Loop alignment: Self-organized Weight Transpose in Predictive Coding through Independent Hebbian Plasticity.

Backpropagation (BP) requires feedback (FB) weights to equal the transpose of feedforward (FF) weights at every layer and at every step. This issue is called the weight transport problem, and has been considered to be biologically implausible. Neural circuits are not built from anatomically paired FF/FB axons but from asymmetric projections that form recurrent loops shared across many computations. In multi-nuclei circuits (e.g., thalamo-cortical, cortico-cerebellar, basal-ganglia loops), the return path traverses entirely distinct nuclear structures, making any mechanism for maintaining synchronized weight transposes anatomically absent. Predictive coding (PC), originally proposed as a biologically plausible substitute for BP, silently inherits this constraint despite its biological motivation. Therefore, we propose two PC variants that eliminate it entirely. PC-DH (Dual Hebbian plasticity model) updates FF and FB weights by independent local Hebbian rules with no inter-pathway coordination. PC-RFB (Random FeedBack) fixes the FB weight with a random matrix, testing the robustness of the learning mechanism. Both match baseline PC accuracy (~85\% on MNIST). We identify a phenomenon we termed \textbf{Loop Alignment}: the spontaneous convergence of FF and FB weights toward a transpose relationship, driven by the structure of the inference loop. We then prove that it holds exactly at every training step in both models. To test whether Loop Alignment holds in the multi-nuclei setting, we conduct a circuit-sharing experiment: two networks learn different tasks (Net 1: MNIST; Net 2: Fashion-MNIST) over a shared pathway, where each network's FB is routed through the other's FF weights. Even when the entire shared pathway is plastic and driven simultaneously by both tasks, each network solves its task independently and FF/FB alignment is stronger than in the isolated case. Loop Alignment explains how the brain can learn effectively without weight transport, and support multi-task learning over shared asymmetric circuits.