The BCPNN learns by estimating probability distributions from data, then turns those estimates into synaptic weights and biases. The six equations below specify the model.

Support — total drive into a unit (lateral input + bias + external cue + noise), low-pass filtered with time constant $\tau_s$:

$$ \tau_s \frac{ds_j}{dt} = \beta_j + \sum_i w_{ij}\, o_i - s_j + G_I\, I_j(t) + \sigma(t) $$

Output — softmax (winner-take-all) across the units of one hypercolumn, sharpness $K$:

$$ o_j = \frac{e^{K s_j}}{\sum_i e^{K s_i}} $$

Marginal probability — running estimate of how often unit $j$ is active, with slow time constant $\tau_p$:

$$ \tau_p \frac{dp_j}{dt} = o_j - p_j $$

Joint probability — running estimate of how often units $i$ and $j$ co-fire (Hebbian trace):

$$ \tau_p \frac{dP_{ij}}{dt} = o_i\, o_j - P_{ij} $$

Synaptic weight — log-ratio of joint to independent probability (pointwise mutual information):

$$ w_{ij} = \log\!\frac{P_{ij}}{p_i\, p_j} $$

Bias — log-prior probability of the unit:

$$ \beta_j = \log p_j $$

The weight rule is online and local: each synapse needs only its pre- and post-synaptic activity. The softmax enforces competition within a hypercolumn, mirroring the cortical-minicolumn picture above.