Cable theory

The mathematics of a signal spreading down a leaky insulated conductor was worked out for undersea telegraph cables, where it explained why long cables smeared pulses into each other. Rall’s contribution was noticing that a dendrite is the same problem: a resistive core, a leaky capacitive membrane, and no amplification along the way.

$$\lambda^2 \frac{\partial^2 V}{\partial x^2} = \tau \frac{\partial V}{\partial t} + V$$

Two constants govern everything. The length constant $\lambda$ sets how far a signal travels before decaying; the time constant $\tau$ sets how much it is smoothed. Both fall out of geometry and membrane properties.

Why it changed the picture

Before Rall, the working assumption was that a neuron summed its synaptic inputs and the dendrites were plumbing. Cable theory made that untenable.

A synapse far out on a dendrite delivers a smaller, slower, more smeared signal to the soma than an identical synapse near it. So where an input lands is part of what it means — location is a parameter of the computation, not an implementation detail. That single consequence opened dendritic computation as a field, and it came from applying a telegraph equation.

It also explained a methodological headache: the same cable properties that filter synaptic inputs distort recordings made at the soma, meaning a lot of what an electrode sees has been low-pass filtered by the cell’s own geometry before it arrived.

Both directions

Like the Hodgkin–Huxley model, this is neuroscience built out of borrowed circuit theory — and the borrowing was explicit, not analogical. The return trip is the compartmental model: chop a dendritic tree into short cable segments, and you have a circuit netlist you can simulate. Modern neural simulators are, structurally, circuit simulators: they solve the same kind of stiff, netlist-defined system, with the same class of implicit methods, that SPICE was written for.