59 Hebbian plasticity
A guide to the interactive demonstration
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60 What Hebb actually proposed
Donald Hebb’s 1949 formulation is usually compressed into “cells that fire together, wire together,” which is close enough to be memorable and loose enough to be misleading. His own sentence is more careful:
When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.
Three conditions are buried in that sentence, and the demonstration is built to separate them:
- Presynaptic activity is not sufficient. Cell A must be active — but activity alone does nothing.
- The postsynaptic cell must fire. A’s activity has to coincide with B firing.
- Order matters. A must take part in firing B. A synapse that becomes active after the spike cannot have contributed to it, and should not be credited.
The third point was implicit in Hebb and made explicit half a century later by experiments on spike-timing dependent plasticity (STDP): pairing a presynaptic spike a few milliseconds before a postsynaptic spike potentiates the synapse, while the reverse order depresses it, with a window on the order of tens of milliseconds.
60.1 Why a naive Hebbian rule cannot work
If a rule only ever strengthens synapses, it is unstable. Stronger synapses make the cell easier to fire; a cell that fires more often triggers more strengthening; every synapse saturates and the cell loses all selectivity. Any working model needs a counterweight — depression, a decay term, weight normalisation, or a sliding threshold. The rule used here is stable by construction, and Section 61.3 explains how.
61 What is being simulated
61.1 The cell
A leaky integrate-and-fire neuron. The membrane potential \(V\) drifts back toward rest with a time constant \(\tau_m\), and each incoming presynaptic spike adds an instantaneous EPSP equal to that synapse’s weight:
\[ \tau_m \frac{dV}{dt} = (V_{\text{rest}} - V), \qquad V \leftarrow V + w_i \ \text{ on a spike at synapse } i \]
When \(V\) reaches threshold the cell emits a spike, is clamped to a hyperpolarised reset value, and is refractory for a few milliseconds.
| Parameter | Value |
|---|---|
| Resting potential | −70 mV |
| Spike threshold | −54 mV |
| Reset / after-hyperpolarisation | −75 mV |
| Membrane time constant \(\tau_m\) | 20 ms |
| Absolute refractory period | 3 ms |
| Integration step | 1 ms, fixed |
The timestep is fixed at 1 ms and decoupled from your display’s refresh rate, so the simulation behaves identically on a 60 Hz laptop and a 120 Hz monitor. The speed slider changes how many milliseconds of simulated time elapse per second of real time, not the physics.
61.2 The inputs
Twelve excitatory synapses in three groups of four:
- Group A (orange) — on one apical branch
- Group B (teal) — on the other
- Background (grey) — on the basal dendrites
Every synapse fires independently at about 8 Hz, all of the time. On top of that, whichever pattern is selected fires its four synapses in synchrony roughly five times a second.
The starting weight is deliberately chosen so that a synchronous pattern cannot fire the cell. Four coincident EPSPs of 1.2 mV lift the membrane about 7 mV above rest, against a threshold 16 mV above rest — roughly seven standard deviations of the background fluctuation away. The pattern is subthreshold until it learns not to be. This is what makes protocol 2 a real control rather than a formality.
61.3 The learning rule
Each synapse carries a presynaptic trace \(x_i\) that jumps by 1 on a presynaptic spike, and the cell carries a postsynaptic trace \(y\) that jumps by 1 on a postsynaptic spike. Both decay exponentially with \(\tau = 20\) ms. These traces are the model’s memory of “what happened recently.”
Potentiation — evaluated when the cell fires. Every synapse is credited in proportion to how recently it was active, with a soft ceiling so the increment shrinks as the weight approaches its maximum:
\[ \Delta w_i = +A_{+} \, x_i \, (w_{\max} - w_i) \]
Depression — evaluated when a synapse fires. It is penalised in proportion to how recently the cell fired, i.e. for arriving too late to have helped:
\[ \Delta w_i = -A_{-} \, y \]
Because \(x_i\) decays, a synapse active 5 ms before the spike is credited far more than one active 60 ms before. Because depression is triggered by presynaptic spikes that follow postsynaptic ones, the rule is asymmetric in time. Together these reproduce the STDP window.
