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Summary: This paper introduces a new learning framework in leaky integrate and fire neurons, which permits a recurrent network to efficiently learn linear dynamical systems. The approach uses weight changes at two timescales: fast weight changes quickly balance excitation and inhibition, while slower weight changes learn the structure of the LDS. A key insight is that the fast plasticity which balances excitation and inhibition distributes a global signal about the network's performance to all neurons, enabling error driven learning of the LDS with a local learning rule. Major comments: This paper presents the intriguing idea of using the balance of excitation and inhibition to distribute global error information throughout a neural network, permitting supervised learning with a local learning rule. Moreover, the scheme introduced is based on predictive coding, which as the paper shows, naturally leads to sparse irregular spiking activity. On this subtle view, neural firing in response to an identical input will not yield identical precise spike times; but the particular spike times for each input presentation are nonetheless precisely arranged, and cannot be replaced by a rate coded approximation without a drop in fidelity or efficiency.