iaf_psc_exp – Leaky integrate-and-fire neuron model with exponential PSCs
=========================================================================

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

iaf_psc_exp is an implementation of a leaky integrate-and-fire model
with exponential shaped postsynaptic currents (PSCs) according to [1]_.
Thus, postsynaptic currents have an infinitely short rise time.

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

The linear subthreshold dynamics is integrated by the Exact
Integration scheme [2]_. The neuron dynamics is solved on the time
grid given by the computation step size. Incoming as well as emitted
spikes are forced to that grid.

An additional state variable and the corresponding differential
equation represents a piecewise constant external current.

The general framework for the consistent formulation of systems with
neuron like dynamics interacting by point events is described in
[2]_. A flow chart can be found in [3]_.

Spiking in this model can be either deterministic (delta=0) or stochastic (delta
> 0). In the stochastic case this model implements a type of spike response
model with escape noise [4]_.

Remarks:

The present implementation uses individual variables for the
components of the state vector and the non-zero matrix elements of
the propagator. Because the propagator is a lower triangular matrix,
no full matrix multiplication needs to be carried out and the
computation can be done "in place", i.e. no temporary state vector
object is required.

The template support of recent C++ compilers enables a more succinct
formulation without loss of runtime performance already at minimal
optimization levels. A future version of iaf_psc_exp will probably
address the problem of efficient usage of appropriate vector and
matrix objects.

.. note::

  If `tau_m` is very close to `tau_syn_ex` or `tau_syn_in`, the model
  will numerically behave as if `tau_m` is equal to `tau_syn_ex` or
  `tau_syn_in`, respectively, to avoid numerical instabilities.

  For implementation details see the
  `IAF_neurons_singularity <../neuron_docs/model_details/IAF_neurons_singularity.ipynb>`_ notebook.

iaf_psc_exp can handle current input in two ways: Current input
through receptor_type 0 are handled as stepwise constant current
input as in other iaf models, i.e., this current directly enters
the membrane potential equation. Current input through
receptor_type 1, in contrast, is filtered through an exponential
kernel with the time constant of the excitatory synapse,
tau_syn_ex. For an example application, see [4]_.

For conversion between postsynaptic potentials (PSPs) and PSCs,
please refer to the ``postsynaptic_potential_to_current`` function in
:doc:`PyNEST Microcircuit: Helper Functions <../auto_examples/Potjans_2014/helpers>`.

Parameters
++++++++++

The following parameters can be set in the status dictionary.

===========  =======  ========================================================
 E_L          mV      Resting membrane potential
 C_m          pF      Capacity of the membrane
 tau_m        ms      Membrane time constant
 tau_syn_ex   ms      Exponential decay time constant of excitatory synaptic
                      current kernel
 tau_syn_in   ms      Exponential decay time constant of inhibitory synaptic
                      current kernel
 t_ref        ms      Duration of refractory period (V_m = V_reset)
 V_m          mV      Membrane potential in mV
 V_th         mV      Spike threshold in mV
 V_reset      mV      Reset membrane potential after a spike
 I_e          pA      Constant input current
 t_spike      ms      Point in time of last spike
===========  =======  ========================================================

References
++++++++++

.. [1] Tsodyks M, Uziel A, Markram H (2000). Synchrony generation in recurrent
       networks with frequency-dependent synapses. The Journal of Neuroscience,
       20,RC50:1-5. URL: https://infoscience.epfl.ch/record/183402
.. [2] 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
.. [3] 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
.. [4] Schuecker J, Diesmann M, Helias M (2015). Modulated escape from a
       metastable state driven by colored noise. Physical Review E 92:052119
       DOI: https://doi.org/10.1103/PhysRevE.92.052119

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest

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