  The brain efficiently performs nonlinear computations through its intricate
networks of spiking neurons, but how this is done remains elusive. While
nonlinear computations can be implemented successfully in spiking neural
networks, this requires supervised training and the resulting connectivity can
be hard to interpret. In contrast, the required connectivity for any
computation in the form of a linear dynamical system can be directly derived
and understood with the spike coding network (SCN) framework. These networks
also have biologically realistic activity patterns and are highly robust to
cell death. Here we extend the SCN framework to directly implement any
polynomial dynamical system, without the need for training. This results in
networks requiring a mix of synapse types (fast, slow, and multiplicative),
which we term multiplicative spike coding networks (mSCNs). Using mSCNs, we
demonstrate how to directly derive the required connectivity for several
nonlinear dynamical systems. We also show how to carry out higher-order
polynomials with coupled networks that use only pair-wise multiplicative
synapses, and provide expected numbers of connections for each synapse type.
Overall, our work demonstrates a novel method for implementing nonlinear
computations in spiking neural networks, while keeping the attractive features
of standard SCNs (robustness, realistic activity patterns, and interpretable
connectivity). Finally, we discuss the biological plausibility of our approach,
and how the high accuracy and robustness of the approach may be of interest for
neuromorphic computing.
