Hebbian learning

That is the whole idea, and the phrasing is most of why everyone knows it. A synapse between two cells strengthens when those cells are active together. Repeat it enough and the connection has recorded something about the world — no instructor, no answer key, nothing but the coincidence itself.

The line is not Hebb’s. It is a mnemonic generally credited to Carla Shatz some four decades later, and it is looser than what he proposed: Hebb specified repeatedly, and he specified a direction — cell A taking part in firing cell B, rather than the two merely coinciding. Both matter, and the mnemonic drops both. It has survived anyway, because it gets the idea across in six words and the corrections take a paragraph.

Her own wording, in a 1992 Scientific American article on the developing brain, was “cells that fire together wire together”. She was describing the inputs from the two eyes sorting themselves into separate columns of visual cortex, a sorting that stimulating both optic nerves in synchrony can prevent. Löwel and Singer had written “neurons wire together if they fire together” in Science eight months earlier, about horizontal connections in the cortex of kittens raised with a squint, which came to link mainly cells driven by the same eye. Neither was writing about memory. Both were writing about how a visual system wires itself.

What Hebb actually wrote

Hebb began The Organization of Behavior at the Yerkes primate laboratories in Florida and finished it at McGill. He meant it as a theory of behaviour that would bridge neurophysiology and psychology, and the synapse entered it to solve one problem: persistence. Some memories are established instantly and are permanent. Activity reverberating around a closed loop of cells could hold a trace for a second or so, but it would be unstable — refractoriness or any outside event could interrupt it. A structural change could make it permanent, but growth takes time. So he let the reverberation carry the memory until the growth was made, and stated on page 62 when the growth would happen: when an axon of a cell A near enough to excite a cell B “repeatedly or persistently takes part in firing it”, some growth or metabolic change in one or both cells raises A’s efficiency at firing B.

His guess at the mechanism followed immediately — synaptic knobs growing to enlarge the area of contact — with a warning that the rest of the argument depended only on the general statement. The statement was there to build the cell assembly: a diffuse group of cells that repeated stimulation binds into something able to act briefly as a closed system. A series of assembly activations, a “phase sequence”, was in his theory the thought process itself.

Nor did he claim the core idea. On page 70 he calls it an old one: cells repeatedly active at the same time become associated, so that activity in one helps fire the other. Jerzy Konorski had published closely similar proposals a year earlier, which is why some researchers write Hebb–Konorski plasticity. He did not claim the synaptic assumption either; by his own account it had been made repeatedly before (page 60). What he listed as new (page 61) was its setting: connections between single cells, none able to fire another alone; no direct stimulus–response links in the adult; and a reverberating trace that holds the memory while the growth is made.

The rule

Strip it to its arithmetic and it is about as simple as a learning rule gets: the change in a connection is proportional to the product of activity on each side of it.

$$\Delta w_{ij} \propto x_i \, y_j$$

Here $w_{ij}$ is the strength of the connection from cell $i$ to cell $j$, and $x_i$ and $y_j$ are the activities of the sending and receiving cells — firing rates, in the usual reading.

Everything on the right-hand side is available at the synapse. Nothing has to be computed elsewhere and delivered — no global error term, no signal that has to travel backwards through the network.

Two things are missing, and both come back. Firing rates cannot be negative, so neither can their product: as written, the rule only ever strengthens. And there is no time in it — together is left undefined.

Why an engineer cares about locality

That property is the whole reason this crossed into hardware.

A learning rule requiring a global error signal implies routing: some quantity must be computed centrally and distributed to every adjustable element. On a chip that means wires, memory traffic, and energy — the same costs that make conventional accelerators expensive to run. A rule that needs only the two signals already present at a connection can be implemented in that connection.

This is why spike-timing-dependent plasticity — Hebb’s rule refined so that the relative timing of pre- and post-synaptic spikes sets the sign of the change, not just their coincidence — keeps appearing in neuromorphic designs. Not because it out-performs the alternatives, but because it is the kind of rule you can build out of local circuitry rather than move data around to evaluate.

The instability, which is the interesting part

Pure Hebbian learning does not work. Left alone it is positive feedback: strong weights make the post-synaptic cell more likely to fire, which strengthens those weights further, without bound. Every weight saturates and the representation collapses.

