To predict sensory inputs or control motor trajectories, the brain must constantly learn temporal dynamics based on error feedback.

However, it remains unclear how such supervised learning is implemented in biological neural net-works. Learning in recurrent spiking networks is notoriously difficult because lo-cal changes in connectivity may have an unpredictable effect on the global dynamics. The most commonly used learning rules, such as temporal back-propagation,are not local and thus not biologically plausible. Furthermore, reproducing thePoisson-like statistics of neural responses requires the use of networks with balanced excitation and inhibition. Such balance is easily destroyed during learning.Using a top-down approach, we show how networks of integrate-and-fire neurons can learn arbitrary linear dynamical systems by feeding back their error asa feed-forward input. The network uses two types of recurrent connections: fast and slow. The fast connections learn to balance excitation and inhibition using a voltage-based plasticity rule. The slow connections are trained to minimize the error feedback using a current-based Hebbian learning rule. Importantly, the balance maintained by fast connections is crucial to ensure that global error signals are available locally in each neuron, in turn resulting in a local learning rule for the slow connections. This demonstrates that spiking networks can learn complex dynamics using purely local learning rules, using E/I balance as the key rather than an additional constraint. The resulting network implements a given function within the predictive coding scheme, with minimal dimensions and activity."