Computational Neuroscience: Hodgkin-Huxley
Computational Neuroscience: Hodgkin-Huxley
The Hodgkin-Huxley model (1952) describes how voltage-gated ion channels generate the action potential. The membrane is modelled as a capacitor in parallel with conductances for Na$^+$, K$^+$, and leak, each driven by the voltage minus the ion\'s reversal potential.
Gating variables $m$, $h$ (Na$^+$) and $n$ (K$^+$) obey first-order kinetics with voltage-dependent rates $\alpha_x(V)$ and $\beta_x(V)$, calibrated on the squid giant axon. Reduced two-variable models (FitzHugh-Nagumo) preserve the qualitative bifurcation structure while being analytically tractable.
Hodgkin-Huxley Current Balance
$$C_m\dot{V}=I_\text{ext}-g_\text{Na}m^3h(V-E_\text{Na})-g_K n^4(V-E_K)-g_L(V-E_L).$$ Gating: $\dot{m}=\alpha_m(1-m)-\beta_m m$, and similarly for $h,n$.
Action Potential Threshold
Below threshold the Na$^+$ inward current is overcome by outward currents and the perturbation decays. Above threshold, Na$^+$ current regenerates — a positive-feedback spike followed by K$^+$-driven repolarisation and a refractory period set by $h$ inactivation.
Example 1
At rest ($V\approx-65\,\text{mV}$), which gates are open?
Solution: $m\approx0$ (Na$^+$ activation closed), $h\approx1$ (Na$^+$ inactivation open), $n\approx0.3$ (K$^+$ partially open); resting state dominated by leak current.
Example 2
Why is there a refractory period after an action potential?
Solution: The Na$^+$ inactivation gate $h$ closes during the spike and recovers slowly; elevated K$^+$ conductance also opposes re-firing until both gates reset.
Practice
- Write the HH current-balance equation including an applied current $I_\text{ext}$.
- What is the role of the $h$ gate in Na$^+$ channel inactivation?
- How does the FitzHugh-Nagumo model reduce the HH system?
- Explain how myelination increases action-potential conduction velocity.
Show Answer Key
1. $C_m \frac{dV}{dt} = I_{\text{ext}} - \bar{g}_{\text{Na}} m^3 h (V - E_{\text{Na}}) - \bar{g}_K n^4 (V - E_K) - g_L (V - E_L)$, where $m, h, n$ obey $dx/dt = \alpha_x(V)(1-x) - \beta_x(V)x$.
2. The $h$ gate (inactivation variable) closes slowly after depolarization, shutting off Na$^+$ current even while $m$ gates are open. This creates the refractory period: after a spike, Na$^+$ channels are inactivated and cannot reopen until $h$ recovers, preventing immediate re-firing.
3. FHN replaces the three gating variables $(m,h,n)$ with two variables: a fast excitatory variable $v$ (voltage-like) and a slow recovery variable $w$. It captures the essential excitable dynamics (threshold, spike, refractory period) as a 2D system with a cubic nullcline, enabling phase-plane analysis.
4. Myelin sheaths insulate axon segments, forcing current to jump between nodes of Ranvier (saltatory conduction). This reduces membrane capacitance and increases membrane resistance between nodes, increasing the length constant $\lambda$. Conduction velocity increases from ~1 m/s (unmyelinated) to ~100 m/s (myelinated).