gif_cond_exp – Conductance-based generalized integrate-and-fire neuron (GIF) model (from the Gerstner lab)
==========================================================================================================

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

``gif_psc_exp`` is the generalized integrate-and-fire neuron according to
Mensi et al. (2012) [1]_ and Pozzorini et al. (2015) [2]_, with postsynaptic
conductances in the form of truncated exponentials.

This model features both an adaptation current and a dynamic threshold for
spike-frequency adaptation. The membrane potential (V) is described by the
differential equation:

.. math::

 C*dV(t)/dt = -g_L\cdot(V(t)-E_L) - \eta_1(t) - \eta_2(t) - \ldots
    - \eta_n(t) + I(t)

where each :math:`\eta_i` is a spike-triggered current (stc), and the neuron
model can have arbitrary number of them.
Dynamic of each :math:`\eta_i` is described by:

.. math::

 \tau_\eta{_i}\cdot d{\eta_i}/dt = -\eta_i

and in case of spike emission, its value increased by a constant (which can be
positive or negative):

.. math::

 \eta_i = \eta_i + q_{\eta_i} \text{ (in case of spike emission).}

Neuron produces spikes stochastically according to a point process with the
firing intensity:

.. math::

 \lambda(t) = \lambda_0 \cdot \exp (V(t)-V_T(t)) / \Delta_V

where :math:`V_T(t)` is a time-dependent firing threshold:

.. math::

   V_T(t) = V_{T_{star}} + \gamma_1(t) + \gamma_2(t) + \ldots + \gamma_m(t)

where :math:`\gamma_i` is a kernel of spike-frequency adaptation (sfa), and the
neuron model can have arbitrary number of them.
Dynamic of each :math:`\gamma_i` is described by:

.. math::

   \tau_{\gamma_i} \cdot d\gamma_i/dt = -\gamma_i

and in case of spike emission, its value increased by a constant (which can be
positive or negative):

.. math::

   \gamma_i = \gamma_i + q_{\gamma_i}  \text{ (in case of spike emission).}


Note:

In the current implementation of the model, the values of
:math:`\eta_i` and :math:`\gamma_i` are affected
immediately after spike emission. However, `GIF toolbox <http://wiki.epfl.ch/giftoolbox>`_,
which fits the model using experimental data, requires a different set of
:math:`\eta_i` and :math:`\gamma_i`. It applies the jump of
:math:`\eta_i` and :math:`\gamma_i` after the refractory period. One can
easily convert between :math:`q_\eta/\gamma` of these two approaches:

.. math::

  q{_\eta}_{giftoolbox} = q_{\eta_{NEST}} \cdot (1 - \exp( -\tau_{ref} /
   \tau_\eta ))

The same formula applies for :math:`q_{\gamma}`.

The shape of synaptic conductance is exponential.

Parameters
++++++++++

The following parameters can be set in the status dictionary.


======== ======= =======================================
**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
======== ======= =======================================

=========  =================  ===============================================
**Spike adaptation and firing intensity parameters**
-----------------------------------------------------------------------------
 q_stc      list of nA        Values added to spike-triggered currents (stc)
                              after each spike emission
 tau_stc    list of ms        Time constants of stc variables
 q_sfa      list of mV        Values added to spike-frequency adaptation
                              (sfa) after each spike emission
 tau_sfa    list of ms        Time constants of sfa variables
 Delta_V    mV                Stochasticity level
 lambda_0   real              Stochastic intensity at firing threshold V_T i
                              n 1/s.
 V_T_star   mV                Base threshold
=========  =================  ===============================================

=========== ======= ===========================================================
**Synaptic parameters**
-------------------------------------------------------------------------------
 E_ex       mV      Excitatory reversal potential
 tau_syn_ex ms      Decay time of excitatory synaptic conductance
 E_in       mV      Inhibitory reversal potential
 tau_syn_in ms      Decay time of the inhibitory synaptic conductance
=========== ======= ===========================================================

============= ======= =========================================================
**Integration parameters**
-------------------------------------------------------------------------------
gsl_error_tol real    This parameter controls the admissible error of the
                      GSL integrator. Reduce it if NEST complains about
                      numerical instabilities.
============= ======= =========================================================


References
++++++++++

.. [1] Mensi S, Naud R, Pozzorini C, Avermann M, Petersen CC, Gerstner W (2012)
       Parameter extraction and classification of three cortical neuron types
       reveals two distinct adaptation mechanisms. Journal of
       Neurophysiology, 107(6):1756-1775.
       DOI: https://doi.org/10.1152/jn.00408.2011
.. [2] Pozzorini C, Mensi S, Hagens O, Naud R, Koch C, Gerstner W (2015).
       Automated high-throughput characterization of single neurons by means of
       simplified spiking models. PLoS Computational Biology, 11(6), e1004275.
       DOI: https://doi.org/10.1371/journal.pcbi.1004275


Sends
+++++

SpikeEvent

Receives
++++++++

SpikeEvent, CurrentEvent, DataLoggingRequest

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