**Name:**

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

**Synopsis:**

(test_aeif_cond_beta_multisynapse) run

**Description:**

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

I(t)=g(t)[Vrest-Eex]

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

v(t)=G(t-tau_m)

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.

References

A. Roth and M.C.W. van Rossum, Modeling synapses, in Computational

Modeling Methods for Neuroscientists, MIT Press 2013, Chapter 6, pp. 139-159

**Author:**

Bruno Golosio

**FirstVersion:**

August 2016

**SeeAlso:**

**Source:**

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