This is what I have seen at different places:
1>> Some books (e.g. Jack, Noble & Tsien, 1975) and papers (e.g. Purves, 1976) implement alpha-function as:
I(t) = (alpha^2) * t * exp(-alpha*t)
2>> Rall (1967) suggests something of the form:
f(t) = (t/tp) * exp (1 - t/tp)
i.e. f(t) = (alpha*t) * exp (1 - alpha*t)
3>> NEURON, for its AlphaSynapse, uses (let tact = 0):
g(t) = gmax * (t/tau) * exp(-(t-tau)/tau)
i.e. g(t) = gmax * (alpha*t) * exp(1 - alpha*t)
The latter two are basically the same. But I could not figure out why/how the first one is used/dervied?
Does it have anything to do with modeling current or the underlying conductance? Are they equivalent in some manner?
p.s. the documentation for AlphaSynapse might need a slight correction:
I suppose it should be:this has the property that the maximum value is gmax and occurs at
t = delay + tau.
t = onset + tau