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We present an approach to map neuronal models onto neuromorphic hardware using mathematical insights from dynamical system theory. Quantitatively accurate mappings are important for neuromorphic systems to both leverage and extend existing theoretical and numerical cortical modeling results. In the present study, we first calibrate the on-chip bias generators on our custom hardware. Then, taking advantage of the hardware's high-throughput spike communication, we rapidly estimate key mapping parameters with a set of linear relationships for static inputs derived from dynamical system theory. We apply this mapping procedure to three different chips, and show close matching to the neuronal model and between chips-the Jenson-Shannon divergence was reduced to at least one tenth that of the shuffled control. We confirm that our mapping procedure generalizes to dynamic inputs: Silicon neurons match spike timings of a simulated neuron with a standard deviation of 3.4% of the average inter-spike interval.