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

Thanks!