  How physical networks of neurons, bound by spatio-temporal locality
constraints, can perform efficient credit assignment, remains, to a large
extent, an open question. In machine learning, the answer is almost universally
given by the error backpropagation algorithm, through both space and time.
However, this algorithm is well-known to rely on biologically implausible
assumptions, in particular with respect to spatio-temporal (non-)locality.
Alternative forward-propagation models such as real-time recurrent learning
only partially solve the locality problem, but only at the cost of scaling, due
to prohibitive storage requirements.
  We introduce Generalized Latent Equilibrium (GLE), a computational framework
for fully local spatio-temporal credit assignment in physical, dynamical
networks of neurons. We start by defining an energy based on neuron-local
mismatches, from which we derive both neuronal dynamics via stationarity and
parameter dynamics via gradient descent. The resulting dynamics can be
interpreted as a real-time, biologically plausible approximation of
backpropagation through space and time in deep cortical networks with
continuous-time neuronal dynamics and continuously active, local synaptic
plasticity. In particular, GLE exploits the morphology of dendritic trees to
enable more complex information storage and processing in single neurons, as
well as the ability of biological neurons to phase-shift their output rate with
respect to their membrane potential, which is essential in both directions of
information propagation. For the forward computation, it enables the mapping of
time-continuous inputs to neuronal space, effectively performing a
spatio-temporal convolution. For the backward computation, it permits the
temporal inversion of feedback signals, which consequently approximate the
adjoint variables necessary for useful parameter updates.
