We present a multi-input multi-output neural circuit architecture for nonlinear processing and encoding of stimuli in the spike domain. In this architecture a bank of dendritic stimulus processors implements nonlinear transformations of multiple temporal or spatio-temporal signals such as spike trains or auditory and visual stimuli in the analog domain. Dendritic stimulus processors may act on both individual stimuli and on groups of stimuli, thereby executing complex computations that arise as a result of interactions between concurrently received signals. The results of the analog-domain computations are then encoded into a multi-dimensional spike train by a population of spiking neurons modeled as nonlinear dynamical systems. We investigate general conditions under which such circuits faithfully represent stimuli and demonstrate algorithms for (i) stimulus recovery, or decoding, and (ii) identification of dendritic stimulus processors from the observed spikes. Taken together, our results demonstrate a fundamental duality between the identification of the dendritic stimulus processor of a single neuron and the decoding of stimuli encoded by a population of neurons with a bank of dendritic stimulus processors. This duality result enabled us to derive lower bounds on the number of experiments to be performed and the total number of spikes that need to be recorded for identifying a neural circuit.