How a voltage leaks and spreads down a dendrite or an axon — borrowed, wire for wire, from the transatlantic telegraph. A passive cable attenuates and delays; it never regenerates. Rendered, not quoted.
source Rall, W. (1959) “Branching dendritic trees and motoneuron membrane resistivity,” Exp. Neurol. 1(5):491–527 · doi:10.1016/0014-4886(59)90046-9 · from Kelvin (W. Thomson) 1855 telegraph-cable theory. Illustrative parameters marked amber; no medical advice.
The passive cable, one line of physics:
λ² · d²V/dx² = τ · dV/dt + V
At steady state (dV/dt = 0) it collapses to λ²V″ = V, whose solution is the exponential
V(x) = V₀·e^(−x/λ).
The space constant λ = √(rm/ri) is the distance over which the signal falls to 1/e (≈37%). The time constant τ = rm·cm sets how fast the membrane charges.
rm = membrane resistance, ri = axial resistance, cm = membrane capacitance. Values here are illustrative amber — the ratios and identities are exact.
Voltage down a wire. Kelvin (1855) solved the sea-cable so telegraph pulses could reach America; Rall (1959) carried the same equation into the dendritic tree.
This is the passive spread that the-hodgkin-huxley spike must overcome — HH's active, all-or-none regeneration is exactly what a bare cable cannot do. Neighbour sphere: the-hodgkin-huxley (active membrane) ⇄ the-cable-equation (passive membrane).
Re-runs the full selfcheck live and reports. Flips red the instant window 6 tampers with λ.
witness idle…
A voltage clamp V₀ injected at x = 0, and the cable's passive constants:
V₀ = 1.0 · rm · ri · cm λ = √(rm/ri) τ = rm·cm
Lower ri (thicker fibre) → larger λ → the signal spreads farther. Raise ri and watch the curve pull in. The dashed line marks x = λ, where V = V₀/e.
Proven, live: exponential decay with space constant λ; 63% charge at t = τ; strictly monotone, no amplification.
booting…
“A passive dendrite carries a signal to the soma undistorted.”
Wall: FALSE. The cable is a low-pass filter — it both attenuates (amplitude falls as e^(−x/λ)) and delays (τ smears the edge). A distant synapse arrives smaller and slower. This attenuation is why real neurons need active channels; the passive cable alone cannot reach across a long dendrite.
“λ = √(ri/rm), so a higher membrane resistance shortens the spread.”
Correction: λ = √(rm/ri). Higher rm (better insulation) lengthens λ; lower ri (thicker core) also lengthens it. The inverted ratio is the exact bug planted in window 6.
“Making a dendrite thicker always speeds conduction proportionally.”
Correction: λ ∝ √(1/ri) and ri ∝ 1/area, so λ ∝ √(diameter) — a square-root gain, not linear.
“A passive cable can boost a weak signal.”
Correction: passive means monotone decay — V never exceeds V₀ anywhere. Amplification requires active (HH) channels. amber exact param values are illustrative.
Swap in the inverted space constant λ = √(ri/rm). The math still runs — but higher rm now wrongly shortens the spread. The Witness (7) catches it live.