Synaptic plasticity — the activity-dependent modification of synaptic strength — is the primary neural mechanism for learning and memory. Mathematical models of synaptic plasticity formalize the rules by which synaptic weights change as a function of pre- and postsynaptic activity, providing the bridge between neural circuit dynamics and the learning rules used in mathematical psychology. The history of these models begins with Donald Hebb's (1949) verbal postulate and progresses through increasingly sophisticated formalizations that account for a wider range of experimental phenomena.
From Hebb's Rule to BCM Theory
(Strengthen when pre and post are both active)
Covariance rule: Δwᵢⱼ = η · (rᵢ − θᵢ) · (rⱼ − θⱼ)
(Strengthen for correlated activity, weaken for uncorrelated)
BCM rule (Bienenstock, Cooper, & Munro, 1982):
Δwᵢⱼ = η · rᵢ · rⱼ · (rⱼ − θ_M)
θ_M = E[rⱼ²] (sliding modification threshold)
rⱼ > θ_M → LTP; rⱼ < θ_M → LTD
Hebb's rule ("neurons that fire together wire together") states that a synapse is strengthened when both the presynaptic and postsynaptic neurons are active simultaneously. This simple rule is computationally powerful — it implements a form of correlation learning — but is unstable: weights grow without bound. The BCM rule (Bienenstock, Cooper, & Munro, 1982) solved this instability problem by introducing a sliding modification threshold θ_M. When the postsynaptic neuron's activity is above θ_M, the synapse is strengthened (long-term potentiation, LTP); when below θ_M, it is weakened (long-term depression, LTD). Crucially, θ_M itself depends on the recent average activity of the postsynaptic neuron, providing homeostatic self-regulation.
Spike-Timing-Dependent Plasticity (STDP)
The discovery of spike-timing-dependent plasticity (Markram, Lübke, Frotscher, & Sakmann, 1997; Bi & Poo, 1998) revealed that the precise temporal order of pre- and postsynaptic spikes determines the sign and magnitude of synaptic change. When the presynaptic spike precedes the postsynaptic spike by a few milliseconds (pre→post), the synapse is strengthened (LTP). When the order is reversed (post→pre), the synapse is weakened (LTD). The magnitude of the change decays exponentially with the time difference.
Δw = −A₋ · exp(Δt/τ₋) if Δt < 0 (post before pre → LTD)
Δt = t_post − t_pre
τ₊ ≈ 20 ms, τ₋ ≈ 20 ms (time constants)
A₊, A₋ = amplitude parameters
In addition to long-term plasticity (LTP/LTD, persisting hours to years), synapses exhibit short-term plasticity on timescales of milliseconds to seconds. Short-term facilitation increases synaptic strength following a spike, enhancing transmission for rapidly repeated inputs. Short-term depression decreases synaptic strength, reducing transmission during sustained activity. The Tsodyks-Markram model (Tsodyks & Markram, 1997) provides a compact mathematical description of short-term plasticity using variables for available neurotransmitter resources and utilization probability. Short-term plasticity is important for neural models of working memory, adaptation, and temporal filtering.
Mathematical models of synaptic plasticity underpin virtually all neural learning models in mathematical psychology. Hebbian learning is the mechanism behind Hopfield networks and self-organizing maps. Error-driven learning rules (like the delta rule and backpropagation) can be seen as extensions of Hebbian principles with a teaching signal. BCM theory connects to models of experience-dependent cortical development, and STDP provides a biophysical basis for temporal-difference learning in reinforcement learning models. Understanding these plasticity rules is essential for bridging between the abstract learning equations of mathematical psychology and their neural implementation.