Neural plasticity · learning rules

Spike-Timing-Dependent Plasticity

Hebb said “fire together, wire together”, but he did not say in which order. STDP is the experimental answer: a synapse reads the sign of the interval between the pre-synaptic spike and the post-synaptic one, potentiating when the input helped cause the output and depressing when it arrived too late to have mattered.

Set Δt, then run the pairing protocol.

Pairing

Δt = t_post − t_pre +10 ms Positive: the input spike arrives before the cell fires, so it could have contributed. Negative: it arrives after, too late to be a cause.

Pairings60 Bi & Poo paired 60 times at 1 Hz.

Pairing frequency1.0 Hz Above about 10 Hz the real synapse stops obeying the pair-based rule entirely; see the note below.

Starting weight w₀0.50 Where the synapse starts, between 0 and w_max = 1. Start near a bound and half the window has nowhere to go.

Learning window

A₊ potentiation amplitude 0.008 Weight change from a perfectly coincident pair, in units of w_max.

τ₊ potentiation constant 17 ms How far into the past a spike still counts as a cause. Measured at roughly 17 ms in hippocampal culture.

A₋ depression amplitude 0.005 The depression branch is shallower than the potentiation branch but roughly twice as wide.

τ₋ depression constant 34 ms Together with A₋ this sets the depression area, and so the stability of the synapse under uncorrelated firing.

Weight dependence Additive: every pairing moves w by the same amount until it hits a bound. Multiplicative: potentiation shrinks as w approaches w_max.

Playback speed20× Display only. The protocol itself still runs at the pairing frequency above.

The window for a single pairing. The marker sits at the Δt you have selected; the vertical bar is the resulting Δw.

LTP (Δt > 0) LTD (Δt < 0)

Net change after the whole protocol, one run per Δt. This is the plot an experimenter actually measures. Press Sweep Δt to fill it.

final w − w₀

The protocol as the synapse sees it: a pre-synaptic spike and a post-synaptic spike, Δt apart, repeated at the pairing frequency.

pre-synaptic post-synaptic

Weight against pairing number. Additive STDP walks in a straight line until it hits a bound; the multiplicative rule bends over as it approaches one.

weight w0.50

Δw per pairing+0.0044

pairing0 / 60

net change+0.000

window integral−0.034

What the figure is doing

One synapse, one number: the weight w, bounded between 0 and 1. Every time the pre-synaptic cell and the post-synaptic cell both spike, the synapse measures the interval Δt = t_post − t_pre and changes its weight by

Δw = +A₊·exp(−Δt / τ₊) for Δt > 0 Δw = −A₋·exp(Δt / τ₋) for Δt < 0

Both branches decay away from Δt = 0, so only near-coincident spikes change anything. The sign flips at zero, and that discontinuity is the whole point: the synapse is not detecting correlation, it is detecting order. An input that arrives a few milliseconds before the cell fires is evidence that the input helped cause the spike, and it is rewarded. The same input arriving a few milliseconds after the cell has already fired cannot have contributed, and it is punished.

Things worth doing with it

Reverse the order. Run the protocol at Δt = +10 ms, reset, then run it at −10 ms. Same two cells, same number of spikes, same firing rates: opposite outcome. No rate-based learning rule can produce that.
Find the dead zone. Push Δt out past 100 ms and the weight barely moves. The window is tens of milliseconds wide because that is the timescale over which the biochemical traces left by each spike are still detectable, and the NMDA receptor is the canonical candidate: it needs glutamate (the pre-synaptic spike) and depolarisation to expel its Mg²⁺ block (the back-propagating post-synaptic spike), and it only counts them as coincident if they land within its own time constant.
Watch the bounds. Switch between additive and multiplicative and run the same protocol. Additive STDP drives every synapse to 0 or to w_max and produces a bimodal weight distribution; the multiplicative rule settles at intermediate values. Which one the real synapse uses is still argued about, and the argument matters because it decides whether a population of synapses can hold a graded memory at all.
Check the stability condition. The window integral A₊τ₊ − A₋τ₋ is the average weight change per spike pair when pre and post fire independently, with Δt uniformly distributed. The measured values make it negative, which is what keeps uncorrelated input from potentiating a cell to saturation. Drag A₋ down until the integral turns positive and you have built a runaway synapse.
Sweep. The sweep panel re-runs the whole protocol at every Δt and plots the net change, which is the figure from Bi & Poo (1998) that put STDP on the map. Under the additive rule it is just the single-pair window multiplied by the number of pairings, until the bounds clip it. Under the multiplicative rule it is visibly flattened at the top.

Where the model stops being true

This is the pair-based rule, and it is a first approximation that is known to fail in specific, well-documented ways.

It assumes every pre-post pair contributes independently. Real synapses do not add up pairs linearly: in a burst, the later spikes count for less. Triplet and suppression models were built to fix exactly this.
It has no rate dependence, so it cannot reproduce the classic result that the same pairing protocol gives depression at low frequency and potentiation above roughly 10 Hz. Turn the pairing frequency up here and nothing qualitative changes, which is the model telling you where its edge is.
It has no dendritic location, no voltage dependence and no neuromodulation. The measured window changes shape with distance from the soma, and dopamine or acetylcholine can invert its sign outright, which is how a purely local rule ends up serving a global goal.
The weight is a bare number. It stands in for a receptor count, a release probability and a spine volume, all of which have their own dynamics and their own time constants for consolidation.