Bienenstock–Cooper–Munro Theory

• fundamental and foundational rule for synaptic plasticity for describing how neurons develop selectively, things like visual orientation and ocular dominance.
• Traditional Hebbian Plasticity has runaway dynamics, where correlated/high activity strenghthens a synapse, raising future activity, strengthening it further.
  • no mechanism for connections to get weaker, and no upper bound for how strong they can get
• BCM formed as gradual modifications and additions to traditional hebb rules, normalizing it and allowing for decay of synapses, where no activity or unsynchronized activity between neurons results in a loss of connection strength
  • formalized mathematically in Bienenstock, Cooper, and Munro’s 1982 paper

Formally,

$$\frac{d{m}_j(t)}{dt}=\varphi (\mathbf{c}(t))j_j(t)-ϵm_j(t)$$

• $m_j$ is synaptic weight of jth synapse
• $d_j$ is the jth synapse input current
• $c(t)=\mathbf{w}(t)\mathbf{d}(t)={\sum }_j{w}_j(t)d_j(t)$ is weighted sum of currents
• $\varphi (c)$ non linear function that must change sign at some threshold ${\theta }_M$, $\varphi (c)<0⟺c<{\theta }_M$
• $ϵ$ time constant for uniform synapse decay