Echo learning enables biologically plausible temporal credit assignment

Biologically plausible learning requires temporal credit assignment without backpropagation through time (BPTT). Hamiltonian Echo Learning (HEL) achieves this by running a neural system twice --- an \emph{inference} phase then a time-reversed \emph{echo} phase --- recovering exact BPTT gradients. We identify spatio-temporal sum-separability of the Hamiltonian as the structural condition making HEL's learning rule both spatially and temporally local, and exploit it to extend HEL beyond diagonal recurrences: a Hopfield-inspired oscillatory RNN with dense recurrent connectivity yields a contrastive Hebbian rule with constant memory and only two forward passes. This rule matches full BPTT and outperforms e-prop and truncated BPTT on time-series classification and regression. We then relax HEL's Hamiltonian-reversibility constraint --- which forces unbiological connectivity and activation patterns --- and derive \emph{Echo Learning} (EL), exact for arbitrary smooth dynamics whenever the network is reversible and self-adjoint with respect to a readout involution. Both conditions are enforced by a spatio-temporally local homeostatic loss, recovering near-BPTT gradient quality without architectural constraints.