**Name:**

testsuite::test_iaf_psp_peak - test of closed form expression for peak

**Synopsis:**

(test_iaf_psp_peak) run -> compare expression with numerics

**Description:**

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 )

**References:**

[1] Weisstein, Lambert W-function

Author: July 2009, Diesmann

**SeeAlso:**

**Source:**

/home/graber/work-nest/nest-git/nest-simulator/testsuite/unittests/test_iaf_psp_peak.sli