Resonant model—A new paradigm for modeling an action potential of biological cells

Figures

• 

  Fig 1. The modeling workflow. Schematic illustration of the reproduction of a biological cell action potential by the Resonant Model. The model coefficients are determined with the fitting algorithm that provides the best match for the reference action potential waveshape with a minimum least square error. The state controller is designed to respond to the changes in the normal operating conditions of a cell.

• 

  Fig 2. Rabbit sinoatrial node action potential waveshape and it’s frequency spectrum. A: Kurata et al.’s model-generated behavior of spontaneous APs [22]. B: Single sided amplitude spectrum of the AP waveshape generated from Kurata et al.’s model [22] output. 5 PS represents the width of the frequency spectrum covered by the Fourier series in the fifth partial sum, and 8 PS represents the width of the frequency spectrum covered by the Fourier series model in the eighth partial sum.

• 

  Fig 3. Resonant model of a biological cell. A: Schematic representation of a resonant model. B: An oscillator implemented with two integrators in series generating sine term and cosine term of the same frequency simultaneously. C: The waveshape generator implemented in SIMULINK with eight Fourier terms. D: Pictorial representation of the compositional property (series and parallel connections) of the RM cells.

• 

  Fig 4. Coupling between two Resonant model sinoatrial node cells. A: Parameters of the action potential of sinoatrial node cell. MDP is a maximal diastolic potential, Vmax is the maximum potential, and VTh is the threshold potential. B: State controller of the Resonant model of a sinoatrial node cell. f0 is the intrinsic beating frequency of a cell and fc is the entrained frequency C: Diagram of the two Resonant model cells (Cell 1 and Cell 2) coupled with a gap junction.

• 

  Fig 5. Minimum root mean squared error piecewise linear fits to the coefficients of the Resonant model (RM) of a human sinoatrial node cell at different levels of If blockage. The black diamonds represent the RM coefficients obtained after fitting to AP waveshapes of Fabbri et al. model for specific percentage values of If blockage. Dotted lines represent the piecewise linear fits to the coefficients. Different colors represent different linear equation fits. Coefficients of every oscillator in the RM are placed together in a black rectangular box. ω represents the fundamental frequency.

• 

  Fig 6. Selection of the small error and resource utilization friendly fitting curve to the coefficients for the Resonant model optimization. A: Percentage errors for different levels of If block and root mean squared errors (displayed at the top of every fit) for various types of fits to the resonant model coefficients. Piecewise linear with one breakpoint (PL2), piecewise linear with two breakpoints (PL3), piecewise linear with three breakpoints (PL4), piecewise cubic with one breakpoint (PC2), degree 7 polynomial (PO7), and degree 8 polynomial (PO8). B: Percentage root mean squared error (RMSE) for 20 If blockage percentage values when: 1) all the coefficients were fitted with minimum RMSE piecewise linear fits only (dark blue), 2) when all the coefficients were fitted with minimum RMSE fits, i.e., mixed fit types (light blue).

• 

  Fig 7. State controller of Resonant model for ventricular myocyte. Vm = membrane potential, Vrest = resting membrane potential, Vsti = voltage stimulus to a cell, Vth = threshold membrane potential, Vmax = maximum membrane potential, and Vwgout = membrane potential from waveshape generator.

• 

  Fig 8. The contribution of fundamental and seven harmonics to produce an autorhythmic cell AP. AP waveshape (red) is the result of the superposition of fundamental and seven higher frequency components (blue) of Fourier series.

• 

  Fig 9. AP morphologies of autorhythmic cells. A: Comparison of rabbit SAN cell AP morphology (black solid) from Kurata model and AP morphologies obtained from RM with 5 (green dotted), 8 (blue dashed), and 20 (red two dashes) harmonics. B: Violin plots of errors with the variation in the no. of harmonics in the finite Fourier series for reproducing a rabbit SAN cell AP. Also shown is the root mean squared error corresponding to each harmonic. w5 and w20 indicates the maximum thickness of the violin plot of errors for 5 and 20 partial sums, respectively. C: RM (12 harmonics) fit to experimental human sinoatrial node cell’s AP. Note: The experimental data has been regenerated from Fig 5A in [33] D: Comparison between experimental and simulated AP features of human SAN cell. Here, MDP is Maximum diastolic potential; APD20, APD50, and APD90 are action potential durations at 20, 50 and 90% repolarization, respectively; OS is the overshoot; APA is the action potential amplitude; DDR100 is the diastolic depolarization rate over the 100 ms time interval starting at MDP.

• 

  Fig 10. Mutual entrainment of SAN cells. A: Simultaneous recordings of membrane potential of two rabbit SAN Resonant model cells, with the cells coupled for the period indicated by the horizontal two-headed arrow. B: Common interbeat interval (IBI) (as a percentage of the IBI of the intrinsically faster beating cell, IBIf) versus the IBI of the slower beating cell of the coupled cell pair (also as a percentage of the IBIf). Results obtained from a simulation of two Kurata et al. model pacing cells are represented by blue stars [32]. Magenta circles show the experimental results of isolated real rabbit SAN cell pair; green plus represents the results from coupling of real rabbit SAN cell with the model cell [35], and the solid red line shows the common IBI for resonant model SAN cell pair.

• 

  Fig 11. Human sinoatrial node detailed electrophysiology model and human sinoatrial node Resonant model responses to perturbation in an ion transport pathway. A: Action potential waveforms simulated in human sinoatrial node cell electrophysiology model. B: Action potential waveforms simulated in human sinoatrial resonant model under control conditions (CTRL) (dark black solid line) and upon 10% (dotted line), 25% (dashed line), 65% (dash-dot line), and 85% (black solid line) block of If. C: Simulated effects of different levels of If blockage on cycle length (CL), maximum diastolic potential (MDP) and diastolic depolarization rate (DDR100).

• 

  Fig 12. Extension of the Resonant model to additional cell types. A: Simulated resonant model AP morphology (solid red line) in comparison with experimental trace (black dots) of squid giant axon membrane potential. B: Human ventricular myocyte APs generated by the simulation of resonant model (red solid line) and O’Hara et al. model (black dashed line) C: Comparison of simulated O’Hara et. al (blue hatched), resonant model (red) and experimentally measured (black) basic AP parameters of human ventricular myocyte for a single paced beat from quiescence. Shown, from left to right, are the negative resting membrane potential (Vrest), maximum upstroke potential (Vmax), and action potential duration at 30, 50, 70, and 90% repolarization.