Cells that fire together
An output neuron receives two synapses, A and B, both starting at $w = 3$ mV. You repeatedly pair A with the output — forcing both to fire at the same instant — while B stays silent the whole time. Which weight grows?
Before training, firing A alone lifts $V$ by just $3$ mV — a ripple that decays far from threshold, so the output stays silent and $w_A$ stays put. Each pairing fires input and output together, and each coincidence adds exactly $1$ mV: after $9$ pairings the readout shows $w_A = 12$ mV, where the bar hits its cap. $w_B$ has not moved. Fire A alone now and the bump is $12$ mV — four times the synapse you started with.
This is Hebb's rule in its simplest form: a synapse strengthens when its input and its output fire together, and nothing else changes it. There is no depression in this version — an input firing without the output writes zero, never a decrease.
The weight is a counter of correlation: B never fired, so its synapse recorded nothing. But growth feeds back on itself — a stronger synapse fires the output more easily, earning more coincidences — so an up-only rule runs away. The hard cap at $12$ mV stands in for the stabilizers real synapses have.
About this model
What it represents: a two-synapse cell under the plainest Hebbian coincidence rule — fire together, wire together, one millivolt per co-firing, with an imposed ceiling.
What it leaves out: long-term depression (real weights can shrink), spike-timing dependence (STDP — real plasticity cares whether the input led or lagged the output by milliseconds, and flips sign accordingly), and normalization or homeostasis across synapses. The cap is needed precisely because of these omissions: with potentiation only, strengthening makes future strengthening more likely, and the positive feedback loop has no brake — so a hard cap at $12$ mV substitutes for the depression and homeostatic scaling this model deletes.
Source: Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics, Ch. 19.1 (EPFL, free online) — Hebb rules, their instability, and the stabilizing terms this lesson omits.
Reset, then pair A with the output four times. What is $w_A$?
Through all nine pairings, $w_B$ never moves. Why?
Transfer: delete the $12$ mV cap but keep the rule. What does the model predict for a synapse that keeps getting paired?
A Hebbian synapse is a correlation detector: its weight counts the times its input and its output fired together, so the paired path strengthens and the silent one does not. Growth-only learning is unstable by construction — some opposing force, here a hard cap, has to hold the loop down.