['Long-Term Potentiation (LTP) of synapses is thought to be due in part to a change in AMPA Receptor trafficking leading to an increase in the number of AMPA Receptors at the synapse. LTP onset occurs within seconds after the induction signal. A particle-based stochastic simulation software is used to investigate the effect of Brownian diffusion of glutamate receptors on receptor incorporation into the synaptic specialisation and the time-course of LTP expression. The model of the dendritic spine includes receptors diffusing within the membrane, scaffold molecules within the synaptic specialisation capable of binding receptors and a molecular picket-fence surrounding the synaptic membrane area, all features found within the biological system.', 'During simulations, receptors accumulate rapidly at the post-synaptic density (PSD) from the extra-synaptic membrane under a number of biologically observed conditions. The time of half-saturation, t1/2, defined as the time-point at which half the available scaffold proteins are occupied with receptors, is found to be 710 ms. Different scaffold distributions are shown to have little effect on this time-course. Decreasing the probability of escape of receptors from the PSD domain, thus localising receptors closer to the scaffold proteins, substantially decreases t1/2. A decrease of escape probability from 1 to 0 brings about a non-linear decrease in t1/2 from 710 ms to 390 ms. Release-location of receptors within the spine is found to affect the initial rate of receptor incorporation. We simulate three possible sources of receptors: (i) receptors distributed within the spine extra-synaptic membrane; (ii) receptors from exocytotic vesicles released to the synaptic spine; and (iii) receptors entering the spine from the dendritic shaft through the spine neck. Receptors released from exocytotic vesicles initially accumulate faster than receptors released from the other two sources. A model of glutamate release and glutamate-receptor interaction shows that newly inserted receptors make a substantial contribution to a glutamate evoked response within the observed time-frame.'] Is brownian diffusion of AMPA receptors sufficient to explain fast onset of LTP?