# Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model
## 2 Review of the Hodgkin-Huxley Model Equations  
The Hodgkin-Huxley model provided the first quantitative description of electrical excitability in nerve cells [24], involving detailed mathematical equations to describe the voltage-dependent and time-dependent properties of the sodium and potassium conductances [1]. Each piece of the circuit shown in Figure 1 corresponds to a different electrophysiological component of the model. Capacitors represent the charge storage capacity of each gating variable; resistors represent the sodium, potassium, and leakage ion channels in the neuron; and batteries represent the electrochemical potentials that each gating variable has to let ions in and out of the charged cell. From this analogy, the Hodgkin-Huxley equation modeling total membrane current is given by  
\[I=C_{M}\frac{dV}{dt}+I_{ion} \tag{1}\]  
where \(I=I(t)\) is the total membrane current in the axon (with a positive inward current), \(C_{M}\) is the membrane capacity (assumed to be constant), and \(V=V(t)\) is the displacement of the membrane potential from its resting value (assumed to have a negative depolarization). Table 1 lists each model component, along with its corresponding units. For simplicity of terminology, we will refer interchangeably to \(V(t)\) as the voltage within this paper. Note that the voltage \(V\) is related to the membrane potential \(E\) via the relationship \(V=E-E_{r}\), where \(E_{r}\) denotes the absolute value of the resting potential [1]. The ionic current density \(I_{ion}\) is represented as the sum of the three currents  
\[I_{ion}=I_{Na}+I_{K}+I_{\ell} \tag{2}\]  
where \(I_{Na}\), \(I_{K}\), and \(I_{\ell}\) model the currents relating to the sodium, potassium, and leakage channels occurring in the neuron, respectively. The form of the current for each ion channel follows from [PERSON]'s law, where  
\[I_{i}=g_{i}(t,V)(V-V_{i}) \tag{3}\]  
for \(i=Na,K,\ell\). Here \(g_{i}(t,V)\) represents the gate for each channel generally as a function of time and voltage, and \(V-V_{i}\) represents the difference between the overall voltage \(V\) of the system and the channel-specific voltages \(V_{i}\). We describe each of the three ionic currents in more detail as follows. Figure 1: The Hodgkin-Huxley model represented as a circuit. Here the capacitor (\(C_{M}\)) represents the charge storage capacity, the resistors (\(g_{\ell}\), \(g_{K}\), and \(g_{Na}\)) act as the ion channels, and the batteries (\(E_{\ell}\), \(E_{K}\), and \(E_{Na}\)) act as the electrochemical potentials. Sodium currentThe sodium current is given by  
\[I_{Na}=g_{Na}(t,V)(V-V_{Na}) \tag{4}\]  
where the sodium gate  
\[g_{Na}(t,V)=m^{3}h\tilde{g}_{Na} \tag{5}\]  
is impacted by depolarization, which causes an increase in sodium conductance [1]. Here \(\tilde{g}_{Na}\) is a constant (conductance/cm\({}^{2}\)), \(m=m(t)\) is the proportion of active sodium gates open (dimensionless variable which varies over time between 0 and 1), and \(h=h(t)\) is the proportion of inactive gates open (similarly dimensionless, varying between 0 and 1). The sodium voltage is given by \(V_{Na}=E_{Na}-E_{r}\), where \(E_{Na}\) is an equilibrium potential for sodium. Table 2 lists the constant values of \(V_{Na}\) and \(\tilde{g}_{Na}\). The dynamics of the sodium gating variables \(m(t)\) and \(h(t)\) are governed by the following differential equations:  
\[\frac{dm}{dt} = \alpha_{m}(V)(1-m)-\beta_{m}(V)m \tag{6}\] \[\frac{dh}{dt} = \alpha_{h}(V)(1-h)-\beta_{h}(V)h \tag{7}\]  
where the voltage-dependent rate constants (msec\({}^{-1}\)) \(\alpha_{m}\) and \(\alpha_{h}\) represent the rate of flow of ions into the cell and \(\beta_{m}\) and \(\beta_{h}\) represent the flow out.