So something has to bound it, and the candidates are all recognisable control problems — normalising the total synaptic weight per cell, making the threshold for strengthening slide with recent activity, or scaling synapses homeostatically over slower timescales. Real neurons appear to use several of these at once, on different timescales.

Normalisation. Hold the size of each cell’s weight vector fixed, so that one synapse grows only at the others’ expense. Done literally it is a global step — after every update, divide each weight by a norm taken over all of the cell’s synapses. Erkki Oja showed in 1982 that for small updates the division reduces, to first order, to a decay term inside the rule:

$$\Delta w_i = \eta \, y \, (x_i - y \, w_i)$$

where $\eta$ is a small learning rate, $x_i$ the input on synapse $i$, $w_i$ its weight, and $y$ the cell’s output. The subtracted term, which Oja described as an internal feedback in the neuron, makes every synapse leak faster the harder the cell responds. The sum of the squared weights settles near one with no normalisation step anywhere, and each synapse needs only its own weight and the cell’s output — the stabiliser is as local as the rule. Fed a stationary input, the neuron tends to extract its principal component: Karhunen–Loève feature extraction, done by one cell.

The sliding threshold. Bienenstock, Cooper and Munro made the sign of the change depend on how strongly the receiving cell responds, also in 1982. Below a modification threshold an active synapse weakens; above it, it strengthens. The threshold moves, more than proportionally, with a slow running average of the cell’s own activity, so a busy cell finds strengthening harder and a quiet one finds it easier — an adaptive threshold, a slow loop that keeps a fast nonlinearity near its useful operating point. The theory was built to explain how cells in visual cortex become orientation-selective and binocular, the receptive fields Hubel and Wiesel had mapped. In 1996 Kirkwood, Rioult and Bear caught the threshold moving: in the visual cortex of light-deprived rats, potentiation was enhanced and depression diminished, and two days of light reversed it.

Synaptic scaling. Turrigiano and colleagues found a slow loop in living neurons in 1998. Blocking activity in cultured cortical neurons scaled up the strength of all of a neuron’s excitatory inputs; raising activity, by blocking inhibition, scaled them down over 48 hours until firing rates returned close to where they started. Each synapse apparently changed in proportion to its strength. That multiplicative form is what makes it compatible with Hebbian learning: an automatic gain control that sets the overall level without disturbing the ratios between synapses, which is where the learning is stored.

The awkward part is speed. A few seconds of high-frequency stimulation was enough to induce lasting potentiation in Bliss and Lømo’s rabbits; scaling played out over two days. Zenke, Gerstner and Ganguli argued in 2017 that at the rates measured, homeostatic plasticity is too slow to stop the runaway on its own, and that faster compensatory processes are needed alongside it.

Which is a nice illustration of a pattern that recurs across this site: the borrowed idea was not sufficient on its own, and what made it usable was stability engineering.

From conjecture to physiology

Worth being clear that Hebb was speculating. He had no evidence for the mechanism in 1949 — it was a postulate offered because it would explain behaviour, not because anyone had seen a synapse change. He conceded as much himself: there was no direct evidence for the growing knobs.

Bliss and Lømo’s demonstration of long-term potentiation in 1973 gave it a physiological basis, twenty-four years later. That gap, and the fact that the conjecture survived it, is most of why the idea carries the weight it does.

Lømo had first seen the effect in Per Andersen’s laboratory in Oslo in 1966, and Bliss joined him there in 1968. Trains of stimuli lasting seconds, delivered to the perforant path into the dentate area of the hippocampus, left the response to single pulses potentiated for between thirty minutes and ten hours in fifteen of eighteen anaesthetised rabbits. A companion paper by Bliss and Gardner-Medwin found it lasting up to three days in unanaesthetised ones.

What 1973 established was persistence; the conjunction came after. Holding the postsynaptic cell hyperpolarised during the conditioning train blocked potentiation (Malinow and Miller, 1986), and pairing input with postsynaptic spiking strengthened only the synapses active at the time (Kelso, Ganong and Brown, in a 1986 paper titled “Hebbian synapses in hippocampus”).

