Community effect of cardiomyocytes in beating rhythms is determined by stable cells

Figures

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  Figure 1. Schematic diagram of the trajectory of the state variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}(t)(i=\mathrm{1,2)}$$\end{document}φi(t)(i=1,2). The circle represents the trajectory of a state variable in the phase space of cardiomyocytes from one firing to the next firing. If cell-2 fires at a time t (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{2}(t)$$\end{document}φ2(t) = 0) and cell-1 is not in the refractory period, then cell-1 fires at the retardation time τ after cell-2 fires.

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  Figure 2. The mean beating rate and beating fluctuation after synchronisation. Numerical simulations for the 14 pairs of caridiomyocytes (28 cardiomyocytes) before synchronisation in the experiments reported previously29 were performed using our integrate-and -fire model. Experimental values (circles) and theoretical values (filled circles) are plotted for (a) the mean beating rate and (b) beating fluctuation (CV%). For all numerical simulations, we used the same parameter values τ = 0 and μ = 6.5. The fluctuation in beating rhythm is expressed by the CV. Mean beating rate and fluctuation for the 14 pairs after synchronisation are also shown in the Supplementary Information (Supplementary Table S2).

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  Figure 3. Comparison of experimental data and the two models. The change in beating fluctuation before and after synchronisation is shown. The blue circles and brown squares represent the corresponding mean values for 1 min of beating fluctuation of cell-1 and cell-2, respectively. Panels a–c show the results for Case (i), which was a case of synchronisation to a cardiomyocyte with a fast and stable beating rhythm. (a) The experimental result, (b) the numerical result of our model, and (c) the numerical result of the Kuramoto model with (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\omega }}_{1},{\bar{\sigma }}_{1}$$\end{document}ω¯1,σ¯1) = (9.80, 0.94), (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\omega }}_{2},{\bar{\sigma }}_{2}$$\end{document}ω¯2,σ¯2) = (5.09, 1.45). Panels d–f show the results for Case (ii),…

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  Figure 4. Configurations and the order of placing cells. Three types of networks of configuration are shown. (a) Star network, (b) 2D lattice network, and (c) 1D lattice network. A cardiomyocyte is represented as a circle and it interacts with another cardiomyocyte if they are connected by a line. Cardiomyocytes are connected in ascending order according to the numbers in the circles from 1 to 20.

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  Figure 5. Size dependence of fluctuation for three types of configuration and for a large network. Size dependence of fluctuation is shown in double logarithmic graphs. The components of the network are model cardiomyocytes with the same characteristics. Panels a–c show the size dependence of fluctuation for three types of configurations: (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =\mathrm{9.80,}\,\sigma =\mathrm{0.69,}\,\theta =2.94$$\end{document}ω=9.80,σ=0.69,θ=2.94, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =\mathrm{5.00,}\,\sigma =\mathrm{1.01,}\,\theta =1.50$$\end{document}ω=5.00,σ=1.01,θ=1.50 and (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

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  Figure 6. Configurations of a combination between a referential network and a single cell or an assembly of cells. In panels a and b, the filled circles denote a single cell, which adds to the referential network, of which cells are denoted by open circles. (a) Referential network of four cells + a single cell and (b) that of nine cells + a single cell. Panels c and d show the configurations of combined subsystems of four cells and those of nine cells. The cells in referential networks are denoted by open circles and those in counterparts are denoted by filled circles. (c) Referential network of four cells + four cells and (d) that of nine cells + nine cells.

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  Figure 7. Change in the beating fluctuation before and after synchronisation. (a) The referential subsystem is the 2D lattice network and the counterpart is the single cell with a fast and unstable beating rhythm. (b) The change in mean value of beating fluctuation. The data for the referential networks and the counterparts are shown by circles and squares, respectively. The circles and squares show the corresponding mean values for 1 min of beating fluctuation. The results for the other combined systems (c,e,g) are shown similarly in (d,f,h), respectively.

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  Figure 8. Schematic explanation of why the beating rhythm tend to be synchronised to that of more stable cardiomyocytes after connection of two cardiomyocytes. A stable cardiomyocyte can be compared with a heavy pendulum and an unstable cardiomyocyte with a light pendulum. (a) External fluctuation has little effect on a pendulum’s period of swing if its weight is heavy, but has strong effects if its weight is light. (b) When two pendulums are coupled and synchronised, their period of swing is close to that of the heavier pendulum, and fluctuation will be reduced because the total mass of weight increases.