Abstract
This study uses a probabilistic cellular automata (PCA) to model the spatial lattice arrangement of calcium release units (CRUs) within cardiac myocytes. The CRUs are subject to random activation, nearest-neighbor recruitment, and temporal refractoriness, and their interactions produce a physiologically-important condition called calcium alternans, a beat-to-beat oscillation in the amount of calcium released. In the PCA this manifests as a transition to period-2 behavior in the fraction of activated lattice sites. We investigate this phenomenon using PCA simulations and moment-closure approximation methods of zero order (mean-field), first order (pair), and second order (quartet). We show that only the quartet approximation (QA) accurately predicts the thresholds of the activation and recruitment probabilities for the onset of periodic behavior (alternans), as the lower-order approximations do not sufficiently account for important spatial correlations. The QA also accurately predicts the emergence of spatio-temporal clustering in the PCA, providing an analytical framework for investigating pattern formation dynamics in such models. Our analysis demonstrates a systematic approach to efficiently handling the increased combinatorial complexity of the QA, whose required computation time is nontrivially larger compared to the mean-field approximation but remains an order of magnitude lower than the numerical PCA simulations.
Highlights
Probabilistic cellular automata (PCA), a class of binary lattice models, are fairly ubiquitous in modeling complex systems, in fields ranging from physics to ecology to economics [for a selection of examples see [1]]
Plots of the observed and predicted amplitude A and clustering coefficient Q are given in Figure 9 and Figure 10, allowing us to compare the performance of the three approximations against the PCA simulation results
In this study we found that the moment-closure technique using quartet approximation provides a very good prediction of several macroscopic quantities of a probabilistic cellular automata model with two-stage update rules
Summary
Probabilistic cellular automata (PCA), a class of binary lattice models, are fairly ubiquitous in modeling complex systems (along with agent-based models), in fields ranging from physics to ecology to economics [for a selection of examples see [1]] While their appeal stems from their intuitive, bottom-up approach in which the local interactions of randomly-acting particles, agents, or other units are explicitly represented, their importance is in revealing large-scale (macroscopic) phenomena that result in unexpected ways from the various local (microscopic) actions. In higher-order versions, the joint state of larger structures is explicitly modeled using coupled difference equations, while still treating yet more distant neighbors as average These larger structures can be a pair of sites (first-order nearest-neighbor correlations), or a triplet or quartet of sites (second-order neighbor-of-neighbor correlations), or even larger. Higher-order approximations have been successfully used in models for, e.g., ecological dispersion [8, 9] and viral spread in epidemiology [10]
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