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= Hodgkin-Huxley Equations =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 8 minutes
| prerequisites = ODEs; basic electrochemistry; Nernst potential.
| if_too_advanced_see = [[FitzHugh-Nagumo_Equations]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Mostly-plus (irrelevant here).
| units = SI for biological observables. Voltages in mV; conductances in mS/cm2; currents in μA/cm2; time in ms.
}}

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 ==

:<math>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}}</math>

with the gating variables:

:<math>\frac{dm}{dt} = \alpha_m(V)\,(1 - m) - \beta_m(V)\,m</math>

:<math>\frac{dh}{dt} = \alpha_h(V)\,(1 - h) - \beta_h(V)\,h</math>

:<math>\frac{dn}{dt} = \alpha_n(V)\,(1 - n) - \beta_n(V)\,n</math>

== Components ==

{| class="wikitable"
! 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 gNa m3 h — three independent "activation" gates and one "inactivation" gate. The K+ conductance has the form gK n4 — four independent activation gates. These exponents fit the empirical voltage-clamp data.

== Derivation sketch ==

# '''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.
# '''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.
# '''Inactivation of Na+''' required a separate h variable that decreases with voltage.
# '''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):

# m rapidly increases (Na+ activation opens). gNa rises; Na+ flows in; V rises further. Positive feedback.
# V reaches near VNa ≈ +50 mV.
# h slowly decreases (Na+ inactivation). gNa drops.
# n increases (K+ activation). gK rises; K+ flows out; V falls back toward VK.
# 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 ==

* '''Iext = 0''' and start at rest: V remains at rest. ✓
* '''Block Na+''' (gNa → 0, simulating tetrodotoxin): no action potential, but passive voltage response intact. ✓
* '''Block K+''' (gK → 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_Model|Wilson-Cowan]] and [[Amari_Neural_Field|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 Vth.
* '''Channel kinetics''': potentially small ψ-dependent modulation of αx(V), βx(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.

[[Category:Psionics]]
[[Category:Equations]]
[[Category:Neuroscience]]

Hodgkin-Huxley Equations - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$