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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.