After the pulse turns off at t=t1,
Thus, the synapse rises exponentially with a time constant and decays with a time constant This simple form has led many modelers to dispense with the differential equations altogether and use the so-called ``alpha'' functions for s(t) which have the form
where ts is the time of the presynaptic spike (when the presynaptic voltage crosses some set threshold), is the rise time of the synapse (approximately ) and is the decay of the synapse (approximately ). In particular, if the rise time is very fast, then
while if the decay and rise times are close,
The problem with using the so called ``alpha'' functions is that there is some question what to do when there are multiple spikes. Multiple spikes can either be added or the most recent taken. This approach requires monitoring the presynaptic cells and then setting/resetting the synaptic time-courses, s(t) for each synapse. This method of modeling has the advantage that no real dynamics must be computed; once the synapse is set in motion, it follows the prescribed time course. However, Destexhe et al show that using the actual differential equations and the assumption that the pulse of transmitter released is a square pulse, then the formulae above for s(t) lead to a computationally efficient scheme for computing the synaptic gates without having to keep track of all prior spikes. In my opinion, the ``alpha'' functions are useful only for certain types of exactly solvable models called integrate and fire models.
The AMPA synapses can be very fast. For example in some auditory nuclei, they have submillisecond rise and decay times. In typical cortical cells, the rise time is 0.4 to 0.8 milliseconds. Using the above model with a transmitter concentration of 1 mM, the rise time would be 1/(1.1+.19)=.8 msec. Decay is about 5 msec. As a final note, AMPA receptors onto inhibitory interneurons are about twice as fast in rise and fall times.