iaf_psc_delta – Leaky integrate-and-fire model with delta-shaped input currents
===============================================================================

Description
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``iaf_psc_delta`` is a leaky integrate-and-fire neuron model with

* a hard threshold,
* a fixed refractory period,
* Dirac delta (:math:`\delta`)-shaped synaptic input currents.

Membrane potential evolution, spike emission, and refractoriness
................................................................

The membrane potential evolves according to

.. math::

   \frac{dV_\text{m}}{dt} = -\frac{V_{\text{m}} - E_\text{L}}{\tau_{\text{m}}} + \dot{\Delta}_{\text{syn}} + \frac{I_{\text{syn}} + I_\text{e}}{C_{\text{m}}}

where the derivative of change in voltage due to synaptic input :math:`\dot{\Delta}_{\text{syn}}(t)` is discussed below and :math:`I_\text{e}` is
a constant input current set as a model parameter.

A spike is emitted at time step :math:`t^*=t_{k+1}` if

.. math::

   V_\text{m}(t_k) < V_{th} \quad\text{and}\quad V_\text{m}(t_{k+1})\geq V_\text{th} \;.

Subsequently,

.. math::

   V_\text{m}(t) = V_{\text{reset}} \quad\text{for}\quad t^* \leq t < t^* + t_{\text{ref}} \;,

that is, the membrane potential is clamped to :math:`V_{\text{reset}}` during the refractory period.

Synaptic input
..............

The change in membrane potential due to synaptic inputs can be formulated as:

.. math::

   \dot{\Delta}_{\text{syn}}(t) = \sum_{j} w_j \sum_k \delta(t-t_j^k-d_j) \;,

where :math:`j` indexes either excitatory (:math:`w_j > 0`)
or inhibitory (:math:`w_j < 0`) presynaptic neurons,
:math:`k` indexes the spike times of neuron :math:`j`, :math:`d_j`
is the delay from neuron :math:`j`, and :math:`\delta` is the Dirac delta distribution.
This implies that the jump in voltage upon a single synaptic input spike is

.. math::

   \Delta_{\text{syn}} = w \;,

where :math:`w` is the corresponding synaptic weight in mV.


The change in voltage caused by the synaptic input can be interpreted as being caused
by individual post-synaptic currents (PSCs) given by

.. math::

   i_{\text{syn}}(t) = C_{\text{m}} \cdot w \cdot \delta(t).

As a consequence, the total charge :math:`q` transferred by a single PSC is

.. math::

   q = \int_0^{\infty}  i_{\text{syn, X}}(t) dt = C_{\text{m}} \cdot w \;.

By default, :math:`V_\text{m}` is not bounded from below. To limit
hyperpolarization to biophysically plausible values, set parameter
:math:`V_{\text{min}}` as lower bound of :math:`V_\text{m}`.





.. note::
   Spikes arriving while the neuron is refractory, are discarded by
   default. If the property ``refractory_input`` is set to True, such
   spikes are added to the membrane potential at the end of the
   refractory period, dampened according to the interval between
   arrival and end of refractoriness.

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

==================== ================== =============================== ==================================================================================
**Parameter**        **Default**        **Math equivalent**             **Description**
==================== ================== =============================== ==================================================================================
``E_L``              -70 mV             :math:`E_\text{L}`              Resting membrane potential
``C_m``              250 pF             :math:`C_{\text{m}}`            Capacitance of the membrane
``tau_m``            10 ms              :math:`\tau_{\text{m}}`         Membrane time constant
``t_ref``            2 ms               :math:`t_{\text{ref}}`          Duration of refractory period
``V_th``             -55 mV             :math:`V_{\text{th}}`           Spike threshold
``V_reset``          -70 mV             :math:`V_{\text{reset}}`        Reset potential of the membrane
``I_e``              0 pA               :math:`I_\text{e}`              Constant input current
``V_min``            :math:`-\infty` mV :math:`V_{\text{min}}`          Absolute lower value for the membrane potential
``refractory_input`` ``False``          None                            If set to True, spikes arriving during refractory period are integrated afterwards
==================== ================== =============================== ==================================================================================


References
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.. [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

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest

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

Examples using this model
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.. listexamples:: iaf_psc_delta