STDP — Spike-Timing Dependent Plasticity

Picture two friends, A and B. If A always shouts just before B, you'd bet A is cueing B, and you'd trust that link more. If A only ever shouts just after B, then A clearly didn't cause B, and you'd trust the link less. Synapses in the brain do exactly this: a connection grows stronger when the input neuron fires just before the output neuron, and weaker when it fires just after. That ordering — not mere togetherness — is the heart of spike-timing-dependent plasticity (STDP).

The key quantity is the timing gap $\Delta t = t_{\text{post}} - t_{\text{pre}}$: how long after the input spike the output neuron fires. A small positive $\Delta t$ strengthens the synapse (LTP, long-term potentiation); a negative one weakens it (LTD). The change fades the longer the gap, following an exponential. The simplest form is $\Delta w = A_+\,e^{-\Delta t/\tau_+}$ for $\Delta t>0$. With $A_+=0.01$ and $\tau_+=20$ ms, a 3 ms gap gives $\Delta w = 0.01\,e^{-3/20}\approx +0.0086$ — a small boost. A 40 ms gap gives only $0.01\,e^{-2}\approx +0.0014$: too slow to count as causal.

Precisely, STDP is two exponentials glued at $\Delta t=0$: LTP $A_+e^{-\Delta t/\tau_+}$ for $\Delta t>0$, and LTD $-A_-e^{+\Delta t/\tau_-}$ for $\Delta t<0$. Real neurons don't store every spike pair — each spike leaves a decaying trace, and the weight nudges whenever a partner spike arrives. Averaged over random Poisson firing, the drift is $\langle\Delta w\rangle = \nu_{\text{pre}}\,\nu_{\text{post}}\,(A_+\tau_+ - A_-\tau_-)$. Making $A_-\tau_- > A_+\tau_+$ (here $0.24>0.20$) lets noise gently depress synapses, so only well-timed, genuinely causal pairs survive. The sliders map straight onto the math: $A_+,A_-$ set the heights, $\tau_+,\tau_-$ the widths of the window, and the pre/post rates set $\nu$.

(1) On the STDP Curve tab, slide $\tau_+$ up and watch the LTP side widen, so even late inputs still strengthen. (2) On Weight Evolution, set $A_+ = A_-$ with $\tau_+ = \tau_-$: the weight now wanders randomly instead of settling — that is why a depressive bias is needed. (3) Switch the preset to Anti-Hebbian and watch the curve flip, so now firing first weakens the link.

STDP was experimentally demonstrated in hippocampal cultures: pre before post (within ~20 ms) → LTP (up to +50% EPSP); post before pre → LTD (up to −25%). The asymmetric window implements Hebb's postulate with temporal causality — a presynaptic neuron is potentiated if it fired BEFORE (and potentially caused) the postsynaptic spike.

The trace x_pre jumps by 1 at each pre spike and decays exponentially. At each post spike, w += A+ × x_pre. This is biologically plausible: x_pre models the intracellular calcium concentration triggered by NMDA receptor activation when pre-release occurs just before backpropagating post-spike. The trace correctly handles multiple spike pairings with O(1) memory per synapse.

Hard bounds: clip w to [0, wmax]. Produces bimodal weight distribution (weights cluster at 0 or wmax). Soft (multiplicative) bounds: ΔwLTP = A+(wmax − w) × x_pre; ΔwLTD = −A− × w × x_post. Produces unimodal weight distribution, weight-normalising. Soft bounds are more biologically realistic and prevent runaway potentiation.

For random (uncorrelated) Poisson inputs: mean weight change = ν² × (A+τ+ − A−τ−). Setting A−τ− slightly > A+τ+ creates a net depressive bias for uncorrelated inputs — only strongly correlated (causal) spike pairs cause net potentiation. Standard parameters: A+τ+ = 0.01×20 = 0.2 vs A−τ− = 0.012×20 = 0.24. Net bias = −0.04 per pairwise interaction unit.

Three-factor rule: Δw = d(t) × eligibility trace, where d(t) is a dopamine/reward signal. The eligibility trace is the standard STDP kernel stored in a slow decaying memory. Synapses that were active before reward are potentiated; those active without reward are depressed. Implemented in Intel Loihi and enables unsupervised reinforcement learning without gradient computation.