Neuro simulations · Episode 01

Hodgkin–Huxley, 1952

The squid giant axon model, run from the original equations and constants. Each experiment below repeats a protocol from the five 1952 papers in The Journal of Physiology and compares the computed result with the published value.

PAPER IMeasurement of current–voltage relations (with Katz)116:424–448
PAPER IICurrents carried by sodium and potassium ions116:449–472
PAPER IIIThe components of membrane conductance116:473–496
PAPER IVThe dual effect of membrane potential on sodium conductance116:497–506
PAPER VA quantitative description of membrane current…117:500–544

Voltage axis

Verification status. Equations and constants follow Paper V (the rate equations and Table 3). They are checked here by reproducing three of its published results: the 6–7 mV threshold, the 18.8 m/s calculated conduction velocity, and zero ionic current at rest. They have not yet been checked line by line against the PDF; that check is pending.

Paper V · membrane action potential

An all-or-none spike from a quick shock

With the axon space-clamped, a brief shock displaces the membrane by a set amount and the model runs freely. Paper V computes responses to initial depolarisations of 6, 7, 15 and 90 mV at 6.3 °C: 6 mV decays, 7 mV fires.

Initial depolarisation

Temperature

Overlay the paper’s 6, 7, 15 and 90 mV shocks

Comparison with Paper V

Threshold, computed: –
Paper V, 6.3 °C: 6 mV no spike · 7 mV spike
Spike peak: –

Membrane potentialselected shockpaper’s four shocks

Conductancesg_Na = ḡ_Nam³hg_K = ḡ_Kn⁴

Gating variablesmhn

Paper V · refractoriness, anode break, repetitive firing

How the same equations explain excitability

Protocol

Shock size

Interval between shocks

Hyperpolarisation before release

Applied current (depolarising)

Temperature

Computed

Membrane potentialV

Stimulus

Papers I–III · voltage clamp

Holding the voltage separates the two currents

Under voltage clamp the membrane potential is stepped and held, so each gate relaxes exponentially to its new steady state. Sodium conductance rises and then inactivates. Potassium conductance rises with a delay, following n⁴, and stays up.

Depolarising step

Temperature

Show a family of steps (10–130 mV)

At the selected step

Conductancesg_Nag_K

Ionic currents at the selected stepI_NaI_KI_i total

Paper IV · sodium inactivation

A conditioning step sets how much sodium current is available

The membrane is held at a conditioning voltage long enough for h to settle, then stepped to a fixed test depolarisation. The peak sodium current, normalised to its largest value, traces the steady-state inactivation curve h_∞(V).

Conditioning duration

Test depolarisation

Temperature

Computed

Available sodium current vs conditioning voltageh∞ from the rate equationssimulated pulse protocol

Test-pulse I_Na after five conditioning voltages

Paper V · propagated action potential

The spike travels down the axon

Adding the axon’s cable properties turns the membrane equation into a partial differential equation in space and time. Paper V solved it for a fibre of radius 238 µm and axoplasm resistivity 35.4 Ω cm at 18.5 °C, giving a calculated velocity of 18.8 m/s against 21.2 m/s measured.

Temperature

Axon radius

Axoplasm resistivity R₂

Time

Comparison with Paper V

Velocity, computed: –
Paper V, calculated: 18.8 m/s
Paper V, measured: 21.2 m/s

Membrane potential along 6 cm of axonV(x) nowrecording sites

Recorded at two sitesx = 2 cmx = 4 cm

Putting it together

From five papers to one spike

The papers form a single argument. Measure the currents with the voltage held fixed. Separate them by ion. Describe each with gating variables. Then, with nothing left to adjust, compute the action potential and see whether it matches the real one.

STEP 1 · PAPER IMeasure
The voltage clamp holds V fixed, so the current that flows is purely ionic plus a brief capacitive transient.
See the clamp ↑
STEP 2 · PAPER IISeparate
Replacing external sodium with choline removes I_Na, so the total current splits into a sodium part and a potassium part.
See I_Na and I_K ↑
STEP 3 · PAPERS III–IVDescribe
Each current is a conductance times a driving force. g_K follows n⁴. g_Na follows m³h: fast activation and slower inactivation.
See inactivation ↑
STEP 4 · PAPER VPredict the spike
The fitted equations are combined with the membrane capacitance and solved. Threshold, shape and refractoriness all come out of them.
See the spike ↑
STEP 5 · PAPER VPredict propagation
Adding cable properties gives a travelling wave: 18.8 m/s calculated against 21.2 m/s measured.
See it travel ↑

Anatomy of one action potential

A 15 mV quick shock at 6.3 °C, as in the first experiment. Upper panel: the dashed lines are V_Na and V_K, the limits each current pulls the membrane toward. The shaded band marks the current phase.