61.3.1 Why the rule does not run away
The two constants are locked together:
\[ A_{-} = A_{+}\,(w_{\max} - w_0) \]
With that choice, activity in which pre and post spikes are statistically independent drives every weight toward \(w_0\) — from above as well as from below, and regardless of the firing rates involved. The baseline is a genuine attractor rather than a floor. Only presynaptic activity that is reliably correlated with, and earlier than, the postsynaptic spike can lift a weight above \(w_0\) and hold it there.
This is why protocols 1 and 2 leave every weight reading exactly 1.20 after a minute of heavy synaptic bombardment, and it is the single most important difference between this model and a textbook “increase the weight when both are active” rule.
| Parameter | Value |
|---|---|
| Starting weight \(w_0\) | 1.2 mV |
| Maximum weight \(w_{\max}\) | 5.0 mV |
| Minimum weight | 0.3 mV |
| Learning rate \(A_{+}\) | 0.04 (adjustable) |
| Trace time constants | 20 ms |
61.4 The current pulse
Protocols 3–5 add what a slice electrophysiologist would do: a brief depolarising current injection through the recording electrode that forces the cell to spike at a controlled time relative to the pattern. This is the classic pairing protocol, and it is the only way to satisfy Hebb’s second condition while the synapses are still too weak to satisfy it themselves.
The pulse is marked with a violet triangle (▲) on the trace. It is suppressed whenever the cell has already fired within the last 12 ms — so as training proceeds and the synapses take over, the triangles quietly disappear from the display. That disappearance is one of the more satisfying things to point out.
62 Reading the display
62.1 Membrane potential and input raster
The upper panel is a scrolling 2-second window of the membrane potential, drawn the way an intracellular recording is normally presented.
The voltage axis is broken. All of the interesting behaviour lives between −75 and −54 mV, so the subthreshold band is given most of the height and the spike is compressed above the break, which is marked with a zigzag on the axis. Without this, subthreshold summation would be invisible.
Below the trace, each of the twelve synapses gets a row in a raster plot: one tick per presynaptic spike, coloured by group. Vertical lines running the full height of the panel mark each pattern presentation. Four ticks lining up on the marker, and the EPSP directly above them, is the whole story of temporal summation in one glance.
62.2 The cell
The morphology panel shows where the synapses sit. Circle size is the synaptic weight and updates continuously. The violet wave running out along the dendrites on each spike is the backpropagating action potential — the physical signal that tells a distal synapse the cell has fired, and the biological reason a spike can act as a global instruction to a thousand synapses at once.
62.3 Synaptic weights
Twelve bars in three clusters, with the group mean above each and a dashed line at the starting value. This is the learning outcome, and it is where you should be looking most of the time.
Note that individual bars fluctuate around the group mean. This is real behaviour of the rule, not a display artefact: with only four synapses per group, chance coincidences produce a visible random walk. The group mean is the stable quantity. Lowering the learning rate slider shrinks the spread at the cost of slower learning.
62.4 The response strip
In the readout panel, twenty tick marks record the last twenty pattern presentations. A tick is filled if the synapses alone drove the cell across threshold within 9 ms of the pattern — before any current pulse could have contributed. Empty means the pattern arrived and nothing happened.
This strip is the demonstration’s actual measurement. It converts “the weights went up” into “the cell now responds to a stimulus it used to ignore,” which is the claim that matters.
63 The five protocols
They are designed to be run in order. Weights carry over between protocols until you press Reset weights — protocols 4 and 5 depend on this.
63.1 Background only
No pattern, no pairing. Twelve synapses fire at random; the summed input wanders a few millivolts above rest and never reaches threshold. The cell is silent, and every weight sits exactly where it started.
The point: plasticity here requires a postsynaptic spike. With no spikes, there is no learning, no matter how much presynaptic activity there is.
63.2 Present pattern A
Group A now fires in synchrony five times a second. The four coincident EPSPs sum into a clear step in the trace — and stop several millivolts short of threshold. Leave it running for a minute; the weights do not move.
