testsuite::test_iaf_psp_peak - test of closed form expression for peak
(test_iaf_psp_peak) run -> compare expression with numerics  


Several NEST neuron models have an alpha-shaped post-synaptic current (PSC).
In these models the PSC is normalized to unit amplitude. Thus, a synaptic weight
w leads to a PSC with amplitude w in units of pA.
In order to adjust the amplitude of the post-synaptic potential (PSP) of a
neuron model with an alpha-shaped post-synaptic current (PSC) to a particular
amplitude we need to first find the location of the maximum tmax of the PSP.
Here, this is done in two different ways:
1. We numerically search for the root of the derivative of the PSP
2. We used a closed form expression to compute the position of the maximum
The test verifies that the methods lead to the same result. The test file
test_iaf_psp_normalized shows how this value is used to specify w such that a
PSP with a desired amplitude u in units of mV results.

The closed form expression can be found by first transforming the expression
d psp(t) / dt = 0
into the normal form
exp(s) = 1 + a * s,
where s is the scaled time s=bt and a and b depend on the time constants
a = tau_m/tau_alpha, b = 1/tau_alpha - 1/tau_m .

The solution for s can then be expressed with the help of the Lambert W-function W
which is the inverse of x=W*exp(W) and reads

s = 1/a * ( -a W(-exp(-1/a)/a) - 1 )

[1] Weisstein, Lambert W-function

Author: July 2009, Diesmann
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