Rate coding

By 1914 there was good reason to think a nerve impulse comes in one size. Keith Lucas had shown that a frog muscle supplied by eight or nine motor fibres contracts in a few discrete steps as the stimulus is raised, and his pupil Edgar Adrian argued that the impulse in a nerve fibre is likewise all or nothing: its size is set by the fibre, not by the stimulus. That leaves an awkward question for anyone who can tell a touch from a push. If the size of the signal is fixed, where does the intensity go? In 1926 Adrian and Yngve Zotterman recorded from a single sensory fibre and found out. A stronger stimulus does not make bigger impulses. It makes more of them each second.

One fibre, one stretch receptor

Adrian’s first paper, published alone in March 1926, tried a new recording instrument on whole nerves. A weight hung on a frog’s calf muscle set up impulses in the sciatic nerve, more often with heavier weights and less often the longer it hung. But a nerve trunk holds many fibres, and a rising count could mean faster fibres or more of them. The record could not say which.

The second paper, with Zotterman five weeks later, settled it by subtraction. They loaded the sterno-cutaneus, a small muscle Lucas had mentioned in a footnote, with weights from a quarter of a gram to five grams, and cut away strips until the record resolved into one regular rhythm: a single end-organ driving a single fibre.

• The impulses did not change size over an eightfold range of load.
• The frequency rose with the load, over roughly 5 to 100 a second.
• The frequency fell with time. Under a steady 1 g load the interval between impulses grew from 26 ms after 5 seconds to 80 ms after 90. The receptor adapts.

Their explanation needed no new physiology. After each impulse the end-organ is refractory and its excitability recovers gradually; a steady stimulus fires the next impulse when recovery reaches the level that stimulus can trigger, and a stronger stimulus gets there sooner. An engineer would call it a relaxation oscillator whose period is set by its input — a voltage-to-frequency converter, with the refractory period as its reset.

The amplifier was the experiment

The physiology could have been guessed. The measurement could not. An isolated impulse in Adrian’s records was 15–25 microvolts, about a thousand times smaller than the response of a whole nerve shocked electrically, and over in a few milliseconds. The string galvanometer most physiologists used had a string with mass, which distorted a brief nerve response in a way that, Adrian wrote, “no amount of amplification can overcome”.

The gain came from radio. Alexander Forbes and Catharine Thacher, at Harvard, described putting an electron tube in front of a string galvanometer in 1920, and noted that amplifying with the tube had been “standard practice for years” in radio communication. How much it helped depended on impedance: a tube whose grid is biased negative draws almost no current from a high-resistance nerve, so the system “operates as an electrometer rather than a galvanometer”. On a frog’s sciatic nerve the gain was about 45-fold.

Adrian paired the gain with a recorder that had almost no inertia. The capillary electrometer is insensitive, but its mercury is so heavily damped that it behaves as a first-order system, and its lag can be undone by adding a constant times the slope of the trace. In front of it went three Marconi D.E. 5b valves, resistance–capacity coupled, built by W. G. Pye of Cambridge on the lines of an amplifier Herbert Gasser used in America: a voltage gain of 1,850, holding up to about 5 kHz.

Adrian later put the principle in a sentence any instrument designer would recognise: “The recording instruments used nowadays are actually far less sensitive than their predecessors.” With the valves supplying the energy, the recorder is chosen for speed and the limit moves to the amplifying circuits, noise floor included. A microvolt or two was within useful amplification; the thinnest fibres, suspected of carrying pain, were probably lost in “the random fluctuations due to the operation of the thermionic valve”.

The valve brought its vocabulary too: Justin Garson dates Adrian’s shift to describing impulses as “messages”, “signals” and even “codes” to around 1926, after a war in which valves had been used to amplify and intercept coded messages. It also brought the ear. In 1928 Adrian and Bronk added a loudspeaker to the amplifier’s output so that a discharge could be judged “by the ear instead of the eye”; fibres too small to see on the electrometer still made “a series of faint clicks”. Hubel and Wiesel’s 1962 paper on receptive fields still tracks background activity “audible over the monitor as a crackling noise”. Adrian shared the 1932 Nobel Prize with Charles Sherrington, and his lecture opened by crediting “the advent of the triode valve amplifier”.

Read as a signal

A train of identical pulses whose rate carries a value is pulse-frequency modulation, and its attraction is that a pulse of known shape can be regenerated. Adrian had argued, in 1912 and 1914, that the size of the impulse at any point in a fibre “depends only upon the local condition of that point and not upon the previous history of the disturbance” — which is what a repeater does. Whatever happens to the amplitude along the way is discarded; what survives is the timing. The Hodgkin–Huxley model supplied the mechanism in 1952.

Decoding is the mirror image: integrate. Each motor impulse produces a twitch lasting far longer than itself, so a train sums into a force that grows with the rate; the muscle is the low-pass filter that turns pulse rate back into a level. Force is also graded by recruiting more motor units, and because they fire out of step their ripples partly cancel — interleaving, to a power engineer. Adrian’s Nobel lecture made the point audibly: a gramophone record of a sensory nerve let the audience hear “the two kinds of gradation”, frequency in each unit and the number of units in action. That population half of the code is what intracortical brain–computer interfaces read: Georgopoulos and colleagues found in 1986 that motor cortex cells only broadly tuned to the direction of an arm movement predict it together.

What counting costs

A rate is not an instantaneous quantity: to know how often something happens you have to wait for it to happen more than once. Adrian saw the cost — “the briefer the discharge the less opportunity will there be for signalling by change of frequency” — and a fibre firing 20 times a second cannot give a rate in less than one interval, 50 ms. Adaptation adds an ambiguity, since the frequency “depends on the rate of development of the stimulus, as well as on its intensity”: a high rate can mean a strong stimulus or merely a fresh one.

Irregular firing makes it worse. Adrian’s stretch receptor, under a steady load, fired like clockwork; cortical neurons are far more variable. Take the extreme case, spikes arriving at random at rate $r$. A count over a window $T$ then has mean $\bar N = rT$ and standard deviation $\sigma_N = \sqrt{rT}$, so its relative error is

$$\frac{\sigma_N}{\bar N} = \frac{1}{\sqrt{rT}}$$

Thorpe and colleagues asked people whether a photograph flashed for 20 ms contained an animal, and the EEG responses to the two kinds of image diverged about 150 ms after the flash. A neuron firing 50 times a second fits seven or eight spikes into 150 ms, a relative error of more than a third — and that budget has to cover all the visual processing the task needs, not one stage of it.

There are two ways out. One is to count across cells instead of across time, which is where Adrian himself looked: the limitation was “really a small matter”, he said, “for in the body the nervous units do not act in isolation”. A hundred neurons with independent noise cut the relative error tenfold, and Shadlen and Newsome estimated that about 100 cortical neurons give a reliable rate in one interspike interval, 10–50 ms. The other is to stop throwing the timing away. Mainen and Sejnowski found that a cortical neuron repeats its spike times to within a millisecond when driven by a fluctuating, synapse-like input rather than a constant one. A count discards that precision; a temporal code keeps it.