Hodgkin-Huxley Equations

Hodgkin-Huxley Equations

Audience

Difficulty	Intermediate

Notation on this page

The Hodgkin-Huxley equations (1952) are the canonical biophysical model of the neuronal action potential. They were derived from voltage-clamp experiments on the giant axon of the squid (Loligo forbesii), in work that earned Alan Hodgkin and Andrew Huxley the 1963 Nobel Prize in Physiology or Medicine.

The model describes the membrane voltage V(t) of a neuron in terms of:

• Capacitive currents charging the lipid bilayer.
• Ionic currents through voltage-gated Na⁺ and K⁺ channels.
• Leak currents through non-gated channels.

Three auxiliary "gating" variables (m, h, n) describe the voltage-dependent opening and closing of the ion channels.

Statement

$ C_{m}\,{\frac {dV}{dt}}=-g_{\text{Na}}\,m^{3}\,h\,(V-V_{\text{Na}})-g_{K}\,n^{4}\,(V-V_{K})-g_{L}\,(V-V_{L})+I_{\text{ext}} $

with the gating variables:

$ {\frac {dm}{dt}}=\alpha _{m}(V)\,(1-m)-\beta _{m}(V)\,m $

$ {\frac {dh}{dt}}=\alpha _{h}(V)\,(1-h)-\beta _{h}(V)\,h $

$ {\frac {dn}{dt}}=\alpha _{n}(V)\,(1-n)-\beta _{n}(V)\,n $

Components

Symbol	Meaning	Typical value (squid axon)
V	Membrane voltage	− 70 mV (rest), + 50 mV (peak)
Cm	Membrane capacitance	1.0 μF/cm2
gNa	Max Na+ conductance	120 mS/cm2
gK	Max K+ conductance	36 mS/cm2
gL	Leak conductance	0.3 mS/cm2
VNa, VK, VL	Reversal potentials	+50, −77, −54.4 mV
m, h	Na+ activation / inactivation	dimensionless ∈ [0,1]
n	K+ activation	dimensionless ∈ [0,1]
αx(V), βx(V)	Voltage-dependent rate constants	1/ms (empirical exponentials)
Iext	External stimulus current	μA/cm2

The Na⁺ conductance has the form g_Na m³ h — three independent "activation" gates and one "inactivation" gate. The K⁺ conductance has the form g_K n⁴ — four independent activation gates. These exponents fit the empirical voltage-clamp data.

Derivation sketch

1. Voltage-clamp experiments (Hodgkin-Huxley-Katz 1949–1952) measured the ionic current as a function of voltage step. Separating Na⁺ and K⁺ by selective blockers (TTX, TEA) revealed two distinct time courses.
2. Empirical fit to channel kinetics: rising time courses ∝ (1 − exp(−t/τ))ⁿ required n = 3 for Na⁺ activation and n = 4 for K⁺. Hodgkin-Huxley interpreted this as multiple independent gating particles.
3. Inactivation of Na⁺ required a separate h variable that decreases with voltage.
4. Empirical fitting yields α and β as exponential functions of V.

The exponent-of-4 form (n⁴) was later vindicated when potassium channels were found to be tetrameric proteins (4 identical subunits, each contributing a gate). The exponent-of-3 plus inactivation form for Na⁺ mapped onto sodium channels with 4 domains, of which 3 act as activation gates and the 4th as the inactivation gate.

Action-potential dynamics

At rest (V ≈ −70 mV):

• m is small (Na⁺ activation gates closed).
• h is large (Na⁺ inactivation gates open).
• n is small (K⁺ activation gates closed).

When V depolarises above threshold (~ −55 mV):

1. m rapidly increases (Na⁺ activation opens). g_Na rises; Na⁺ flows in; V rises further. Positive feedback.
2. V reaches near V_Na ≈ +50 mV.
3. h slowly decreases (Na⁺ inactivation). g_Na drops.
4. n increases (K⁺ activation). g_K rises; K⁺ flows out; V falls back toward V_K.
5. V undershoots (hyperpolarisation) below rest, then h recovers and the cycle can repeat.

The full cycle takes ~ 1–2 ms — the classical action potential.

Sanity-check limits

• I_ext = 0 and start at rest: V remains at rest. ✓
• Block Na⁺ (g_Na → 0, simulating tetrodotoxin): no action potential, but passive voltage response intact. ✓
• Block K⁺ (g_K → 0, simulating tetraethylammonium): action potential lacks repolarisation; sustained depolarisation. ✓
• Cold temperature (Q₁₀ scaling of rate constants): action potentials slow down without losing form. ✓ (Hodgkin-Huxley validated this directly with the squid axon at different temperatures.)
• Strong sustained input: repetitive firing at a frequency that depends on input magnitude. ✓ (Standard behaviour.)

Connection to ψ

Hodgkin-Huxley is a single-neuron model; ψ-coupling enters at the population level via Wilson-Cowan and Amari extensions. At the single-neuron level, ψ-coupling would modify:

• Threshold: a small ψ-dependent shift in the firing threshold via β · ψ added to V_th.
• Channel kinetics: potentially small ψ-dependent modulation of α_x(V), β_x(V) — but this is below the noise floor of single-neuron measurements.

In the framework, individual-neuron HH dynamics are essentially the standard biophysics; ψ effects emerge at the population scale through coherent collective coupling.

Experimental status

Hodgkin-Huxley is gold-standard mainstream physiology, verified by:

• Direct match to squid-axon voltage-clamp data (the original 1952 series).
• Match to mammalian central-nervous-system neurons with appropriate parameter changes.
• Successful prediction of refractory periods, accommodation, anode-break excitation, propagation velocities along axons.
• Reproduction in patch-clamp recordings down to single-channel level (Sakmann-Neher 1976+).

The HH equations are not in dispute. They are the foundation of modern computational neurophysiology.

See Also

• FitzHugh-Nagumo_Equations
• Wilson-Cowan_Model
• Amari_Neural_Field
• Neural_Field_Equations
• Jansen-Rit_Neural_Mass

References

• Hodgkin, A. L., Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve." Journal of Physiology 117: 500–544.
• Hodgkin, A. L., Huxley, A. F., Katz, B. (1952). "Measurement of current-voltage relations in the membrane of the giant axon of Loligo." Journal of Physiology 116: 424–448.
• Sakmann, B., Neher, E. (1976). "Single-channel currents recorded from membrane of denervated frog muscle fibres." Nature 260: 799–802.
• Koch, C., Segev, I. (eds.) (1998). Methods in Neuronal Modeling. MIT Press.