Editing Openai/69371729-f090-8004-b69b-aa9f25ca0b40 (section)

=== 6 — Putting it together: a compact algorithm ===

Parameters: η\etaη (Hebb rate), λ\lambdaλ (decay), ρ\rhoρ (trace smoothing), cgrowc_{\text{grow}}cgrow, cprunec_{\text{prune}}cprune, θsmall\theta_{\text{small}}θsmall, Kmax⁡K_{\max}Kmax.

For each time step with input x(t)\mathbf{x}(t)x(t) and resulting postsyn y(t)\mathbf{y}(t)y(t):
# Update local correlation traces: cij←(1−ρ)cij+ρ xj yi.c_{ij} \leftarrow (1-\rho)c_{ij} + \rho\, x_j\, y_i.cij←(1−ρ)cij+ρxjyi.
# Weight update (Hebbian + decay): wij←wij+η xj yi−λ wij.w_{ij} \leftarrow w_{ij} + \eta\, x_j\, y_i - \lambda\, w_{ij}.wij←wij+ηxjyi−λwij. (Or use Oja: wij←wij+η(xjyi−yi2wij)w_{ij} \leftarrow w_{ij} + \eta(x_j y_i - y_i^2 w_{ij})wij←wij+η(xjyi−yi2wij).)
# Prune if: - ∣wij∣<θsmall|w_{ij}| < \theta_{\text{small}}∣wij∣<θsmall and cij<cprunec_{ij} < c_{\text{prune}}cij<cprune (for some time), then set wij=0w_{ij}=0wij=0 and mark as absent.
# Grow if: - wij=0w_{ij}=0wij=0 and cij>cgrowc_{ij} > c_{\text{grow}}cij>cgrow and total connections <Kmax⁡<K_{\max}<Kmax, then create connection with wij←γ cijw_{ij} \leftarrow \gamma\, c_{ij}wij←γcij (small γ\gammaγ). - If Kmax⁡K_{\max}Kmax reached, optionally replace weakest existing connection if the candidate's cijc_{ij}cij is larger.

This is fully local: each synapse needs only its cijc_{ij}cij, wijw_{ij}wij, and maybe a small timer.