Annihilation dynamics during spiral defect chaos revealed by particle models

Abstract

Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention. Many physical systems exhibit annihilation events during which pairs of objects collide and are removed from the system. These events occur in a number of soft-matter and active-matter systems that exhibit spatiotemporal patterning. For example, topological defects in nematic liquid crystals can develop motile topological defects that annihilate when they meet ^1,2. Pair-wise annihilation of defects or singularities also plays a role in a number of biological systems. In bacterial biofilms, for instance, imperfect cell alignment results in point-like defects that carry half-integer topological charge and can annihilate in pairs. These topological defects explain the formation of layers and have been proposed as a model for the buckling of biofilms in colonies of nematically ordered cells^3,4.

FIG. 1.

A Grayscale snapshots of $u$ showing spiral defect chaos in (top) the LR model and (bottom) the FK model with $A=25{\mathrm{c}\mathrm{m}}^2$. Indicated are the tips of clockwise (black stars) and counterclockwise (yellow stars) rotating spiral waves. Snapshots were taken at (left) $t^{\prime }=8\mathrm{m}\mathrm{s}$, (middle) $t^{\prime }=4\mathrm{m}\mathrm{s}$, and (right) $t^{\prime }=0\mathrm{m}\mathrm{s}$ before an annihilation event. Annihilation can be explained by a wave-block resulting from a depolarized area acting as a wall to spiral tip motion. B Single tip trajectories are shown from the FK model and the LR model (orange). The trajectories in the LR model show fewer pivots, supporting our result that the LR model has a larger diffusion coefficient than the FK model (see also Table I).

FIG. 2.

A Probability density of lifetime $\mathrm{\Gamma }$ of annihilating pairs in the two cardiac models. Inset is the probability density of $\mathrm{\Gamma }$ to visualize the short-lived tips, revealing that the abundance of short-lived spiral tips (< 15ms) was somewhat greater for the LR model relative to the FK model. This log-log histogram has 10 bins per decade. B Probability density of the termination times of simulations of the two cardiac models, along with an exponential fit (solid line).

FIG. 3.

A MSD of spiral tips versus temporal lag. Black lines indicate Brownian motion. B Mean radial velocity versus inverse distance between annihilating tips. Shaded bands represent 95% confidence intervals. Dashed line represents $dR/dt=0$.

FIG. 4.

Exponents of power laws fit to MSD plotted as a function of lag window for the FK model (left) and the LR model (right). The resulting exponents are visualized using a color scale and 95% confidence intervals were determined by ordinary least squares. Gray shaded regions indicate where the 95% confidence interval contained the slope of 1, which corresponds to Brownian motion. The remaining regions exhibited a statistically significant difference from Brownian behavior ($p<0.05$). Indicated by the black dots are the fits reported in the text.

FIG. 5.

MSR between annihilating tips versus time until annihilation from simulations of the FK and LR models with shaded regions corresponding to 95% confidence intervals. Also shown are the fits of MSR from the OPM (solid lines) and the LPM (dashed lines).

FIG. 6.

A MSR between annihilating tips versus time until annihilation from simulations of the FK and LR models, using a larger domain size of $A=39.0625{\mathrm{c}\mathrm{m}}^2$, with shaded regions corresponding to 95% confidence intervals. The solid lines correspond to fits from the OPM while the dashed lines correspond to fits from the LPM. B Computed attraction coefficient versus domain size. C Sum of attractive and diffusive forces versus domain size. Error bars indicate 95% confidence.

FIG. 7.

Mean annihilation rate versus number density for spiral tips from the cardiac models (symbols) and their linear particle model fits (dashed lines).

FIG. 8.

A Mean annihilation rate versus number density obtained from the LPM using parameters corresponding to the FK model for different values of a (indicated by the inset color bar). Black lines are guides to the eyes, corresponding to power laws with exponent 4/3 (upper curve) and 2 (lower curve). B Power law exponent as a function of a computed using the LPM with parameters corresponding to both the FK and LR model. C Corresponding power law magnitude versus a. Black circles in B&C represent values of a corresponding to the cardiac models. Fits considered ordinary least squares over the interval $n\in \left[0.2,1\right]{\mathrm{c}\mathrm{m}}^{-2}$.

FIG. 9.

A Mean creation rate (triangles) and mean annihilation rate (dots) versus number density for spiral tips from the cardiac models using different domain sizes. Dashed lines correspond to power law fits (Table III. Black dots correspond to the mean particle density. B Probability density of termination times of the LPM for increasing values of $a$ equally spaced from $a=1{\mathrm{c}\mathrm{m}}^2/\mathrm{s}$ to $a=5{\mathrm{c}\mathrm{m}}^2/\mathrm{s}$. Parameter values correspond to the FK model and $A=25{\mathrm{c}\mathrm{m}}^2$.

FIG. 10.

A Average tip number as a function of $A$ computed using the cardiac models (symbols), along with the linear prediction of Eq. $12 B Average tip number as a function of $a$ computed using the LPM with parameter values corresponding to $A=25{\mathrm{c}\mathrm{m}}^2$. The darkened symbols correspond to the value of $a$ representing the cardiac models. C Mean termination time versus $A$ computed using Eq. $15 (dashed lines) and separately obtained from the cardiac models (symbols). D Mean termination time as a function of $a$ (using parameter values for $A=25{\mathrm{c}\mathrm{m}}^2$). Black circles correspond to $a$ obtained by fitting the cardiac models.

FIG. 11.

A MSR from the FK model with the excitability parameter, ${\tau }_d$, increased by 20%. The blue shaded region represents 95% confidence intervals estimated via bootstrap. The black line represents the fit to the OPM. B MSR from the LR model with the extracellular potassium concentration, ${\left[K^{+}\right]}_o$, decreased by 29.6%. The orange shaded region represents 95% confidence intervals estimated via bootstrap. The red line represents the fit to the OPM.