Four Ways to Fit an Ion Channel Model

Figures

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  Figure 1. Voltage protocols, currents measured in cell #5, and simulated phase diagrams for all six protocols (Pr2–7) used in this study. Simulations for the phase diagrams were performed with the cell #5 parameters obtained in Beattie et al. (20). (A) Pr2 is used to measure the experimental time constant of activation. We show the voltage-step protocol (top left), the resulting current as measured in cell #5 (lower left), and a phase plane diagram (right). The protocol is repeated six times, with an increasing duration of the P1 step for each repeat. This is indicated in the plots by using a different color for each repeat. (B) Pr3 is used to measure the experimental steady state activation curve. It is repeated seven times, with a different voltage for the P1 step at each repeat. (C) Pr4 is used to measure experimental time constants of inactivation; it repeats 16 times with a different voltage for step P3. (D) Pr5 is used to measure experimental time constants of activation and inactivation, as well as steady-state inactivation. It has nine repeats with a different voltage for step P2. (E) Pr6 consists of several (healthy and pathological) AP wave forms and was used as an independent…

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  Figure 2. Summary curves calculated from the voltage-step protocols Pr2–5 for all nine cells. The mean for all cells is indicated with a dotted line. Note how the general trend, not always obvious from the single-cell results, is clearly captured by the line connecting the means. (A) The experimental activation curve ã∞ derived from Pr3 is shown. (B) The experimental steady state of inactivation r˜∞ derived from Pr5. (C) The IV curve (peak current during the P2 step of Pr5) plotted against the P2 voltage. (D) The experimental time constant of activation τ˜a. Values for V < 0 mV were estimated from Pr5, the single value at 40 mV from Pr2. (E) The experimental time constant of inactivation τ˜r. Values for V < −30 mV were estimated from Pr5, values for −40 mV and upwards from Pr4. To see this figure in color, go online.

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  Figure 3. Boundaries and transforms on the parameter space and the effect of solver tolerances. (A) Constraining the transition rates k1–k4 (over physiologically relevant voltages) to relevant timescales creates a two-dimensional boundary on each parameter pair (Supporting Materials and Methods, Section S2.2). The gray region indicates valid samples, with an upper rate constraint in orange and a lower rate constraint in blue. The horizontal and vertical gray lines indicate the lower and upper bounds for the parameters p1–p8. (B) The same parameter constraints with a log transform on one of the parameters are shown. Note this increases the size of the feasible region for off-the-shelf optimizers that know nothing about the problem, and we later see the objective function becomes reasonably convex and symmetric under this transform (see Fig. 11). (C) Using default tolerances of adaptive solvers can lead to numerical noise in the function being optimized. This can be remedied by selecting lower tolerances. To see this figure in color, go online.

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  Figure 4. Method 1 goodness of fit and cross-validation on cell #5. Experimental approximations of the steady states (ã∞ and r˜∞) and time constants (τ˜a and τ˜r) are shown, derived from measurements in cell #5. The model curves for a∞(V), r∞(V), τa(V), and τ(V) are shown for models fitted with each of the four methods. Note that only method 1 was trained on this data, making this a goodness-of-fit figure for method 1, whereas for the other methods, this figure shows a prediction. To see this figure in color, go online.

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  Figure 5. Method 2 goodness of fit and cross-validation on cell #5. The experimental steady state of activation, the IV curve, and both experimental time constants are shown. By simulating the protocols and performing the same analysis, similar summary curves were derived for each of the four methods. The fit for method 2 is shown, along with the predictions from models made with methods 1, 3, and 4. To see this figure in color, go online.

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  Figure 6. Method 3 goodness of fit and cross-validation on cell #5. The top row shows the voltage protocols Pr2–5, with different colors used for each sweep. The same colors are used in each of the panels below to show the corresponding experimental data, whereas simulation results are shown in black. In the top row, the full time span of each protocol is shown, but the plots shown below each zoom in on a specific region, indicated by the shaded area in the top row. The fit made with method 3 is highlighted with blue borders (fit 3) while the other rows show predictions from models made with methods 1, 2, and 4. To see this figure in color, go online.

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  Figure 7. Method 4 goodness of fit and cross-validation on cell #5. The top panel shows the voltage protocol (Pr7) while the three panels below show the same current recording (gray line), and each row shows a prediction from a model made with methods 1–3, respectively (colored lines). The bottom panel, highlighted with blue borders, shows the fit for method 4 (fit 4). To see this figure in color, go online.

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  Figure 8. Validation on the AP waveform signal (Pr6). The voltage protocol is shown on the top row, with subsequent rows colored traces showing predictions from methods 1–4, respectively. The current measured in cell #5 in response to Pr6 is shown in gray (same in each row). None of the models were trained on Pr6 data, making this an independent validation. The left panels show the full duration of the signal, and the panels on the right zoom in on selected regions (2.41–2.48, 5.5–7.3, and 7.25–7.9 s). To see this figure in color, go online.

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  Figure 9. Validation and cross-validation results. (Top) Results for cell #5 are shown. Each row shows the relative RMSE for a fitting method, scaled so that the best performing method is indicated by 1, whereas a method with a relative score of, e.g., 1.2 had an RMSE that was 1.2 times larger. The top row shows the relative EAP for each method, with the remaining rows showing EM1, EM2, EM3, and EM4, respectively. (Bottom) Mean results over all nine cells are shown, with SDs given in parentheses. To see this figure in color, go online.

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  Figure 10. The percentage of optimizations from different starting points that found the “best result.” Darker bars in the background indicate the percentage of optimizations that returned a result with an RMSE within 1% of the best result found. The lighter bars in the foreground indicate the percentage that was also close in parameter space, (such that no single parameter varied by more than 1% from the best result found). Note that for methods 3 and 4, the bars overlap exactly, but the method 2 results for some cells contain “‘best” points with similar low RMSEs but different parameter values. To see this figure in color, go online.

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  Figure 11. An exploration of two-dimensional slices of the objective functions. (Top) EM2 for cell #5 is shown, near the optimum found by method 2 (indicated by an orange “×”). Each panel shows the result of varying two parameters, with the other seven held constant. The parameters varied in the leftmost panels determine the rate of activation, followed by deactivation in the second panel, inactivation in the third, and finally, the rate of recovery. The white lines indicate the boundaries used during optimization. (Lower) An exploration of EM4 for cell #5 is shown, near the optimum found by method 4. The darkest purple color in the figure represents areas where either summary curve derivation (top panel) or simulation (lower panel) failed. To see this figure in color, go online.

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  Figure 12. Best solutions returned by the four methods for all cells. The activation rate parameters p1 and p2 are shown in (A), deactivation in (B), and inactivation and recovery in (C) and (D), respectively. To see this figure in color, go online.