izhikevich_cond_beta – Conductance-based Izhikevich neuron model
================================================================

Description
+++++++++++

Implementation of the simple spiking neuron model introduced by Izhikevich
[1]_ with synaptic conductance modeled by a beta function, as described in [2]_.
The dynamics are given by:

.. math::

  \frac{dV_m}{dt} &= 0.04 V_m^2 + 5 V_m + 140 - u + I \\
  \frac{du}{dt} &= a (b V_m - u))


.. math::

   &\text{if}\;\;\; V_m \geq V_{th}:\\
   &\;\;\;\; V_m \text{ is set to } c\\
   &\;\;\;\; u \text{ is incremented by } d\\
   & \, \\
   &v \text{ jumps on each spike arrival by the weight of the spike}

This implementation uses the standard technique for forward Euler integration.
This model is multisynapse, so it allows an arbitrary number of synaptic
rise time and decay time constants. The number of receptor ports must be
specified at neuron creation (default value is 1) and the receptor index starts
from 0 (and not from 1 as in NEST multisynapse models). The time constants are
supplied by by two arrays, ``tau_rise`` and ``tau_decay`` for the synaptic rise
time and decay time, respectively. The synaptic reversal potentials are supplied
by the array ``E_rev``. Port numbers are automatically assigned in the range
from 0 to ``n_receptors-1``. During connection, the ports are selected with the
synapse property ``receptor``.

Parameters
++++++++++

The following parameters can be set in the status dictionary.

======================= =======  ==============================================
 V_m                    mV       Membrane potential
 u                      mV       Membrane potential recovery variable
 V_th                   mV       Spike threshold
 a                      real     Describes time scale of recovery variable
 b                      real     Sensitivity of recovery variable
 c                      mV       After-spike reset value of V_m
 d                      mV       After-spike reset value of u
 I_e                    pA       Constant input current
 t_ref                  ms       Refractory time
 den_delay              ms       Dendritic delay
 E_rev                  mV       Leak reversal potential
 tau_rise               ms       Rise time constant of synaptic conductance
 tau_decay              ms       Decay time constant of synaptic conductance
 h_min_rel              real     Starting step in ODE integration relative to
                                 time resolution
 h0_rel                 real     Minimum step in ODE integration relative to
                                 time resolution
======================= =======  ==============================================

References
++++++++++

.. [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] A. Roth and M. C. W. van Rossum, Computational Modeling Methods
       for Neuroscientists, MIT Press 2013, Chapter 6.
       DOI: https://doi.org/10.7551/mitpress/9780262013277.003.0007


See also
++++++++
:doc:`Neuron <index_neuron>`, :doc:`Integrate-And-Fire <index_integrate-and-fire>`