Drift and breakup of spiral waves in reaction-diffusion-mechanics systems

Abstract

Rotating spiral waves organize excitation in various biological, physical, and chemical systems. They underpin a variety of important phenomena, such as cardiac arrhythmias, morphogenesis processes, and spatial patterns in chemical reactions. Important insights into spiral wave dynamics have been obtained from theoretical studies of the reaction-diffusion (RD) partial differential equations. However, most of these studies have ignored the fact that spiral wave rotation is often accompanied by substantial deformations of the medium. Here, we show that joint consideration of the RD equations with the equations of continuum mechanics for tissue deformations (RD-mechanics systems), yield important effects on spiral wave dynamics. We show that deformation can induce the breakup of spiral waves into complex spatiotemporal patterns. We also show that mechanics leads to spiral wave drift throughout the medium approaching dynamical attractors, which are determined by the parameters of the model and the size of the medium. We study mechanisms of these effects and discuss their applicability to the theory of cardiac arrhythmias. Overall, we demonstrate the importance of RD-mechanics systems for mathematics applied to life sciences.

Fig. 1.

Spiral wave breakup caused by mechanical activity. (a) Spiral wave rotation in a RD system based on Eqs. 1–3 and in the absence of deformation with u_c = 0.15, τ₁ = τ₂ = τ₃ = 30, and τ₄ = 3. (b) Similar computations in a deforming medium using Eqs. 1–8 with G_s = 0.03. Both snapshots are taken at 1,800 [t.u.]. The medium consists of 513 × 513 grid points and 16 × 16 mechanical elements, each containing 33 × 33 grid points.

Fig. 2.

Breakup in a model with biophysical activation and inactivation of the fast inward current I_fi. (a) The scaled activation and inactivation curves from the TNNP model (24) (black lines). (The values are scaled from the original values of voltage to the interval of voltage between 0 and 1 for the Fenton–Karma model.) Plotted are the scaled curves of h_∞(u) × j_∞(u) (solid black line) and of m³_∞(u) (dashed black line) from ref. . The gray lines show the Heaviside description of the activation (dashed gray line) and inactivation (solid gray line) curves in Eqs. 1–3. (b) Spiral wave breakup in a model with biophysical activation and inactivation of the fast inward current I_fi (see text for details). The snapshot is taken at 1,200 [t.u.], g_fi = 14.2, and all other parameters and initial conditions are the same as in Fig. 1.

Fig. 3.

Mechanism of breakup caused by mechanical activity. (a) Time courses of voltage u (black trace in Upper), stretch-activated current I_s multiplied by 100 (green trace in Lower), the fast inward current I_fi (blue trace in Lower), and the variable v (red trace in Lower), which are responsible for the inactivation of I_fi at the point marked by the filled square in b. The dashed line is located at 560 [t.u.], and the pink line is at u = 0.15. (b) Fragmentation of the spiral wave, showing the same simulation as in Fig. 1b but at time 560 [t.u.]. All parameters and conditions are the same as in Fig. 1b. (c) Magnification of the dashed rectangular region from a. (d) A similar computation as in a but in the absence of I_s (G_s = 0 in Eq. 8).

Fig. 4.

Spiral wave drift caused by mechanical activity. (a) Initial position of a spiral wave tip (black circle) and the final state of the spiral wave after 120 rotations (shaded image) in a medium containing 141 × 141 grid points with 11 × 11 points per mechanics element and G_s = 0.01. The arrow indicates the direction of spiral wave drift, and the solid curve illustrates the trajectory of the spiral tip. (b) A similar simulation to that in a but in a medium containing 151 × 151 grid points. Other parameter values were the same as those for Fig. 1b, except u_c = 0.25, τ₁ = 95, and τ₃ = 300.

Fig. 5.

Characteristics of meander patterns. (a) R₀ and R₁ in [s.u.] versus medium size (in grid points). (b) f₀ and f₁ in [t.u.]⁻¹ versus medium size.