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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. IndexTerms-Dynamical systems, neural simulation, neuromorphic engineering, quadratic integrate-and-fire model, silicon neuron. I. QUANTITATIVE NEUROMORPHIC MAPPING N EUROMORPHIC engineering aims to emulate computations carried out in the nervous system by mimicking neurons and their inter-connectivity in VLSI hardware [1]. Having succeeded in morphing visual [2], [3] and auditory [4], [5] systems into mixed-analog-digital circuits, engineers are entering the arena of cortical modeling [6]-[9]. This is an arena in which neuromorphic systems' parallel operation and low energy consumption give them distinct advantages over software simulation. Reproducing existing theoretical and numerical cortical modeling results using the neuromorphic approach will be facilitated by establishing quantitative links between parameters of neural models and those of their electronic analogs. Furthermore, this will build a foundation for engineers to scale up neuromorphic models beyond the limit of software simulators.Initial attempts to map neural models onto neuromorphic chips took a model-less approach [ Fig. 1(a)]. Key operating voltages or currents, and occasionally their transients, were measured using on-chip analog-to-digital converters or external instruments. These measurements were compared with the desired behavior, and circuit biases were adjusted using intuition Manuscript Fig. 1. Approaches to quantitative neuromorphic mapping. Our approach is highlighted in bold. (a) Adjust circuit biases with heuristics to achieve the desired behavior without a model. (b) Use model simulation as reference to nonlinearly optimize the circuit's behavior. (c) Mathematically analyze the model to derive a set of simple linear parametric relations.or heuristic algorithms until the matching was acceptable [10], [11]. While such procedures capture the hardware operation in detail, they are time-consuming in both data collection and analysis, and space-consuming in terms of hardware, as special circuits have to be included to expose relevant circuit n...

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