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

Heterosynaptic plasticity (HSP) rules combine local activity-dependent potentiation with compensatory decay, a biologically and physically plausible substrate for learning. However, in a learning network the local rule is embedded in a closed loop: perturbations of plastic weights alter neural signals, which in turn alter the plasticity drive, so stability is not guaranteed. We show that linearizing an HSP motif around a balanced operating point yields a return-gain condition: stability depends on the closed-loop sensitivity from a weight back to its own learning signal. We apply this analysis to continuous-time Kolen–Pollack (KP) learning, a well-known HSP-style mechanism for aligning feedback weights with forward weights. While KP's symmetric updates exponentially contract the forward–feedback mismatch, they leave a weight-scale mode governed by the generic HSP return-gain spectrum; feedback alignment is therefore not sufficient for stability, and this causes deep continuous-time KP networks to diverge. We show that bounded activation functions limit the effective return gain and stabilize continuous-time KP in deep multilayer perceptrons, matching or exceeding layer-local normalization. These results identify closed-loop gain control as a central requirement for stable heterosynaptic learning in biological and continuous-time substrates.