Pacemaking function of two simplified cell models

Figures

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  Fig 1. Timeline of simplified physiological cell models development. Solid shapes correspond to excitable cell models, dashed shapes—to pacemaking cell models, respectively.

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  Fig 2. Single-cell nonlinear dynamics. A. AP and pAP models. A.1. Nullclines and phase portrait for the conventional AP model after suprathreshold stimulation. A.2. Nullclines and phase portrait for the pAP model with different bifurcation parameter bAP. Dashed and dotted lines correspond to u- and v-nullclines, respectively. Nullcline u = 0 is not shown. Stable limit cycles for bAP = 0.01, 0.03, and 0.05 are shown with corresponding EPs marked by squares of the same color. A.3. Action potentials, colors correspond to the curves in panel A.2. B. CN and pCN models. B.1. Nullclines and phase portrait for the conventional CN model after suprathreshold stimulation. B.2. Nullclines and phase portrait for the pCN model with different bifurcation parameter bCN. Dashed and dotted lines correspond to u- and h-nullclines, respectively. Nullcline u = 0 is not shown. Stable limit cycles for bCN = 0.12, 0.3, and 0.44 are shown with corresponding EPs marked by squares of the same color. B.3. Action potentials, colors correspond to the curves in panel B.2.

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  Fig 3. Schematics of the simulation examples. A. 2D SAN model with the pAP-AP cells. B. SAN structure with a border of passive tissue and exit pathways. C. 3D intestine model with the pCN-CN cells.

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  Fig 4. Dependence of pAP cell characteristics on various parameters. A. On the bifurcation parameter bAP. B. On the parameter ϵ0. C. On the parameter μ1. D. On the parameter μ2. E. On the external current Iext. Left panels correspond to the bifurcation diagrams, central—to the calculated frequencies, and right panels—to the calculated DI on APD ratios.

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  Fig 5. Dependence of pCN cell characteristics on various parameters. A. On the bifurcation parameter bCN. B. On the slope factor us, C. On the parameter ugate. D. On the external current Iext. Left panels correspond to the bifurcation diagrams, central—to the calculated frequencies, and right panels—to the calculated DI on APD ratios.

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  Fig 6. A. 1D pAP-AP coupled system. A.1. Minimal and maximal 1:1 synchronized frequency dependence on the coupling coefficient d for the pAP cell coupled with n = 2 (red), 1 (green), and 0.5 (blues) strands of 20 AP cells. Inset illustrates a schematic representation of the 1D pAP-AP coupled system. Ellipse symbolizes pacemaker cell and rectangles—excitable cells. A.2. Minimal and maximal values of the parameter bAP vs d corresponding to panel A.1. B. 1D pCN-CN coupled system. B.1. Minimal and maximal 1:1 synchronized frequency dependence on the coupling coefficient d for the pCN cell coupled with n = 2 (red), 1 (green), and 0.5 (blue) strands of 20 CN cells. Inset illustrates a schematic representation of the 1D pCN-CN coupled system. B.2. Minimal and maximal values of the parameter bCN vs d corresponding to panel B.1.

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  Fig 7. Synchronization in the simplified 1D coupled model systems with fixed parameters b. Dependence of synchronization in the pAP-AP (A) and pCN-CN (B) systems with a different number of excitable cell strands n on the coupling parameter d. Solid lines correspond to the pacemaker frequency and the dotted lines—to the frequency at the 16th excitable cell in the strand(s). C. Action potentials for the pAP-AP system for the number of strands n = 0.5. The bold red and blue lines mark the action potentials of the pacemaker cell and 16th cell of a strand, respectively. D. The same as in panel C, but for the pCN-CN system.

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  Fig 8. 2D simulation of SAN. Calculated time sequences of the transmembrane potential u along the bold black line in Fig 3A and activation sequences for SAN structure without (A–C) and with (D–F) walls of passive tissue. A. For diffusion coefficient D = 0.090 mm2ms-1. B. D = 0.048 mm2ms-1. C. Activation sequence corresponding to the case shown in panel A. D. DA = 0.160 mm2ms-1, DS = 0.060 mm2ms-1. E. DA = 0.160 mm2ms-1, DS = 0.052 mm2ms-1. F. Activation sequence corresponding to the case shown in panel D.

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  Fig 9. 3D simulation of intestine. A. Snapshots of spatial distributions of the transmembrane potential uI in the ICC layer at different time moments t. Brighter areas correspond to higher potential. From top to bottom: (1–5) formation of constant frequency plateaus, (6–11) onset and evolution of intestinal dysrhythmia pattern due to temporary conduction block induced at t = 8100 s to both ICC and SMC layers. B. Distributions of the intrinsic frequency (DI = DM = 0, solid black line), entrained frequency of oscillators at t = 8000 s (solid red line), and at the end of the simulation (t = 11600 s, dashed blue line). C–D. uI (red solid lines) and uM (blue dot-dashed lines) action potentials at the distance of 170 mm (C) and 1200 mm (D) at the beginning of the simulation (t ≃ 100 s). E–F. The same as in panels C and D, but at t ≃ 8000 s.