iaf_tum_2000 – Leaky integrate-and-fire neuron model with exponential PSCs and integrated short-term plasticity synapse
=======================================================================================================================

Description
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``iaf_tum_2000`` is a leaky integrate-and-fire neuron model with short-term synaptic
plasticity and exponential shaped postsynaptic currents (PSCs). In particular,
``iaf_tum_2000`` implements short-term depression and short-term facilitation
according to [1]_ by solving Eqs. (3) and (4) from that paper in an exact manner.

``iaf_tum_2000`` differs from :doc:`iaf_psc_exp </models/iaf_psc_exp>` by the addition
of synaptic state variables :math:`x`, :math:`z` and :math:`u`, which together
with the membrane potential :math:`V_\text{m}` and synaptic current :math:`I_\text{syn}`
obey the following dynamics:

.. math::

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

   I_{\text{syn}} &= I_\text{syn,ex} + I_\text{syn,in}

   I_\text{syn,X} &= \sum_{j \in \Gamma_X} w_j y_j

   \frac{dx_j}{dt} &= \frac{z_j}{\tau_{\text{rec}}} - u_j x_j \delta(t - t_j)

   \frac{dy_j}{dt} &= -\frac{y_j}{\tau_{\text{syn},X}} + u_j x_j \delta(t - t_j)

   \frac{dz_j}{dt} &= \frac{y_j}{\tau_{\text{syn},X}} - \frac{y_j}{\tau_{\text{rec}}}

   \frac{du_j}{dt} &= -\frac{u}{\tau_{\text{fac}}} + U(1 - u) \delta(t - t_j)


where :math:`\Gamma_X` is an index set over either excitatory (:math:`\text{X} = \text{ex}`) or inhibitory (:math:`\text{X} = \text{in}`) presynaptic neurons,
:math:`k` indexes the spike times of neuron :math:`j`, and :math:`d_j`
is the delay from neuron :math:`j`.

``iaf_tum_2000`` incorporates the :doc:`tsodyks_synapse </models/tsodyks_synapse>`
computations directly in the presynaptic neuron, that is, the  synaptic state
variables :math:`x,y,z,u` are integrated in the presynaptic neuron instead of
the synapse model. For a presynaptic neuron with :math:`K` outgoing connections
following the ``tsodyks_synapse`` dynamics, ``iaf_tum_2000`` saves :math:`K-1`
integrations of the synaptic ODEs. This makes ``iaf_tum_2000`` very computationally
efficient in network simulations. Since the synaptic ODEs are linear, the
postsynaptic current can be found as the sum of all presynaptic synaptic
currents computed in the presynaptic neurons.

In order for synaptic depression or facilitation to take effect, both the
presynaptic and postsynaptic neuron must be of type ``iaf_tum_2000``.

.. note::

  Connections between ``iaf_tum_2000`` neurons must be through ``receptor_type`` 1.

.. warning::

  ``iaf_tum_2000`` does not support :ref:`precise spike timing <sim_precise_spike_times>`.
  Using precise spike timing will result in incorrect dynamics and must therefore
  be avoided.

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

=============== ======== =============================== ========================================================================
**Parameter**   **Unit** **Math equivalent**             **Description**
=============== ======== =============================== ========================================================================
 ``V_m``         mV       :math:`V_{\text{m}}`           Membrane potential
 ``E_L``         mV       :math:`E_\text{L}`             Resting membrane potential
 ``C_m``         pF       :math:`C_{\text{m}}`           Capacity of the membrane
 ``tau_m``       ms       :math:`\tau_{\text{m}}`        Membrane time constant
 ``t_ref``       ms       :math:`t_{\text{ref}}`         Duration of refractory period
 ``V_th``        mV       :math:`V_{\text{th}}`          Spike threshold
 ``V_reset``     mV       :math:`V_{\text{reset}}`       Reset potential of the membrane
 ``tau_syn_ex``  ms       :math:`\tau_{\text{syn, ex}}`  Excitatory synaptic time constant
 ``tau_syn_in``  ms       :math:`\tau_{\text{syn, in}}`  Inhibitory synaptic time constant
 ``U``           real     :math:`U`                      Parameter determining the increase in u with each spike [0,1]
 ``tau_fac``     ms       :math:`\tau_{\text{fac}}`      Time constant for facilitation
 ``tau_rec``     ms       :math:`\tau_{\text{rec}}`      Time constant for depression
 ``x``           real     :math:`x`                      Initial fraction of synaptic vesicles in the readily releasable pool [0,1]
 ``y``           real     :math:`y`                      Initial fraction of synaptic vesicles in the synaptic cleft [0,1]
 ``u``           real     :math:`u`                      Initial release probability of synaptic vesicles [0,1]
 ``I_e``         pA       :math:`I_\text{e}`             Constant input current
 ``V_min``       mV       :math:`V_{\text{min}}`         Absolute lower value for the membrane potenial (default :math:`-\infty`)
=============== ======== =============================== ========================================================================


References
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.. [1] Tsodyks M, Uziel A, Markram H (2000). Synchrony generation in recurrent
       networks with frequency-dependent synapses. Journal of Neuroscience,
       20 RC50. URL: http://infoscience.epfl.ch/record/183402

Transmits
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SpikeEvent

See also
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:doc:`Neuron <index_neuron>`, :doc:`Integrate-And-Fire <index_integrate-and-fire>`, :doc:`Current-Based <index_current-based>`, :doc:`Synapse <index_synapse>`, :doc:`Short-Term Plasticity <index_short-term plasticity>`

Examples using this model
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.. listexamples:: iaf_tum_2000