# aeif_cond_alpha_multisynapse – Conductance based adaptive exponential integrate-and-fire neuron model¶

## Description¶

aeif_cond_alpha_multisynapse is a conductance-based adaptive exponential integrate-and-fire neuron model according to Brette and Gerstner (2005) with multiple synaptic rise time and decay time constants, and synaptic conductance modeled by an alpha function.

It allows an arbitrary number of synaptic time constants. Synaptic conductance is modeled by an alpha function, as described by A. Roth and M. C. W. van Rossum in Computational Modeling Methods for Neuroscientists, MIT Press 2013, Chapter 6.

The time constants are supplied by an array, “tau_syn”, and the pertaining synaptic reversal potentials are supplied by the array “E_rev”. Port numbers are automatically assigned in the range from 1 to n_receptors. During connection, the ports are selected with the property “receptor_type”.

The membrane potential is given by the following differential equation:

$C dV/dt = -g_L(V-E_L) + g_L*\Delta_T*\exp((V-V_T)/\Delta_T) + I_{syn_{tot}}(V, t)- w + I_e$

where

$I_{syn_{tot}}(V,t) = \sum_i g_i(t) (V - E_{rev,i}) ,$

the synapse i is excitatory or inhibitory depending on the value of $$E_{rev,i}$$ and the differential equation for the spike-adaptation current w is

$\tau_w * dw/dt = a(V - E_L) - w$

When the neuron fires a spike, the adaptation current w <- w + b.

For implementation details see the aeif_models_implementation notebook.

## Parameters¶

The following parameters can be set in the status dictionary.

 Dynamic state variables: V_m mV Membrane potential w pA Spike-adaptation current
 Membrane Parameters C_m pF Capacity of the membrane t_ref ms Duration of refractory period V_reset mV Reset value for V_m after a spike E_L mV Leak reversal potential g_L nS Leak conductance I_e pA Constant external input current Delta_T mV Slope factor V_th mV Spike initiation threshold V_peak mV Spike detection threshold