  Brains need to predict how the body reacts to motor commands. It is an open
question how networks of spiking neurons can learn to reproduce the non-linear
body dynamics caused by motor commands, using local, online and stable learning
rules. Here, we present a supervised learning scheme for the feedforward and
recurrent connections in a network of heterogeneous spiking neurons. The error
in the output is fed back through fixed random connections with a negative
gain, causing the network to follow the desired dynamics, while an online and
local rule changes the weights. The rule for Feedback-based Online Local
Learning Of Weights (FOLLOW) is local in the sense that weight changes depend
on the presynaptic activity and the error signal projected onto the
postsynaptic neuron. We provide examples of learning linear, non-linear and
chaotic dynamics, as well as the dynamics of a two-link arm. Using the Lyapunov
method, and under reasonable assumptions and approximations, we show that
FOLLOW learning is stable uniformly, with the error going to zero
asymptotically.