The point: this is the control that matters. Strong, repeated, perfectly synchronised presynaptic activity, on its own, changes nothing. Most casual statements of Hebb’s rule (“cells that fire together wire together”) would predict potentiation here. They are wrong, and this is why the wording of the original matters.
Worth asking a class to predict the outcome before running it.
63.3 Pair A with a spike
A current pulse now drives the cell 10 ms after each pattern: pre before post. Group A climbs from 1.2 mV toward roughly 4.3 mV over ten to fifteen seconds while the other eight synapses stay on the dashed line.
Watch three things at once:
- the orange bars separating from the teal and grey ones
- the response strip filling in as the pattern begins crossing threshold on its own
- the violet pulse markers disappearing from the trace, because the synapses have taken over the job the electrode was doing
The point: Hebb’s conjunction — pre and post, in the right order — strengthens the specific synapses that participated, and leaves the rest alone. The selectivity is as important as the strengthening.
Once it saturates, switch Plasticity off and leave the protocol running: the learned response persists without further change, confirming the response is stored in the weights and not in some transient state.
63.4 Reverse the order
Exactly the same two events, opposite order: the cell fires 10 ms before the pattern arrives. Group A collapses toward the floor within about ten seconds, and the response strip empties.
The point: Hebb’s rule is about causation, not co-occurrence. Two events that co-occur just as often, in the same numbers, at the same rates, produce the opposite outcome purely because of their order. A synapse that fires after the spike could not have contributed to it.
The trace makes this unusually legible: the spike is visibly to the left of the orange stimulus marker, and each pattern lands on a cell that is still hyperpolarised from its own action potential.
63.5 Train pattern B instead
Pairing moves to group B, which climbs the same way A did. Group A is not reset — it decays back toward baseline over roughly a minute, because its spikes now fall at random times relative to the cell’s, and the rule’s uncorrelated equilibrium pulls it home.
The point: synapses hold their gains only for as long as they keep earning them. Nothing in the mechanism marks a weight as permanent, which is a reasonable first illustration of why long-term memory requires consolidation machinery beyond the plasticity rule itself.
64 Things to try beyond the presets
Sweep the pairing delay. With protocol 3 selected, move the delay slider slowly from +40 ms down through zero to −40 ms, pausing at each value long enough for the weights to settle. You are tracing the STDP window by hand: strong potentiation at small positive delays, weakening as the delay grows, crossover near zero, depression on the negative side.
Turn plasticity off and on. Freezing the weights lets you measure the current response without changing it — the difference between a training block and a test block in a real experiment.
Slow it down. At 0.05× you can watch individual background EPSPs arrive and decay, and see exactly how four coincident inputs sum where four scattered ones do not.
Raise the learning rate to the maximum. Learning becomes fast but the individual weights become visibly noisier, and the group separation less clean — a small, concrete illustration of the speed/stability trade-off that pervades learning rules.
Predict, then run. The demo is most useful when the class commits to an answer first. Protocol 2 in particular tends to produce confident wrong predictions.
65 Where the model simplifies
Worth stating out loud so nobody over-reads the demonstration:
- Pair-based STDP is known to be incomplete. It does not reproduce the frequency dependence of real LTP induction: pairing at 0.5 Hz and at 50 Hz give very different outcomes in slice preparations, and this model will not show that difference. Triplet-STDP and BCM-style rules were developed precisely to fix it.
- There is no biophysics of induction. No NMDA receptors, no calcium influx, no coincidence detection at the molecular level, no protein synthesis, no distinction between early and late LTP. The rule is phenomenological.
- The neuron is a point. Dendrites are drawn but not simulated. In a real cell, synaptic location matters, dendritic branches integrate somewhat independently, and the backpropagating action potential attenuates with distance.
- All synapses are excitatory, and there is no inhibition, no homeostatic scaling, and no structural plasticity.
- EPSPs rise instantaneously. Real ones rise over a few milliseconds. This affects the fine shape of the trace but not the qualitative behaviour.
None of these undermine the three things the demonstration is built to show: that presynaptic activity alone is insufficient, that a postsynaptic spike is required, and that the temporal order of the two determines the sign of the change.