The molecule doing it is a coincidence detector. The NMDA receptor opens when glutamate released by the presynaptic cell binds to it, but near resting potential its channel is blocked by magnesium ions, which depolarisation of the postsynaptic membrane relieves — shown in 1984 by Nowak and colleagues with the patch clamp, and by Mayer, Westbrook and Guthrie. Substantial current needs both, which puts something close to the product in the equation above into one receptor. Collingridge, Kehl and McLennan had found in 1983 that blocking it left ordinary transmission alone but prevented potentiation; it is the receptor that, as Bliss and Collingridge put it, gives long-term potentiation its Hebbian character.

Timing came last. Order effects had been seen with trains of stimuli — Levy and Steward reported one in 1983 — but in 1997–98 the measurement reached single spikes: Markram and colleagues in pairs of neocortical neurons, then Bi and Poo in cultured hippocampal neurons, found that an input arriving up to about 20 ms before the postsynaptic spike was strengthened and one arriving after it weakened.

That is “takes part in firing”, measured. An input early enough to have helped cause the spike is strengthened; one that arrived after it, and so could not have, is weakened. Cells that fire together, it turns out, do not always wire together. Order decides.

The rule in silicon

The constraint that makes the rule plausible in tissue is one chip designers impose on themselves. The 2018 paper describing Intel’s Loihi makes it a design rule: only the destination neuron may read or change a weight, and learning may use only local information such as the pre- and postsynaptic spike trains. That is Hebb’s locality written as a hardware specification, and the interesting differences are in where the weight is kept.

Charge on a floating gate. Diorio, Hasler, Minch and Mead’s single-transistor synapse (1996) stores the weight as charge on an isolated gate, where it stays without power, and moves it either way by hot-electron injection and electron tunnelling. Because the update depends on the stored value and the transistor’s own terminal voltages, the device computes with its weight and adjusts it in the same place: an array of them produces its output and all its local updates in parallel, at subthreshold currents, the regime analogue silicon neurons also run in.

Charge on a capacitor. The conventional memory element, but capacitors leak. In Indiveri’s analogue spike-timing synapses (2002), each spike leaves a decaying waveform that a spike from the other side samples to nudge the weight voltage up or down; on long timescales a comparator built from minimum-size transistors drives that voltage to one of two stable levels. The alternatives he set out show the trade — refresh the value like DRAM, digitise it and store it off chip, or use a floating gate with a control circuit for every synapse. Bistability keeps each synapse small and its memory stable, at the price of resolution: in the long run, one bit.

Numbers in a learning engine. Loihi is digital: 128 neuromorphic cores and 2.07 billion transistors in a 14 nm process, each core holding 1,024 compartments, from which its neurons are built, and its own programmable learning engine. Each spike train is reduced to a trace — a number that jumps at every spike and decays by a fixed factor each time step, a first-order low-pass filter — and learning rules are sums of products of traces, the weight, and a reward trace broadcast to chosen sets of synapses. Short time constants capture spike timing, long ones firing rates. Weights take one to nine bits.

Even there, locality is not free. A presynaptic spike already has to look up its weights, so the depressing half of a spike-timing rule costs only a read turned into a read-modify-write. The potentiating half is triggered by the postsynaptic spike, which has to find every synapse onto its neuron — a reverse lookup the paper names as the one significant difficulty pairwise rules introduce. Loihi sidesteps it by deferring all updates to the end of a learning epoch. Local in the tissue does not mean local in the memory map. A fourth place to keep the weight — a resistance set by the history of the current through it — is the subject of memristive synapses.

What the rule leaves out

Consequence. A Hebbian synapse strengthens whatever co-occurs, useful or not; nothing in the rule knows whether the coincidence mattered. For learning that serves behaviour, something has to say so, and the brain’s candidates are neuromodulators such as dopamine, whose firing carries something close to a reward prediction error. Three-factor learning rules formalise it: pre- and postsynaptic coincidence marks a synapse as eligible, and the change happens only if the modulator arrives. Loihi’s reward trace is the same idea in hardware.

The spike itself. Hebb wrote “takes part in firing”, but the NMDA receptor reads voltage. Pairing input with depolarising current injected into the cell is enough to induce potentiation (Wigström, Gustafsson and colleagues, and independently Sastry, Goh and Auyeung, both in 1986), and Kelso, Ganong and Brown called the mechanism they found pseudo-Hebbian: conjunctive, as Hebb required, but through biophysics unknown when he wrote.