Timothy J Tyree ,Patrick Murphy, and Wouter-Jan RappelPair-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.
A increasing aBC Fig. 3 Results of particle models when changing a\textbf{a} Mean annihilation rate density versus number density.Power laws were fit from the linear particle model for increasingvalues of the attraction coefficient, $a$, as indicated by (inset) the color bar. \textbf{b} Power law exponent as a function of $a$ for (blue) the FKmodel and (orange) the LR model.\textbf{c} Power law magnitude versus $a$. 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 cm2, 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.
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These fits demonstrate that the LPM can accurately replicate the annihilation rates of the cardiac models (MPE <4%). Importantly, as was the case for a, these fits use the same parameter values for both system sizes. Also note that simulations of the particle model are much more efficient than the cardiac models, especially for large domain sizes. Specifically, the particle model simulations use O((L/x)2 N2 avg) fewer operations per time step, where Navg is the average number of particles. For example, for A = 100 cm2, the speed-up exceeded 104-fold.
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This rate was found to increase with increasing values of a, which can be understood by realizing that larger attractive forces result in distant particles coming closer together faster. Importantly, however, we found that the rate was always fitted well by the power law w = Mn (Fig. 8A). For both models, we found that for increasing values of a the exponent became smaller (Fig. 8B) while M increased (Fig. 8C).
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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 [0.2, 1] cm2. FIG. 7. Mean annihilation rate versus number density for spiral tips from the cardiac models (symbols) and their linear particle model fits (dashed lines).
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