  The biological plausibility of the backpropagation algorithm has long been
doubted by neuroscientists. Two major reasons are that neurons would need to
send two different types of signal in the forward and backward phases, and that
pairs of neurons would need to communicate through symmetric bidirectional
connections. We present a simple two-phase learning procedure for fixed point
recurrent networks that addresses both these issues. In our model, neurons
perform leaky integration and synaptic weights are updated through a local
mechanism. Our learning method generalizes Equilibrium Propagation to vector
field dynamics, relaxing the requirement of an energy function. As a
consequence of this generalization, the algorithm does not compute the true
gradient of the objective function, but rather approximates it at a precision
which is proven to be directly related to the degree of symmetry of the
feedforward and feedback weights. We show experimentally that our algorithm
optimizes the objective function.
