Neuron Morphology & Cable Theory

Hodgkin-Huxley action potential propagation along a dendritic cable (Rall's cable theory)

Cable: Membrane Voltage V(x,t)

Membrane resistance r_m: 2.0 kΩ·cm

Axial resistance r_a: 1.0 kΩ/cm

Stim position: 20%

λ = √(r_m/r_a) = √(2.0/1.0) = 1.41 cm | τ_m = r_m·c_m = 2.0 ms

Space constant λ: voltage decays as e^(−x/λ) along passive cable. Larger λ → signals propagate further. Active cable (HH model) supports all-or-none propagation regardless of λ.

Action Potential Waveform V(t)

View position: 20%

Hodgkin-Huxley: V̇ = (1/C_m)[I_ext − g_Na·m³·h·(V−E_Na) − g_K·n⁴·(V−E_K) − g_L·(V−E_L)]. Action potential: depolarization → Na⁺ influx → repolarization → K⁺ efflux → refractory period.

Space Constant & Signal Decay

Passive cable: V(x) = V₀·e^(−|x−x₀|/λ). Active cable: action potential amplitude is constant — regenerative Na⁺ channels overcome attenuation.