  Neuromorphic hardware strives to emulate brain-like neural networks and thus
holds the promise for scalable, low-power information processing on temporal
data streams. Yet, to solve real-world problems, these networks need to be
trained. However, training on neuromorphic substrates creates significant
challenges due to the offline character and the required non-local computations
of gradient-based learning algorithms. This article provides a mathematical
framework for the design of practical online learning algorithms for
neuromorphic substrates. Specifically, we show a direct connection between
Real-Time Recurrent Learning (RTRL), an online algorithm for computing
gradients in conventional Recurrent Neural Networks (RNNs), and biologically
plausible learning rules for training Spiking Neural Networks (SNNs). Further,
we motivate a sparse approximation based on block-diagonal Jacobians, which
reduces the algorithm's computational complexity, diminishes the non-local
information requirements, and empirically leads to good learning performance,
thereby improving its applicability to neuromorphic substrates. In summary, our
framework bridges the gap between synaptic plasticity and gradient-based
approaches from deep learning and lays the foundations for powerful information
processing on future neuromorphic hardware systems.
