Backpropagation through time (BPTT) is how artificial neural networks learn temporal patterns: unroll the network's computation through time, compute the gradient of the loss with respect to every parameter at every timestep, then update. It's effective but biologically implausible — it requires storing all intermediate states, propagating errors backward through time (violating causality for the neurons), and coordinating updates globally across the network.
Brandt et al. (arXiv:2603.09600) derive a biologically plausible approximation to BPTT from first principles — not from engineering constraints but from physics. They start with a prospective energy function of neuronal states and apply the principle of extremal action: the dynamics that extremize the action functional yield the error signals needed for learning. Energy conservation and the stationarity condition together produce equations that look like the adjoint method (the continuous-time equivalent of BPTT) but can be implemented with fully local operations — each neuron and synapse computes its update using only information available at that location and time.
The locality is the key advance. In standard BPTT, computing the gradient at time t requires knowing the error at all future times. The extremal action formulation transforms this into a real-time error signal that propagates forward alongside the computation. The error dynamics run in the same temporal direction as the network dynamics — no backward pass, no storage of intermediate states, no temporal non-locality.
The framework unifies several previous approaches (equilibrium propagation, predictive coding, latent equilibrium) as special cases of the same variational principle applied with different approximations. The physics isn't metaphorical — the energy function, the action, and the extremal principle are the same mathematical objects used in Lagrangian mechanics. If biological neurons implement something like this, the brain learns through the same principle that governs planetary orbits.