Backpropagation through space, time, and the brain
Benjamin Ellenberger Paul Haider Jakob Jordan Kevin Max Ismael Jaras Laura Kriener Federico Benitez Mihai A. Petrovici
Effective learning in neuronal networks requires the adaptation of individual
synapses given their relative contribution to solving a task. However, physical
neuronal systems -- whether biological or artificial -- are constrained by
spatio-temporal locality. How such networks 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 (BP) and time (BPTT). However, BP(TT) is well-known to rely
on biologically implausible assumptions, in particular with respect to
spatiotemporal (non-)locality, while forward-propagation models such as
real-time recurrent learning (RTRL) suffer from prohibitive memory constraints.
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 BPTT in
deep cortical networks with continuous-time neuronal dynamics and continuously
active, local synaptic plasticity. In particular, GLE exploits 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, performing an effective
spatiotemporal convolution. For the backward computation, it permits the
temporal inversion of feedback signals, which consequently approximate the
adjoint states necessary for useful parameter updates.