Metabifurcation analysis of a phenomenological ventricular cell model
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Heart disease is the leading cause of death in America, and arrhythmia is considered one of the most important precursors of heart attacks. We cannot predict arrhythmia directly; however, it has been shown that cardiac alternans is closely related to arrhythmia. Therefore, predicting alternans could be the _rst step in preventing arrhythmia. Fenton and Karma(1998) came up with the Fenton-Karma model (the F-K model) with three variables and 13 parameters that is a relatively simple and basic model that including information on alternans. Our research is based on the F-K model at the cell level rather than the tissue level. We study the parameter space of the F-K model and discover robust correlations between parameters and dynamical responses. The relations cannot be disclosed by some statistical methods like principal component analysis(PCA) or K-means clustering because we cannot see the dynamical behaviors of parameter sets. The links instead emerge when the parameter space is partitioned according to bifur- cation responses. We call this general method \metabifurcation analysis". Concretely, the bifurcation responses are shown by the bifurcation plots, which show the model re- sponses, like action potential duration(APD) restitution curves, to a sequential change in pacing frequency. According to the bifurcation patterns, we partition the parameter space in _ve \parent families". After investigating and characterizing them in depth, we subtly classify the largest parent families into four subfamilies. We discuss their essential differences in qualitative dynamical behaviors like whether they exhibit bistability and how solutions change after alternans appears. The partitioning cannot be achieved by the investigation of only a small number of parameter sets. To obtain a relatively large number of parameter sets to make the results reliable, we implement a particle swarm optimization process with parallel pro- gramming interfaces like OpenMP and Open MPI to search the parameter space and obtain 1,525,833 physiologically admissible parameter sets; then, we randomly choose a representative sample of 270,000 parameter sets to do the bifurcation analysis. After doing the metabifurcation analysis on the the F-K model, some results are listed as follows. First, more than 70% parameter sets from 270,000 meaningful parameter sets do not or only have one period-doubling bifurcation. Second,t-w plays the leading role of identifying subfamilies in the parent family(PF2) with only one period-doubling bifurcation; Moreover, in one subfamily(F3), there is a positive linear relation between t-w and the appearance of bistability.