testsuite::test_aeif_cond_beta_multisynapse - sli script for the aeif multisynapse model with synaptic conductance modeled by a double exponential function
(test_aeif_cond_beta_multisynapse) run  

This test creates a multisynapse neuron with three receptor ports with
different synaptic rise times and decay times, and connect it to
two excitatory and one inhibitory signals. At the end, the script compares
the simulated values of V(t) with an approximate analytical formula, which
can be derived as follows:
For small excitatory inputs the synaptic current can be approximated as
where g(t) is the synaptic conductance, Vrest is the resting potential and
Eex is the excitatory reverse potential (see Roth and van Rossum, p. 144).
Using the LIF model, the differential equation for the membrane potential
can be written as
tau_m dv/dt = -v + G
where tau_m = Cm/gL, v = Vm - Vrest, and G=g(t)(Eex-Vrest)/gL
Using a first-order Taylor expansion of v around a generic time t0:
v(t0+tau_m)=v(t0)+tau_m dv/dt + O(tau_m^2)
and substituting t=t0+tau_m we get
This approximation is valid for small excitatory inputs if tau_m is small
compared to the time scale of variation of G(t). Basically, this happens
when the synaptic rise time and decay time are much greater than tau_m.
An analogous approximation can be derived for small inhibitory inputs.

A. Roth and M.C.W. van Rossum, Modeling synapses, in Computational
Modeling Methods for Neuroscientists, MIT Press 2013, Chapter 6, pp. 139-159

Bruno Golosio  
August 2016