# Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
## 4 Numerical Experiments
### Estimating Time-Varying Applied Current via Parameter Tracking
The results in Figure 7 show that when \(\sigma_{\xi}=10\), although the EnKF mean estimate was able to well track the true solution, the estimated \(\pm 2\) standard deviation curves around the mean are very large around the mean, reflecting a lack of confidence in the estimate. On the other hand, when \(\sigma_{\xi}=0.1\), the filter is unable to well track the true parameter and eventually diverges. In this case, while the filter is unable to track the true parameter, the estimated \(\pm 2\) standard deviation curves are very tight around the mean, implying a high confidence in an incorrect estimate. For this example, the choice of  
Figure 5: Resulting EnKF with parameter tracking estimates of \(V(t)\), \(n(t)\), \(m(t)\), \(h(t)\) (top, from left to right), and applied current \(I(t)\) (bottom) from the data obtained from the pulsing step current in (35). In each panel, the EnKF estimated mean is shown in solid red while the true solution is shown in solid black. The dashed red lines show the estimated \(\pm 2\) standard deviation curves around the mean. Figure 6: Resulting EnKF with parameter tracking estimates of \(V(t)\), \(n(t)\), \(m(t)\), \(h(t)\) (top, from left to right), and applied current \(I(t)\) (bottom) from the data obtained from the sinusoidal current \(I(t)=10\sin(0.2t)+10\)\(\mathrm{mA/cm^{2}}\) in (36). In each panel, the EnKF estimated mean is shown in solid red while the true solution is shown in solid black. The dashed red lines show the estimated \(\pm 2\) standard deviation curves around the mean. \(\sigma_{\xi}=0.5\) in the parameter random walk visually captures the true underlying parameter the best out of the values considered, as it keeps the EnKF estimated mean closely with a small standard deviation around the mean. Frequency of data availableIn addition to the choice of \(\sigma_{\xi}\), the frequency of time series data available also has a significant effect on the resulting parameter tracking estimates. To analyze this on the problem at hand, we subsampled the available voltage data corresponding to the pulsing step current (35) and sinusoidal current (36) every 10, 20, and 50 time points, resulting in data sets containing 201, 101, and 41 equidistant voltage observations over the time interval \([0,200]\). This means that data was considered in the observation step in the EnKF every first, second, or fifth millisecond (as compared to every 0.1 msec using the full data). Note that the filter still performed the prediction step every 0.1 msec as before, but the observation step was only preformed if data was available at that time point. The results in Figures 8 and 9 show that as data becomes more and more sparse, the parameter tracking estimate of the applied current parameter loses more and more characteristics of the underlying deterministic function. In particular, in Figure 8, as less data is available, the parameter tracking estimate of the pulsing step function begins lagging in estimating the steps and is unable to well maintain the shape. Similar results are seen in Figure 9 for the sinusoidal function, where the parameter tracking estimate loses its periodicity as less data becomes available. In both figures, it is clear that the sparser the data set, the more challenging it becomes for the filter to track the true underlying applied current function. Figure 7: Resulting EnKF with parameter tracking estimates of \(I(t)\) using different standard deviations \(\sigma_{\xi}\) for the parameter tracking drift term in (32). In each panel, the solid black line is the true sinusoidal applied current \(I(t)=10\sin(0.2t)+10\)\(\mathrm{mA/cm^{2}}\) in (36) that the filter aims to estimate. The solid red line is the EnKF estimated mean, and the dashed red lines show the estimated \(\pm 2\) standard deviation curves around the mean. Figure 8: Resulting EnKF with parameter tracking estimates of \(I(t)\) when the data generated using the pulsing step current in (35) is subsampled every 10 (**top**), 20 (**middle**), and 50 (**bottom**) time points.