Lateral inhibition

H. Keffer Hartline had taken the receptors of the horseshoe crab’s eye to be independent, until he noticed that stray light in the laboratory often slowed the one he was recording from instead of speeding it up. Its neighbours, seeing the room lights more directly, were inhibiting it. When every receptor in a sheet holds down its neighbours’ output, the sheet reports differences in light more than light: edges are exaggerated, uniform regions suppressed. Ernst Mach had inferred as much from perception in 1865, arguing that the bright and dark bands seen at the two ends of a ramp of light must come from reciprocal inhibition between neighbouring points of the retina. The crab let it be measured.

Two receptors, two equations

Each ommatidium of the crab’s coarsely faceted eye is one receptor unit, its output carried by a single fibre of the optic nerve. Hartline reported the inhibition in a 1949 abstract and described it in full in 1956 with Henry Wagner and Floyd Ratliff: purely inhibitory, stronger the brighter, larger and nearer the lit area, and abolished by cutting the plexus of nerve fibres that joins the ommatidia.

The 1957 paper, with Ratliff alone, made it a model. They recorded two fibres at once, from ommatidia no more than a few millimetres apart, and lit each alone and both together. Each fired less in company, and its loss was a linear function of the other’s firing rate, not of the light on the other, once that rate passed a threshold — about 8 or 9 impulses a second in one pair a millimetre apart, where each further impulse per second from one cost the other 0.15 or 0.17 impulses per second. Hartline later wrote the equations for many receptors as

$$r_p = e_p - \sum_{j \ne p} K_{p,j} \left( r_j - r^{0}_{p,j} \right)$$

where $r_p$ is the firing rate of receptor $p$, $e_p$ its rate when lit alone, $K_{p,j}$ the inhibitory coefficient of receptor $j$ on $p$, and $r^{0}_{p,j}$ the threshold of that action; a term counts only while $r_j$ is above it.

The inhibition is driven by outputs, and every output is itself inhibited, so this is a feedback network. Without thresholds it reads $(I + K)\,r = e$, which the eye solves by settling. Expanding the inverse gives $e - Ke + K^2 e - \dots$, and the second correction is the inhibitors being inhibited. Hartline and Ratliff saw it: a near patch of light cut a receptor’s count by 50 impulses in 10 seconds, a far patch by 5, and both together by only 40, because the far patch slowed the near one. They called it disinhibition. A feedforward network, which subtracts a weighted sum of inputs, stops at $e - Ke$ and cannot do it.

In 1959 they went back to Mach. Moving a photographic plate with a ramp of density across the eye while recording one fibre, they got a faithful trace of the ramp with every other receptor masked, and with the mask off a maximum and a minimum at its two ends, where a person sees Mach bands. Hartline shared the 1967 Nobel Prize in Physiology or Medicine with Ragnar Granit and George Wald.

The filter engineering already had

Subtracting a weighted neighbourhood average from each point is a spatial filter. Image processing uses a gentle version, unsharp masking: blur a copy, subtract it from the original and add a multiple of the difference back, so uniform regions pass unchanged and an edge gains a bright fringe and a dark one — Mach bands, made on purpose. Retinal ganglion cells keep mostly the difference, through the antagonistic centre and surround of their receptive fields. In 1966 Christina Enroth-Cugell and J. G. Robson measured 21 cat ganglion cells with drifting sinusoidal gratings and, taking centre and surround as Gaussians that subtract, as Schade and Rodieck had, fitted every one. The spectrum of a difference of Gaussians is another, so the filter is band-pass: the centre’s width cuts fine detail, and the near-cancellation of centre and surround cuts coarse structure. Their fitted surrounds weighed 73% to 98% as much as the centres, leaving a uniform field 2% to 27% of what the centre alone would deliver.

Srinivasan, Laughlin and Dubs read the subtraction statistically in 1982: the surround predicts the centre from its neighbours, and subtracting the prediction strips out much of the correlation natural images carry, so the cell’s output range is spent on what was not predictable and fine detail survives noise added further on — efficient coding in one operation. Since most of a scene is locally predictable, most outputs sit near zero and the large ones gather where the image changes: a sparse code.

What silicon borrowed

The arithmetic was not new: Mead and Mahowald noted that a difference of Gaussians approximates the Laplacian filters widely used in computer vision. What their 1988 silicon retina took from biology was the architecture. Horizontal cells in many species are joined by gap junctions into a continuous sheet, and Misha Mahowald and Carver Mead copied it as a hexagonal network of transistor resistors, each photoreceptor driving its node through a conductance, so that Kirchhoff’s laws do the averaging. The network is a two-dimensional leaky cable, lateral resistors for the core and the conductances for the membrane, and their ratio sets how far a receptor’s influence reaches. An amplifier in each pixel reports receptor minus network, and the chip’s response to an edge resembled a cat ganglion cell’s. Unlike the crab’s eye, it computed the surround from its inputs: the feedforward form.

Push mutual inhibition hard enough and a network stops reporting differences and starts choosing. In 1989 Lazzaro, Ryckebusch, Mahowald and Mead described that extreme in silicon, the winner-take-all. Wiring $n$ units to inhibit one another takes $n(n-1)$ connections; their circuit used one wire. Each unit, two transistors, pushes current onto the shared wire and reads its voltage back. The largest input sets that voltage and every other output falls to about zero, while the winner’s output encodes the logarithm of its input over four orders of magnitude. Versions with more than 170 inputs worked inside chips for sound localisation and stereo vision. One active unit out of many is as sparse as a code gets.

One subtraction is not a retina

The recurrent and feedforward forms are different machines, and the vertebrate retina’s surround is partly recurrent too: in 1971 Baylor, Fuortes and O’Bryan found that current passed into a turtle’s horizontal cell changed the potential of a nearby cone. A loop with a delay can turn subtraction into addition. Lateral inhibition in Limulus sets in noticeably late, and in 1967 Ratliff, Knight, Toyoda and Hartline found that enlarging the lit area reduced the eye’s response to slow flicker but amplified it at intermediate frequencies, around 3 Hz, the inhibition from one peak arriving in the next trough. The winner-take-all, also a loop, avoids ringing only if its bias current is large enough.

The textbook demonstration has a hole in it. A single centre–surround filter predicts its largest Mach bands at a sharp step in light, where none are seen; Ratliff himself argued in 1984 that sharp edges actively suppress them, which no single subtraction does.

Nor is the surround fixed. Enroth-Cugell and Robson’s measurements suggested that dimming the light widens a cell’s summing regions and leaves its surround relatively ineffective, the behaviour Srinivasan, Laughlin and Dubs later derived from noise: subtracting a neighbour’s estimate removes shared signal but adds the neighbour’s noise, and when photons are scarce averaging is worth more than differencing. Enroth-Cugell and Robson also found cells, which they called Y-cells, that summed very non-linearly, and noted that a divisive centre–surround interaction would be as consistent with their data as a subtractive one.