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We present a heuristic derivation of Gaussian approximations for stochasticchemical reaction systems with distributed delay. In particular we derive thecorresponding chemical Langevin equation. Due to the non-Markovian character ofthe underlying dynamics these equations are integro-differential equations, andthe noise in the Gaussian approximation is coloured. Following on from thechemical Langevin equation a further reduction leads to the linear-noiseapproximation. We apply the formalism to a delay variant of the celebratedBrusselator model, and show how it can be used to characterise noise-drivenquasi-cycles, as well as noise-triggered spiking. We find surprisinglyintricate dependence of the typical frequency of quasi-cycles on the delayperiod.

Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system