Noise

Thermal noise is the same $4kTBR$ in an op-amp and in a pipette seal, which is why electrophysiology is fundamentally an instrumentation problem. What is stranger is how often biology looks less like it is suppressing noise than putting it to work.

The floor under every amplifier

In 1918 Walter Schottky sought the limits of valve amplification. The first, curiously, was the size of the elementary charge: a current of discrete charges fluctuates — shot noise. The second was heat, which he judged the less restrictive. John B. Johnson reported measuring it in 1927, and in 1928 his full paper and Harry Nyquist’s derivation followed as consecutive papers: a mean-square voltage of $4kTBR$, for Boltzmann’s constant $k$, absolute temperature $T$, bandwidth $B$ and resistance $R$ — strictly, as Johnson wrote it, the real part of an impedance. A third, flicker noise, rises towards low frequencies roughly as $1/f$; Johnson found it in valves in 1925, and Schottky named it the flicker effect the next year.

Amplifier noise is quoted referred to the input, as the signal that would match it, and bandwidth is a noise decision: white noise grows as $\sqrt{B}$ in RMS, and $1/f$ noise puts equal power into every decade.

Noise in small numbers

Faisal, Selen and Wolpert’s 2008 review finds noise at every stage, worst where molecules are few. Photons arrive at random, at a photoreceptor as at a two-photon microscope. Ion channels open at random, and a thin axon’s input resistance is high enough that one sodium channel opening by chance can fire a spike: Faisal, White and Laughlin calculated in 2005 that axons below about 0.1 µm would be useless, close to the thinnest found. Small central synapses typically release transmitter with low probability. Yet the spike generator looks quiet: driven by synapse-like fluctuations, a neuron in a slice of rat neocortex repeats its spike times to within a millisecond (Mainen and Sejnowski, 1995).

Some noise may be put to work. In 1993 Douglass and colleagues added noise to a weak periodic stimulus at crayfish mechanoreceptors, and the signal came through best at an intermediate level: stochastic resonance, an effect already shown in physical systems. Paddlefish detect plankton over a wider range in the right amount of electrical noise, which Daphnia swarms produce (Russell, Wilkens and Moss, 1999).

Where they meet, and where they part

They meet at the electrode. The voltage clamp measured milliamps per square centimetre of squid axon. The patch clamp measures picoamps, which is why routine single-channel recording waited for gigaohm seals: at 100 MΩ the seal’s Johnson current noise, $\sqrt{4kTB/R}$, is 0.4 pA RMS over 1 kHz. Above about 10 Hz an electrode’s own noise is mostly thermal, from the real part of its impedance, so smaller microelectrode-array sites make more of it and Neuropixels chose low-impedance ones. They meet in the filter too: in dim light, at low signal-to-noise ratio, the fly’s first visual interneurons weaken and spread their surround, as efficient coding predicts (Srinivasan, Laughlin and Dubs, 1982).

They part over what counts as signal. Johnson noise is only heat; much of an electrode’s noise is other neurons. Bharat Biswal, by his own later account, set out to catalogue the noise sources in fMRI and found the largest was slow, with a $1/f$ shape; in 1995 he and colleagues showed it correlated across the resting motor cortex. The resting EEG falls off as a power law too, $1/f$-like rather than exactly $1/f$. A shared spectrum is not a shared cause.