Alignment Is Not Learning: Loop-Gain Control in Heterosynaptic Plastic Circuits

Heterosynaptic plasticity (HSP) rules combine local activity-dependent potentiation with compensatory decay, providing a biologically and physically plausible substrate for learning. However, in a learning network the local plasticity rule is embedded in a closed loop: perturbations of plastic weights alter neural signals, which in turn alter the plasticity drive. This suggests that stability of networks built with these rules is not guaranteed. Linearizing an HSP motif around a balanced operating point yields a loop-gain condition: stability depends on the closed-loop sensitivity from weights back to their local learning signals. We apply this analysis to continuous-time Kolen–Pollack learning, a well-known HSP-style mechanism for aligning feedback weights with forward weights. Our analysis shows that while KP's symmetric updates exponentially contract the forward–feedback weight mismatch, they critically leave a weight scale mode governed by the generic HSP loop-gain spectrum. Thus feedback alignment is not sufficient for stability. Finally, we show that bounded activation functions limit the effective plastic loop gain and stabilize continuous-time KP learning in deep multilayer perceptrons, outperforming the alternative local mechanisms we test at every depth and enabling training at depths where they fail. Across mechanisms, stability follows loop-gain control: a circuit whose loop gain is left uncontrolled fails, by fast divergence or by unbounded weight drift, even when its feedback weights align and its update magnitudes are bounded. These results identify loop-gain control as a central requirement for stable heterosynaptic learning in biological and continuous-time substrates.