.12 and 1.22. Results are well approximated by x1=1.08+0.08Λ and λs=3.54+0.32Λ (r=1.12) and x1=1.12+0.17Λ and λs=3.16+0.66Λ (r=1.22). Other parameters are α,γ=0.3, β=0, ξ=1, and w=0 with L=200 and Δx=0.04.

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    • Figure 15
      Figure 15

      Alternans profiles c(x) (blue dots) and a(x) (dashed red) for random initial conditions [each point c(x) drawn independently from the uniform distribution U(−0.1,0.1)] for (a) initial CV parameter Λ=30 and (b) after slowly decreasing Λ to 10. Insets: Zoomed-in vein of the first nodal area. Other parameters are r=1.2, α,γ=0.3, β=0, ξ=1, w=0 with L=20 and Δx=0.02.

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    • Figure 16
      Figure 16

      APD and CV restitution curves calculated for the Shiferaw-Fox model for scaling parameter value τ=2.

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    • Figure 17
      Figure 17

      Effect of increasing the scaling parameter τ on the CV restitution curve for the Shiferaw-Fox model. In solid blue, dashed red, and dot-dashed black the CV restitution curves for τ=2, 6, and 10, respectively.

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    • Figure 18
      Figure 18

      CV restitution length scale Λ as computed numerically from the Shiferaw-Fox model over a range of values of the scaling parameter τ.

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    • Figure 19
      Figure 19

      Unidirectional pinning in the Shiferaw-Fox model via changing τ. (a) Calcium profiles c(x) as τ is increased from 9 to 10 and (b) calcium profiles c(x) as τ is restored to 9. (c) Second node location x2 (blue circles) and τ (dashed red) vs beat number.

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    • Figure 20
      Figure 20

      Unidirectional pinning in the Shiferaw-Fox model via changing BCL. (a) Calcium profiles c(x) as BCL is increased from 330 to 340 ms and (b) calcium profiles c(x) as BCL is restored to 330 ms. (c) Second node location x2 (blue circles) and BCL (dashed red) vs beat number.

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    • Figure 21
      Figure 21

      Asymmetry of node shapes in the Shiferaw-Fox model via changing τ. (a) Asymmetry Δ in calcium profiles c(x) of the Shiferaw-Fox model as τ is decreased from 10 to 6 and (b) representative calcium profiles c(x) for three values of Λ: Λ=41.76, 52.68, and 71.74 (blue circles, red crosses, and green triangles).

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    • Figure 22
      Figure 22

      Scaling of the spatial wavelength λs in the Shiferaw-Fox model as a function of the CV restitution length scale Λ as τ is slowly increased from 8 to 10. Results in the discontinuous and smooth regimes are obtained using BCL=330 ms (blue circles) and 345 ms (red crosses), respectively.

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    • Figure 23
      Figure 23

      (a) The critical onset value r1(Λ), (b) spatial wavelength λs, and (c) velocity v of smooth solutions near the onset of alternans as observed from numerical simulations of Eqs. (19) and (20) (blue circles) compared to our theoretical predictions given by Eqs. (A5), (30), and (A6) (dashed red) for nonzero APD restitution β=0.2. Other parameters are α,γ=0.3, ξ=1, and w=0.

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    • Figure 24
      Figure 24

      (a) The critical onset value r1(Λ) and (b) spatial wavelength λs of smooth solutions near the onset of alternans as observed from numerical simulations of Eqs. (19) and (20) (blue circles) compared to our theoretical predictions given by Eqs. (A12) and (A13) (dashed red) for nonzero asymmetry w=0.4. Other parameters are α,γ=0.3, β=0, and ξ=1.

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    • Figure 25
      Figure 25

      (a) Transient dynamics of the first node x1 after node movement is induced by decreasing Λ=50 to Λ=20 for coarse and refined spatial discretizations Δx=0.05 (solid blue) and 0.005 (dashed red). (b) Decay of the transformed variable y1 (see text for the definition of y1). Other parameters are r=1.15, α,γ=0.3, β=0, and ξ=1 with L=20.

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    • Figure 26
      Figure 26

      Example solutions for the limit of flat CV restitution (Λ→∞) on a bi-infinite cable with a node at x=0 for r ranging from 0.7 (blue curve with the smallest amplitude) to 1.3 (red curve with the largest amplitude). Other parameters are α,γ=0.3, β=0, ξ=1, and w=0.

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