Fitted to voltage-clamp data

Maximum conductances ḡ_Na, ḡ_K, ḡ_l
Reversal potentials V_Na, V_K, V_l
The six rate functions α and β for m, h and n
The exponents (m³, n⁴), chosen to reproduce the delayed rise of the conductances

Predicted, never fitted

The all-or-none threshold (6–7 mV for a quick shock)
The shape, overshoot and undershoot of the action potential
Absolute and relative refractory periods
Anode-break excitation and repetitive firing
Conduction velocity: 18.8 m/s against 21.2 m/s measured

Why it still matters

Hodgkin and Huxley showed that excitability follows from two voltage-dependent conductances and the membrane capacitance, with no extra mechanism. Their gating particles predicted voltage-gated ion channels before anyone could see one. Single channels were first recorded with the patch clamp (Neher and Sakmann, 1976). The same conductance-based form, I = C dV/dt + Σ g(V − E), underlies most biophysical neuron models in use today. Hodgkin and Huxley shared the 1963 Nobel Prize in Physiology or Medicine with John Eccles.

Methods

Equations, constants and numerics

Model (Paper V, 1952 sign convention)

I = C_M dV/dt + ḡ_Kn⁴(V − V_K) + ḡ_Nam³h(V − V_Na) + ḡ_l(V − V_l)

dx/dt = α_x(1 − x) − β_xx, x ∈ {m, h, n}

α_n = 0.01(V + 10) / [exp((V + 10)/10) − 1]

β_n = 0.125 exp(V/80)

α_m = 0.1(V + 25) / [exp((V + 25)/10) − 1]

β_m = 4 exp(V/18)

α_h = 0.07 exp(V/20)

β_h = 1 / [exp((V + 30)/10) + 1]

all α, β × φ, φ = 3^{(T − 6.3)/10}

(a/2R₂) ∂²V/∂x² = C_M ∂V/∂t + I_ipropagated case

V is the displacement from the resting potential in mV, with depolarisation negative. Inward current is positive, which the constants imply: with V_Na = −115 mV the sodium current comes out positive. The modern display flips both voltage and current, so that depolarisation and outward current are positive. α and β are in ms⁻¹, t in ms, I in µA/cm², g in mS/cm². At V = −10 and V = −25 the expressions for α_n and α_m are 0/0 and take their limits, 0.1 and 1.0 ms⁻¹.

Constants

Symbol	Value	Meaning
C_M	1.0 µF/cm²	Membrane capacitance
ḡ_Na	120 mS/cm²	Maximum sodium conductance
ḡ_K	36 mS/cm²	Maximum potassium conductance
ḡ_l	0.3 mS/cm²	Leak conductance
V_Na	−115 mV	Sodium equilibrium potential (relative to rest)
V_K	+12 mV	Potassium equilibrium potential
V_l	−10.613 mV	Leak reversal, chosen so total ionic current is zero at rest
T	6.3 °C	Reference temperature for the rate equations
a	238 µm	Axon radius (propagation)
R₂	35.4 Ω cm	Axoplasm resistivity (propagation)
T	18.5 °C	Temperature of the propagated calculation

Numerical methods

Space-clamped runs (experiments 1–2): fourth-order Runge–Kutta, Δt = 0.01/φ ms (0.01 ms at 6.3 °C).
Voltage clamp (3–4): exact solution. With V held fixed each gate follows x(t) = x∞ − (x∞ − x₀)e^−t/τ, with x∞ = α/(α+β) and τ = 1/(α+β). The clamp is ideal: no capacitive transient and no series resistance.
Cable (5): second-order central differences in space, Δx = 0.25 mm over 6 cm (241 nodes), sealed ends. V is advanced by forward Euler with Δt = 0.2 Δx²/D, where D = a/(2R₂C_M). The gates are advanced by the exact exponential (Rush–Larsen) update, with rates tabulated every 0.01 mV and linearly interpolated. The stimulus is 3 mA/cm² for 0.1 ms over the first 2 mm.
Velocity is the inverse slope of a least-squares fit of the time each node passes 50 mV depolarisation, against position, for 1.5–4.5 cm. Halving Δx moves it by less than 0.1 m/s (0.5 → 0.25 → 0.125 mm: 18.55, 18.68, 18.72 m/s at the paper’s settings).
Spike criterion: depolarisation beyond 50 mV. Thresholds are found by bisection to 0.01 mV.

What the model leaves out

Gating is deterministic. There is no channel noise, so firing near threshold is perfectly repeatable.
Concentrations are fixed. There is no Na⁺/K⁺ accumulation or pump current, and repetitive firing runs indefinitely.
Reversal potentials are constants, not computed from the Nernst equation.
The cable is a uniform, unmyelinated cylinder with no extracellular resistance.
The modern voltage axis (V_m = −65 − V) relabels the display only. A resting potential of −65 mV is a common modern choice, not a value from the 1952 papers.