izhikevich – Izhikevich neuron model
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Description
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``izhikevich`` implements the simple spiking neuron model introduced by Izhikevich [1]_.
This model reproduces spiking and bursting behavior of known types of cortical neurons.

Membrane potential evolution, spike emission, and refractoriness
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The model is defined by the following differential equations:

.. math::

   \frac{dV_{\text{m}}}{dt} = 0.04 V_{\text{m}}^2 + 5 V_{\text{m}} + 140 - U_{\text{m}} + I_{\text{e}}

.. math::

   \frac{dU_{\text{m}}}{dt} = a (b V_{\text{m}} - U_{\text{m}})

where :math:`V_{\text{m}}` is the membrane potential, :math:`U_{\text{m}}` is the recovery variable, and
:math:`I_{\text{e}}` is the input current.

A spike is emitted when :math:`V_{\text{m}}` reaches a threshold :math:`V_{\text{th}}`.
At this point, the membrane potential and recovery variable are updated according to:

.. math::

   &\text{if}\;\ V_m \geq V_{th}:\\
   &\;\ V_m \text{ is set to } c\\
   &\;\ U_m \text{ is incremented by } d\\

In addition, each incoming spike increases :math:`V_{\text{m}}` by the synaptic weight associated with the spike.

As published in [1]_, the numerics differs from the standard forward Euler technique in two ways:

 * the recovery variable :math:`U_{\text{m}}` is updated based on the new value of :math:`V_{\text{m}}`, rather than the previous one.
 * the membrane potential :math:`V_{\text{m}}` is updated with a time step half the size of that used for :math:`U_{\text{m}}`.

This model offers both forms of integration, they can be selected using the boolean parameter ``consistent_integration``:

 * ``consistent_integration = false``: use the published form of the dynamics (for replicating published results).
 * ``consistent_integration = true`` *(default)*: use the standard Euler method (recommended for general use).

.. note::

   For a detailed analysis of the numerical differences between these integration schemes and their impact on simulation results, see [2]_.

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

======================= ============ ====================== ===================================================
**Parameter**           **Default**  **Math equivalent**    **Description**
======================= ============ ====================== ===================================================
 V_m                    -65 mV       :math:`V_{\text{m}}`   Membrane potential
 U_m                    -13 mV       :math:`U_{\text{m}}`   Membrane potential recovery variable
 V_th                   30 mV        :math:`V_{\text{th}}`  Spike threshold
 I_e                    0 pA         :math:`I_{\text{e}}`   Constant input current (R=1)
 V_min                  -1.79 mV     :math:`V_{\text{min}}` Absolute lower value for the membrane potential
 a                      0.02 real    :math:`a`              Describes time scale of recovery variable
 b                      0.2 real     :math:`b`              Sensitivity of recovery variable
 c                      -65 mV       :math:`c`              After-spike reset value of V_m
 d                      8 mV         :math:`d`              After-spike reset value of U_m
 consistent_integration ``true``     None                   Use standard integration technique
======================= ============ ====================== ===================================================

References
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.. [1] Izhikevich EM. (2003). Simple model of spiking neurons. IEEE Transactions
       on Neural Networks, 14:1569-1572. DOI: https://doi.org/10.1109/TNN.2003.820440

.. [2] Pauli R, Weidel P, Kunkel S, Morrison A (2018). Reproducing polychronization: A guide to maximizing
       the reproducibility of spiking network models. Frontiers in Neuroinformatics, 12.
       DOI: https://www.frontiersin.org/article/10.3389/fninf.2018.00046

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest

See also
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iaf_psc_delta, mat2_psc_exp

Examples using this model
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.. listexamples:: izhikevich