# Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
## 2 Review of the Hodgkin-Huxley Model Equations
The rate constants are modeled using the following equations, derived from [PERSON] and [PERSON]'s experimental results [1]:  
\[\alpha_{m}(V) = \frac{0.1(V+25)}{\exp(\frac{V+25}{10})-1} \tag{8}\] \[\alpha_{h}(V) = 0.07\exp\left(\frac{V}{20}\right)\] (9) \[I_{ion}=I_{Na}+I_{K}+I_{\ell} \tag{2}\]0 (10) \[I_{ion}=I_{Na}+I_{K}+I_{\ell} \tag{2}\]1  
Note that setting \(\alpha_{m}\) to its limit value of 1 at \(V=-25\) mV avoids the discontinuity at that point. Potassium currentThe potassium current is given by  
\[I_{ion}=I_{Na}+I_{K}+I_{\ell} \tag{2}\]2  
with potassium gate equation  
\[I_{ion}=I_{Na}+I_{K}+I_{\ell} \tag{2}\]3  
Here \(\tilde{g}_{K}\) is a constant (conductance/cm\({}^{2}\)) and \(n=n(t)\) is the proportion of potassium gates open (dimensionless, varying between 0 and 1). The potassium voltage is given by \(V_{K}=E_{K}-E_{r}\), where \begin{table}
\begin{tabular}{l l l} \hline \hline
**Component** & **Description** & **Units** \\ \hline \(I\) & Total membrane current & mA/cm\({}^{2}\) \\ \(C_{M}\) & Membrane capacity & uF/cm\({}^{2}\) \\ \(V\) & Voltage & mV \\ \(I_{ion}\) & Ionic current density & mA/cm\({}^{2}\) \\ \(t\) & Time & msec \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Components of the Hodgkin-Huxley model (1). is an equilibrium potential for potassium, sensitive to the overall outside concentration of charged ions [25]. Table 2 lists the constant values of \(V_{K}\) and \(\bar{g}_{K}\). The dynamics of the gating variable \(n(t)\) are similarly modeled using the differential equation  
\[\frac{dn}{dt}=\alpha_{n}(V)(1-n)-\beta_{n}(V)n \tag{14}\]  
where the rate constants  
\[\alpha_{n}(V) = \frac{0.01(V+10)}{\exp(\frac{V+10}{10})-1} \tag{15}\] \[\beta_{n}(V) = 0.125\exp\Big{(}\frac{V}{80}\Big{)} \tag{16}\]  
were also derived using experimental data [1]. Note the discontinuity in \(\alpha_{n}\) when \(V=-10\) mV can be avoided by setting it equal to its limit value of 0.1 at that point. Leakage currentThe leakage current is a small combined current, accounting mostly for chloride but also other ions. The leakage current is given by  
\[I_{\ell}=\bar{g}_{\ell}(V-V_{\ell}) \tag{17}\]  
with constant conductance \(\bar{g}_{\ell}\) and leakage voltage \(V_{\ell}=E_{\ell}-E_{r}\). Here \(E_{\ell}\) is the potential at which the leak current is zero. The leakage voltage \(V_{\ell}\) is needed for any calculation for threshold, but it is unlikely to give any information about the nature of charged particles [25]. Table 2 lists the constant values of \(V_{\ell}\) and \(\bar{g}_{\ell}\). Model summaryIn summary, the Hodgkin-Huxley model comprises the total membrane current Equation (1), which depends on time, voltage, and the solutions to the transfer Equations (6), (7) and (14). Note that when a constant voltage is applied, \(\frac{dV}{dt}=0\) and (1) simplifies to \(I=I_{ion}\). In this case, Equations (6), (7) and (14) can be solved independently to compute the total ionic current. However, when voltage changes with time due to an applied current, \(\frac{dV}{dt}\
eq 0\) and all four equations must be solved simultaneously. The complete system of coupled ordinary differential equations is given by \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Description** & **Value** & **Units** \\ \hline \(C_{M}\) & Membrane capacity & 1.0 & \(\mu\)F/cm\({}^{2}\) \\ \(V_{Na}\) & Sodium voltage & \(-115\) & mV \\ \(V_{K}\) & Potassium voltage & \(12\) & mV \\ \(V_{\ell}\) & Leakage voltage & \(-10.613\) & mV \\ \(\bar{g}_{Na}\) & Sodium gate constant & \(120\) & m.mho/cm\({}^{2}\) \\ \(\bar{g}_{K}\) & Potassium gate constant & \(36\) & m.mho/cm\({}^{2}\) \\ \(\bar{g}_{\ell}\) & Leakage gate constant & \(0.3\) & m.mho/cm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constant parameter values used in the Hodgkin-Huxley model. Note that m.mho stands for 1/ohm (the reverse of ohm) or amp/volts. It is the unit Siemen and represents the derived unit of electrical conductance.