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Understanding why neural systems can process information extremely fast is a fundamental question in theoretical neuroscience The present study investigates the effect of noise on speeding up neural computation We consider a computational task in which a neuronal network tracks a time-varying stimulus Two network models with varying recurrent structures are explored, namely, neurons have weak sparse connections and have strong balanced interactions It turns out that when the input noise is Poissonian, i.e., the noise strength is proportional to the mean of the input, the network have the best tracking performances This is due to two good properties in the transient dynamics of the network associated with the Poissonian noise, which are: 1) the instant firing rate of the network is proportional to the mean of the external input when the network is at a stationary state; and 2) the stationary state of the network is insensitive to the stimulus change These two properties enable the network to track the stimulus change rapidly Simulation results confirm our theoretical analysis.