Improved Prediction of Periodicity Using Quartet Approximation in a Lattice Model of Intracellular Calcium Release

Abstract

This study uses a probabilistic cellular automata (PCA) to model the spatial and temporal dynamics 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 non-trivially larger compared to the mean-field approximation but remains an order of magnitude lower than the numerical PCA simulations.