Neuron Cable Equation

Voltage propagation along a passive dendrite

τ ∂V/∂t = λ² ∂²V/∂x² − V + I(x,t)
Membrane Voltage V(x,t)
Steady-state V∞(x)
150 μm
20 ms
400 μm
80 pA
Soma (0%)
Cable theory (Wilfrid Rall, 1959): Dendrites act as leaky electrical cables. The length constant λ = √(r_m/r_i) determines how far voltage decays with distance — larger λ means better conduction. The time constant τ = r_m·c_m determines how quickly the membrane charges. Current injected at the soma spreads exponentially: V∞(x) = V₀·e^(−x/λ). Click-drag on the dendrite to inject current at different positions.