# Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
## 5 Discussion  
The aim of this work was to utilize the EnKF with parameter tracking in estimating the time-varying applied current parameter in the Hodgkin-Huxley model. In particular, the parameter tracking algorithm was analyzed in estimating four different deterministic applied currents using synthetically-generated voltage data. We first verified that the algorithm was able to successfully track the underlying applied current, along with the unobserved model states, for each of the four test cases. In addition to tracking the applied current \(I(t)\), baseline results show that the filter is able to accurately estimate the unobserved states \(n(t)\), \(m(t)\), and \(h(t)\) from the generated voltage data. Further numerical experiments were conducted to analyze how the parameter tracking estimates of the applied current parameter are affected under different implementation conditions, namely, when changing the standard deviation of the parameter drift term in (32) and when the algorithm is provided increasingly less voltage data. Overall, using the augmented EnKF with parameter tracking as described in Section 3, we were able to well track the underlying applied current functions in each of the four cases considered, establishing the baseline results shown in Figures 3-6. In each case, the EnKF mean well approximates the true underlying applied current, in cases where the current is constant, a single pulse, multiple pulses, and a sinusoidal function, as well as the corresponding model states. Uncertainty bounds around the EnKF estimates, as reflected in the estimated \(\pm 2\) standard deviation curves around the mean, in each case well contain the true applied current and associated model states, with less uncertainty around the measured voltage than the unobserved gating variables. From the baseline results, we were able to further explore the effects of two important aspects of the parameter tracking implementation: the choice of the standard deviation of the parameter drift term in (32), and the availability of time series voltage data. Numerical experiments showed that using different values for \(\sigma_{\xi}\) results in drastically different levels of accuracy and confidence in the parameter tracking estimates for the applied current. As shown in Figure 7, when using \(\sigma_{\xi}=10\) for the parameter drift standard deviation, the resulting EnKF mean estimate is able to well track the true current; however, the resulting confidence in the estimate is low, as reflected in the wide range between the estimated \(\pm 2\) standard deviation curves. Oppositely, when using \(\sigma_{\xi}=0.1\), the parameter tracking estimate diverges from the true solution, returning an estimate of the applied current parameter that is significantly inaccurate despite having high confidence in the estimate, as reflected in the tight uncertainty bounds. The results in Figure 7 suggest that, for the data considered, \(\sigma_{\xi}=0.5\) is the best choice for the parameter drift standard deviation out of the values tested for this problem, with the corresponding applied current estimate accurately tracking the true solution with less uncertainty reflected in the resulting \(\pm 2\) standard deviation curves around the EnKF mean than for larger \(\sigma_{\xi}\) values. Further numerical tests demonstrate how the accuracy of the parameter tracking estimate is affected when the frequency of available time series voltage data is decreased, resulting in fewer data points being used as updating information in the filter. The results in Figures 8 and 9 show that as less data is made available via subsampling, the parameter tracking algorithm has increasing difficulty in estimating the true underlying applied current function, losing structural features such as the step onset of the pulsing step current and the periodicity of the sinusoidal current.