# Command: testsuite::test_iaf_psp_peak

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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:

- testsuite::test_lambertw
- testsuite::test_iaf_psp_normalized
- LambertWm1

Source:

    /opt/conda/conda-bld/nest_1512418852064/work/nest-2.12.0/testsuite/unittests/test_iaf_psp_peak.sli

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