# Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
## 4 Numerical Experiments
### Estimating Time-Varying Applied Current via Parameter Tracking  
We first establish baseline results for each of the four data sets described in Section 4.1 by applying the augmented EnKF with parameter tracking to estimate the time-varying applied current and Hodgkin-Huxley model states. We then perform numerical experiments to analyze the effects of the standard deviation of the parameter drift term in (32) and the frequency of time series voltage data available on the resulting applied current parameter estimates. Figure 2: Synthetic measurements of voltage \(V(t)\) generated using four different deterministic applied currents \(I(t)\). In each row, the plot on the left depicts the noisy voltage measurements (shown in negative mV) generated using the applied current on the right. From top to bottom, the plots show data generated using the following current functions: constant current with \(I(t)=2\) mA/cm\({}^{2}\) as in (33); one long step function, defined in (34); pulsing step function that alternates every 20 msec, as defined in (35); and the sinusoidal function \(I(t)=10\sin(0.2t)+10\) mA/cm\({}^{2}\) as in (36). Baseline resultsTo establish baseline results for parameter tracking in each of the four data sets described in Section 4.1, we employ the augmented EnKF with parameter tracking using \(N=100\) ensemble members with the standard deviation of the parameter random walk in (32) set to \(\sigma_{\xi}=1\). Initial ensembles for the model states \(x(t)=[V(t);n(t);m(t);h(t)]\) and applied current parameter \(\theta(t)=I(t)\) were drawn from uniform prior distributions, with \(V_{0|0}^{p}\sim\mathcal{U}(-100,0)\); \(n_{0|0}^{p},m_{0|0}^{p},h_{0|0}^{p}\sim\mathcal{U}(0,1)\); and \(l_{0|0}^{p}\sim\mathcal{U}(0,4)\) for each \(p=1,2,\ldots,N\). The results in Figures 3-6 show the parameter tracking estimates of \(I(t)\), along with the time series estimates of the Hodgkin-Huxley model states, for each of the four cases. Note that in each case, the filter mean is able to track the underlying true applied current, along with the unmeasurable system states, with uncertainty bounds represented by the \(\pm 2\) estimated standard deviation curves. Figure 4: 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 one-step current in (34). 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 3: Resulting ensemble Kalman filter (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 constant current \(I(t)=2\) mA/cm\({}^{2}\) in (33). 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. Standard deviation of parameter drift.As noted in Section 3, a carefully chosen standard deviation \(\sigma_{\xi}\) for the random walk in (32) is crucial in maintaining the accuracy of the time-varying parameter estimate and avoiding filter divergence; see, e.g., [21, 22, 23, 27, 23] and references therein. To demonstrate the sensitivity of the parameter tracking algorithm to this choice, we tested the effects of changing \(\sigma_{\xi}\) by letting \(\sigma_{\xi}=10\), 2, 1, 0.5, 0.25, and 0.1 in estimating the applied current, compared with \(\sigma_{\xi}=1\) used in the baseline results above. For sake of demonstration, we focused on the data generated using the sinusoidal current defined in (36); similar results hold in the other cases.