iaf_psc_exp_htum – Leaky integrate-and-fire model with separate relative and absolute refractory period
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Description
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``iaf_psc_exp_htum`` is an implementation of a leaky integrate-and-fire model
with exponential shaped postsynaptic currents (PSCs) according to [1]_.
The postsynaptic currents have an infinitely short rise time.
In particular, this model allows setting an absolute and relative
refractory time separately, as required by [1]_.

The threshold crossing is followed by an absolute refractory period
(``t_ref_abs``) during which the membrane potential is clamped to the resting
potential. During the total refractory period (``t_ref_tot``), the membrane
potential evolves, but the neuron will not emit a spike, even if the
membrane potential reaches threshold. The total refractory time must be
larger or equal to the absolute refractory time. If equal, the
refractoriness of the model if equivalent to the other models of NEST.

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]_.

.. note::

   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_htum 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 <../model_details/IAF_neurons_singularity.ipynb>`_ notebook.


See also [4]_.

Parameters
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The following parameters can be set in the status dictionary.

===========  ====== ========================================================
 E_L          mV     Resting membrane potenial
 C_m          pF     Capacity of the membrane
 tau_m        ms     Membrane time constant
 tau_syn_ex   ms     Time constant of postsynaptic excitatory currents
 tau_syn_in   ms     Time constant of postsynaptic inhibitory currents
 t_ref_abs    ms     Duration of absolute refractory period (V_m = V_reset)
 t_ref_tot    ms     Duration of total refractory period (no spiking)
 V_m          mV     Membrane potential
 V_th         mV     Spike threshold
 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
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.. [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] Hill, A. V. (1936). Excitation and accommodation in nerve. Proceedings of
       the Royal Society of London. Series B-Biological Sciences, 119(814), 305-355.
       DOI: https://doi.org/10.1098/rspb.1936.0012
.. [3] 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
.. [4] 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

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest