Stabilizing Continuous-Time Kolen–Pollack Learning with a Scale-Balance Condition

Kolen–Pollack (KP) is a candidate biologically plausible learning rule whose continuous-time formulation eliminates phase separation but suffers from instability in deep networks. Existing discrete-time KP implementations rely on engineered stabilizers (normalization, weight decay, gradient clipping, and adaptive optimizers) that have no obvious counterpart in a continuous-time biological or analog substrate. We analyze continuous-time KP through heterosynaptic plasticity (HSP) theory and identify a local scale-balance condition between the plasticity drive and the decay term that KP's alignment mechanism does not by itself enforce. We show that bounded activations restore this balance, and matches or exceeds layer-local normalization performance on 5 and 10 layer MLPs.