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. Physiologists had already borrowed it for axons. Rall’s contribution was taking seriously that a dendrite is the same problem: a resistive core, a leaky capacitive membrane, and no amplification along the way.

With $V$ the membrane voltage measured from rest, $x$ the distance along the cable and $t$ the time:

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

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. The time constant belongs to the membrane alone, $\tau = R_m C_m$, the resistance and capacitance of a unit area of it multiplied together. The length constant also depends on the branch, $\lambda = \sqrt{R_m d / 4R_i}$, with $d$ the diameter and $R_i$ the resistivity of the core — so a branch a quarter as thick is electrically twice as long.

From telegraph to dendrite

William Thomson, later Lord Kelvin, worked the problem out in letters to George Stokes in October 1854, with an eye on the 2,000 or 3,000 miles of wire a cable to America would need; the Royal Society received the paper in May 1855. He treated the cable as a resistance and a capacitance per unit length, recognised Fourier’s equation for heat flowing along a bar, and drew the conclusion that mattered for a transatlantic line: the retardation of a signal grows with the square of the length, so each signal on a cable four times longer takes sixteen times as long. He also added a term for imperfect insulation, and that leaky version is, term for term, the equation above: divide it through by his leakage coefficient and $\lambda^2$ and $\tau$ appear. In 1876 Oliver Heaviside added the wire’s self-induction per unit length — the series inductance of the full telegrapher’s equations — and noted that setting it to zero brings back the submarine cable. The equation above is the telegrapher’s in that limit, with Thomson’s leakage kept.

Physiology had that picture of a nerve fibre long before anyone applied it to a dendrite. Ludimar Hermann was treating the fibre as a core conductor in the late nineteenth century. In 1939 Kenneth Cole and Alan Hodgkin measured the longitudinal resistance of a squid giant axon against electrode spacing, fitted it with a curve derived from “the equations of Kelvin”, and extracted its characteristic length, a few millimetres. In 1946 Hodgkin and William Rushton wrote the equations for a fibre in full — membrane resistance and capacitance, axoplasm resistance, external fluid — and tested them on a 75 µm lobster axon: the steady potential fell off exponentially with distance, as predicted. By the time Rall came to the problem, cable theory was established axon physiology, and it had even been pointed at dendrites. In 1955 Coombs, Eccles and Fatt modelled a cat motoneuron as a sphere carrying six unbranched dendrites of “indefinitely great length” and calculated its membrane resistance “on the basis of cable theory”.

What Rall added came in three steps. In 1959 he solved the steady state for a dendritic tree with arbitrary branch lengths and diameters, and found the 3/2 power of diameter at the centre of it: a cylinder’s input conductance goes as $d^{3/2}$, so where the daughters’ values add up to the parent’s, the daughters load the parent exactly as more of the parent would — an impedance match. In 1962 he collapsed trees built that way, with every tip at the same electrotonic distance, onto a single equivalent cylinder with the soma at one end, valid for transients as well as the steady state. In 1964 he chopped that cylinder into ten compartments and integrated it numerically, which is the compartmental model.

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.

It did so with numbers. Eccles and his colleagues put the membrane resistivity at 400–600 Ω cm², and their model gave the dendrites about 2.3 times the input conductance of the soma. Rall’s 1959 analysis put the ratio between about 21 and 35 — electrically, the dendrites are most of the cell — and the resistivity nearer 5,000 Ω cm², mainly because the dendrites were bigger than had been assumed. Coombs, Eccles and Fatt had suspected as much: “the dendritic processes must contribute rather more than has thus far been allowed”.

Because $\lambda$ goes as the square root of $R_m$, a resistivity ten times too low makes every length constant about 3.2 times too short: a tree two length constants long looks more than six long, and synapses at its far end look useless. Rall later traced Eccles’s long electrotonic lengths to exactly that error, and as late as 1964 Eccles was still writing that synapses on the remote parts of dendrites were “virtually ineffective”. By Rall’s account, it was theory and experiment from him and colleagues at the NIH, published in 1967, that persuaded most neurophysiologists otherwise.

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.

Timing compounds it. In the 1964 compartmental model, four equal inputs switched on in sequence from far out on the dendrite towards the soma gave a somatic peak nearly twice as large as the same four switched on from near the soma outwards — a direction-selective response from nothing but passive delay, which Rall noted could serve to detect movement.

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.

Rall’s first intervention, a note to Science in 1957, was exactly this: fast voltage transients recorded from cat motoneurons were being fitted as single exponentials, as if the cell were a soma without dendrites, which gave too short a membrane time constant. Run in reverse, the same physics is the space-clamp problem that the voltage clamp and the patch clamp meet: a clamp at the soma loses its grip on a synapse steeply with distance.

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.

It has been done in SPICE itself. In 1985 Idan Segev and colleagues coded neurons with arbitrarily branched dendritic trees as lists of short cylindrical segments, each a resistor and capacitor in parallel for the membrane with core resistors in series, and on test trees built to Rall’s constraints SPICE’s transients matched the analytical solutions closely.

Purpose-built simulators keep that structure. NEURON integrates by backward Euler by default, or by a variant of Crank–Nicolson, which is built on the trapezoidal rule — SPICE’s default. The stiffness comes with chopping up a cable. A passive cable relaxes as a sum of spatial modes whose decay rates grow with the square of their spatial frequency, and a model keeps one mode per compartment, so refining the mesh tenfold makes its fastest mode about a hundred times faster while the membrane time constant stays put. An explicit integrator’s step has to stay comparable to that fastest mode’s time constant just to remain stable; backward Euler’s does not.

One difference is a gift from anatomy. A dendritic tree has no loops, so, as Michael Hines showed in 1984, Gaussian elimination can be ordered from the leaves inwards to solve it in $O(N)$ operations for $N$ compartments — exactly the cost of an unbranched cable, against $O(N^3)$ for elimination on a general, dense set of $N$ equations. A circuit simulator has to allow for loops; a neuron simulator can exploit their absence.

What the passive cable leaves out

“No amplification along the way” is an assumption, and it has not survived intact. The equation is a small-signal model, and Hodgkin and Rushton used it as one: they made their quantitative measurements with currents of a third to a half of threshold, and as a current approached threshold a local response always appeared. Dendrites carry voltage-gated channels too: in 1994 Greg Stuart and Bert Sakmann patch-clamped the dendrites of neocortical pyramidal cells and found action potentials that start in the axon and then propagate actively back into the dendritic tree, which no passive cable can do. Compartmental models absorb this by giving the compartments excitable membrane, as Rall and Gordon Shepherd were already doing in 1968. The closed-form cable results do not.

The equivalent cylinder asks for more again: the 3/2 rule at every branch point and every tip at the same electrotonic distance. Rall never expected real trees to meet that exactly; the branch diameters of cat motoneurons roughly agree with it, which is why the idealisation earned its keep.

And every prediction is only as good as constants that are hard to pin down. Hodgkin and Rushton’s thirteen lobster experiments gave membrane resistances anywhere from 600 to 7,000 Ω cm². The square root in $\lambda$ blunts a spread like that; the Eccles episode shows that it does not remove it.