iaf_psc_exp_neuron
##################

iaf_psc_exp - Leaky integrate-and-fire neuron model


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

iaf_psc_exp is an implementation of a leaky integrate-and-fire model
with exponentially decaying synaptic currents according to [1]_.
Thus, postsynaptic currents have an infinitely short rise time.

The threshold crossing is followed by an absolute refractory period
during which the membrane potential is clamped to the resting potential
and spiking is prohibited.

The general framework for the consistent formulation of systems with
neuron like dynamics interacting by point events is described in
[1]_.  A flow chart can be found in [2]_.

Critical tests for the formulation of the neuron model are the
comparisons of simulation results for different computation step
sizes.

.. note::

   If tau_m is very close to tau_syn_exc or tau_syn_inh, numerical problems
   may arise due to singularities in the propagator matrics. If this is
   the case, replace equal-valued parameters by a single parameter.

For details, please see ``IAF_neurons_singularity.ipynb`` in
the NEST source code (``docs/model_details``).


References
++++++++++

.. [1] Rotter S, Diesmann M (1999). Exact simulation of
       time-invariant linear systems with applications to neuronal
       modeling. Biologial Cybernetics 81:381-402.
       DOI: https://doi.org/10.1007/s004220050570
.. [2] Diesmann M, Gewaltig M-O, Rotter S, & Aertsen A (2001). State
       space analysis of synchronous spiking in cortical neural
       networks. Neurocomputing 38-40:565-571.
       DOI: https://doi.org/10.1016/S0925-2312(01)00409-X
.. [3] Morrison A, Straube S, Plesser H E, Diesmann M (2006). Exact
       subthreshold integration with continuous spike times in discrete time
       neural network simulations. Neural Computation, in press
       DOI: https://doi.org/10.1162/neco.2007.19.1.47


See also
++++++++

iaf_psc_delta, iaf_psc_alpha, iaf_cond_exp

Copyright statement
+++++++++++++++++++

This file is part of NEST.

Copyright (C) 2004 The NEST Initiative

NEST is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 2 of the License, or
(at your option) any later version.

NEST is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with NEST.  If not, see <http://www.gnu.org/licenses/>.


Parameters
++++++++++
.. csv-table::
    :header: "Name", "Physical unit", "Default value", "Description"
    :widths: auto

    
    "C_m", "pF", "250pF", "Capacitance of the membrane"    
    "tau_m", "ms", "10ms", "Membrane time constant"    
    "tau_syn_inh", "ms", "2ms", "Time constant of inhibitory synaptic current"    
    "tau_syn_exc", "ms", "2ms", "Time constant of excitatory synaptic current"    
    "refr_T", "ms", "2ms", "Duration of refractory period"    
    "E_L", "mV", "-70mV", "Resting potential"    
    "V_reset", "mV", "-70mV", "Reset value of the membrane potential"    
    "V_th", "mV", "-55mV", "Spike threshold potential"    
    "I_e", "pA", "0pA", "constant external input current"



State variables
+++++++++++++++

.. csv-table::
    :header: "Name", "Physical unit", "Default value", "Description"
    :widths: auto

    
    "V_m", "mV", "E_L", "Membrane potential"    
    "refr_t", "ms", "0ms", "Refractory period timer"    
    "I_syn_exc", "pA", "0pA", ""    
    "I_syn_inh", "pA", "0pA", ""




Equations
+++++++++



.. math::
   \frac{ dI_{syn,exc} } { dt }= \frac{ -I_{syn,exc} } { \tau_{syn,exc} }

.. math::
   \frac{ dI_{syn,inh} } { dt }= \frac{ -I_{syn,inh} } { \tau_{syn,inh} }

.. math::
   \frac{ dV_{m} } { dt }= \frac{ -(V_{m} - E_{L}) } { \tau_{m} } + \frac 1 { C_{m} } \left( { (I_{syn,exc} - I_{syn,inh} + I_{e} + I_{stim}) } \right) 

.. math::
   \frac{ drefr_{t} } { dt }= \frac{ -1000.0 \cdot \mathrm{ms} } { \mathrm{s} }



Source code
+++++++++++

The model source code can be found in the NESTML models repository here: `iaf_psc_exp_neuron <https://github.com/nest/nestml/tree/main/models/neurons/iaf_psc_exp_neuron.nestml>`_.

.. include:: iaf_psc_exp_neuron_characterisation.rst


.. footer::

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