  Recurrent neural networks are powerful tools for understanding and modeling
computation and representation by populations of neurons. Continuous-variable
or "rate" model networks have been analyzed and applied extensively for these
purposes. However, neurons fire action potentials, and the discrete nature of
spiking is an important feature of neural circuit dynamics. Despite significant
advances, training recurrently connected spiking neural networks remains a
challenge. We present a procedure for training recurrently connected spiking
networks to generate dynamical patterns autonomously, to produce complex
temporal outputs based on integrating network input, and to model physiological
data. Our procedure makes use of a continuous-variable network to identify
targets for training the inputs to the spiking model neurons. Surprisingly, we
are able to construct spiking networks that duplicate tasks performed by
continuous-variable networks with only a relatively minor expansion in the
number of neurons. Our approach provides a novel view of the significance and
appropriate use of "firing rate" models, and it is a useful approach for
building model spiking networks that can be used to address important questions
about representation and computation in neural systems.
