Replacing $m(V)$ by $m_\infty (V)$

First, recall that the equations for the gating variables can be solved exactly if $V$ is held constant. For example, if $m(0)=0$, then

$$m(t) = m_\infty(V) (1 - e^{-t/\tau_m(V)}),$$

where

$$m_\infty(V) \equiv \frac{\alpha_m}{\alpha_m + \beta_m}, \qquad \tau_m(V) \equiv \frac{1}{\alpha_m + \beta_m},$$

with similar expressions for $n$ and $h$. Figure 1 shows plots of the functions $m_\infty (V), n_\infty (V), h_\infty (V), \tau _m(V), \tau _n(V)$, and $\tau _h(V)$.

Figure 1: The functions $m_\infty (V), n_\infty (V), h_\infty (V), \tau _m(V), \tau _n(V)$, and $\tau _h(V)$.

We see that the time constant $\tau_m$ is much smaller than $\tau_n$ or $\tau_h$ over the entire range of interest of $V$. This means that the $m$ variable evolves faster than the $n$ or $h$ variables. This suggests that we might be able to replace $m(t)$ in the Hodgkin/Huxley model by $m_\infty (V(t))$, i.e., use the instantaneous value of $V$ to compute $m_\infty$, then use this for $m$ in the righthand sides of the $dV/dt$, $dn/dt$, and $dh/dt$ equations. The appropriateness of this approximation is illustrated in Figure 2. This shows the timeseries for input currents of $I=6.5 \mu A/cm^2$ and $I=20 \mu A/cm^2$ after the transient behavior has decayed away.

Figure 2: The blue line shows $m(t)$ as calculated from the full four-dimensional Hodgkin/Huxley model, while the red line shows $m_\infty (V(t))$.

The code used to generate this plot can't be given here because of its similarity to the code you're writing for Homework #1. It should be relatively easy to modify your own code to generate a similar plot - you're encouraged to try this, and also to write appropriate code to generate Figure 1.