aeif_cond_beta_multisynapse – Conductance-based adaptive exponential integrate-and-fire neuron model
====================================================================================================

Description
+++++++++++

``aeif_cond_beta_multisynapse`` is a conductance-based adaptive exponential
integrate-and-fire neuron model according to [1]_ with
multiple synaptic rise time and decay time constants, and synaptic conductance
modeled by a beta function.

This implementation uses the 5th order Runge-Kutta solver with
adaptive step size to integrate the differential equation.

It allows an arbitrary number of synaptic rise time and decay time constants.
Synaptic conductance is modeled by a beta function, as described in [2]_.

The membrane potential is given by the following differential equation:

.. math::

  C_m \frac{dV}{dt} = -g_L(V-E_L) + g_L\Delta_T
\exp\left(\frac{V-V_{th}}{\Delta_T}\right)
  + I_{syn_{tot}}(V, t)- w + I_e

where:

.. math::

 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
:math:`E_{rev,i}` and the differential equation for the
spike-adaptation current `w` is

.. math::

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

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

.. note::

  The number of receptor ports must be specified at neuron creation (default
value is 1) and the receptor index starts from 0 (and not from 1 as in NEST
multisynapse models). The time constants are supplied by by two arrays,
``tau_rise`` and ``tau_decay`` for the synaptic rise time and decay time,
respectively. The synaptic reversal potentials are supplied by the array
``E_rev``. Port numbers are automatically assigned in the range 0 to
``n_receptors-1``. During connection, the ports are selected with the synapse
property ``receptor``.

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**
----------------------------------------------------------
 V_th      mV      Spike initiation threshold
 Delta_T   mV      Slope factor
 g_L       nS      Leak conductance
 E_L       mV      Leak reversal potential
 C_m       pF      Capacity of the membrane
 I_e       pA      Constant external input current
 V_peak    mV      Spike detection threshold
 V_reset   mV      Reset value for V_m after a spike
 t_ref     ms      Duration of refractory period
 den_delay ms      Dendritic delay
========== ======= =======================================

======== ======= ==================================
**Spike adaptation parameters**
---------------------------------------------------
 a       ns      Subthreshold adaptation
 b       pA      Spike-triggered adaptation
 tau_w   ms      Adaptation time constant
======== ======= ==================================

========= ============= ===================================================
**Synaptic parameters**
---------------------------------------------------------------------------
E_rev     list of mV    Reversal potential
tau_rise  list of ms    Rise time constant of synaptic conductance
tau_decay list of ms    Decay time constant of synaptic conductance
========= ============= ===================================================

========= ======= =========================================================
**Integration parameters**
---------------------------------------------------------------------------
h0_rel    real    Starting step in ODE integration relative to time
                  resolution
h_min_rel real    Minimum step in ODE integration relative to time
                  resolution
========= ======= =========================================================

References
++++++++++

.. [1] Brette R and Gerstner W (2005). Adaptive exponential
       integrate-and-fire model as an effective description of neuronal
       activity. Journal of Neurophysiology. 943637-3642
       DOI: https://doi.org/10.1152/jn.00686.2005

.. [2] A. Roth and M. C. W. van Rossum, Computational Modeling Methods
       for Neuroscientists, MIT Press 2013, Chapter 6.
       DOI: https://doi.org/10.7551/mitpress/9780262013277.003.0007

See also
++++++++
:doc:`Neuron <index_neuron>`, :doc:`Adaptive Threshold <index_adaptive threshold>`, :doc:`Integrate-And-Fire <index_integrate-and-fire>`, :doc:`Conductance-Based <index_conductance-based>`