iaf_cond_exp – Leaky integrate-and-fire neuron model with exponentially-shaped synaptic conductances
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Description
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``iaf_cond_exp`` is a leaky integrate-and-fire neuron model with

* a hard threshold,
* a fixed refractory period,
* no adaptation mechanisms,
* exponentially-shaped synaptic conductances according to [1]_, normalized such that an event of weight 1.0 results in a peak conductance of 1 nS.

Membrane potential evolution, spike emission, and refractoriness
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The membrane potential evolves according to

.. math::

   \frac{dV_\text{m}}{dt} = \frac{ -g_{\text{L}} (V_{\text{m}} - E_{\text{L}}) - I_{\text{syn}} + I_\text{e} } {C_{\text{m}}}

where the synaptic input current :math:`I_{\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 synaptic input current has an excitatory and an inhibitory component

.. math::

   I_{\text{syn}}(t) = I_{\text{syn, ex}}(t) + I_{\text{syn, in}}(t)

where

.. math::

   I_{\text{syn, X}}(t) = (V_{\text{m}}(t) - E_{\text{syn, X}}) \sum_{j}  \sum_k g_{\text{j, X}}(t-t_j^k-d_j) \;,

where :math:`j` indexes 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`.

The individual synaptic conductances are given by

.. math::

   g_{\text{j, X}}(t) = w_{\text{j}} \cdot e^{-\frac{t}{\tau_{\text{syn, X}}}} \Theta(t)

where :math:`\Theta(x)` is the Heaviside step function. The conductances are normalized to unit maximum, that is,

.. math::

   g_{\text{j, X}}(t= 0) = w_{\text{j}} \;,

where :math:`w` is a weight (excitatory if :math:`w > 0` or inhibitory if :math:`w < 0`).

Parameters
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=============== ================== =============================== ========================================================================
**Parameter**   **Default**        **Math equivalent**             **Description**
=============== ================== =============================== ========================================================================
``E_L``         -70 mV             :math:`E_\text{L}`              Leak reversal potential
``C_m``         250 pF             :math:`C_{\text{m}}`            Capacity of the membrane
``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
``E_ex``        0 mV               :math:`E_\text{ex}`             Excitatory reversal potential
``E_in``        -85 mV             :math:`E_\text{in}`             Inhibitory reversal potential
``g_L``         -16.6667 nS        :math:`g_\text{L}`              Leak conductance
``tau_syn_ex``  0.2 ms             :math:`\tau_{\text{syn, ex}}`   Exponential decay time constant of excitatory synaptic conductance kernel
``tau_syn_in``  2.0 ms             :math:`\tau_{\text{syn, in}}`   Exponential decay time constant of inhibitory synaptic conductance kernel
``I_e``         0 pA               :math:`I_\text{e}`              Constant input current
=============== ================== =============================== ========================================================================

The following state variables evolve during simulation and are available either as neuron properties or as recordables.

================== ================= ========================== =================================
**State variable** **Initial value** **Math equivalent**        **Description**
================== ================= ========================== =================================
``V_m``            -70 mV            :math:`V_{\text{m}}`       Membrane potential
``g_ex``           0 nS              :math:`g_{\text{ex}}`      Excitatory synaptic conductance
``g_in``           0 nS              :math:`g_{\text{in}}`      Inhibitory synaptic conductance
================== ================= ========================== =================================

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest

References
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.. [1] Meffin H, Burkitt AN, Grayden DB (2004). An analytical
       model for the large, fluctuating synaptic conductance state typical of
       neocortical neurons in vivo. Journal of Computational Neuroscience,
       16:159-175.
       DOI: https://doi.org/10.1023/B:JCNS.0000014108.03012.81

See also
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iaf_psc_delta, iaf_psc_alpha, iaf_psc_exp, iaf_cond_alpha, iaf_cond_beta

Examples using this model
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.. listexamples:: iaf_cond_exp