hh_cond_beta_gap_traub – Hodgkin-Huxley neuron with gap junction support and beta function synaptic conductances
================================================================================================================

Description
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``hh_cond_beta_gap_traub`` is an implementation of a modified Hodgkin-Huxley model
that also supports gap junctions.

This model is derived from the ``hh_conda_exp`` model, but supports double-exponential-shaped
(beta-shaped) synaptic conductances and also supports gap junctions. The model is originally
based on a model of hippocampal pyramidal cells by Traub and Miles [1]_.
The key differences between the current model and the model in [1]_ are:

- This model is a point neuron, not a compartmental model.
- Following [2]_, this model includes only ``I_Na`` and ``I_K``, with simpler ``I_K`` dynamics than
  in [1]_, so it has only three instead of eight gating variables;
  in particular, all Ca dynamics have been removed.
- Incoming spikes induce an instantaneous conductance change followed by
  exponential decay instead of activation over time.
- The model incorporates gap junctions [3]_.

For details on asynchronicity in spike and firing events with Hodgkin Huxley models
see :ref:`here <hh_details>`.


Postsynaptic currents
---------------------

Incoming spike events induce a postsynaptic change of conductance modelled by a
beta function as outlined in [4]_ [5]_. The beta function is normalized such that an
event of weight 1.0 results in a peak current of 1 nS at :math:`t = \tau_{rise,xx}`
where xx is `ex` or `in`.

Spike Detection
---------------

Spike detection is done by a combined threshold-and-local-maximum search: if
there is a local maximum above a certain threshold of the membrane potential,
it is considered a spike.

Gap Junctions
-------------

Gap Junctions are implemented by a gap current of the form
:math:`g_{ij}( V_i - V_j)`.

.. note::
   In this model, a spike is emitted if :math:`V_m \geq V_T + 30` mV and
   :math:`V_m` has fallen during the current time step.

   To avoid multiple spikes from occurring during the falling flank of a
   spike, it is essential to choose a sufficiently long refractory period.
   Traub and Miles used :math:`t_{ref} = 3` ms ([1]_, p 118), while we used
   :math:`t_{ref} = 2` ms in [1]_.

Parameters
++++++++++

The following parameters can be set in the status dictionary.

============ ======  =======================================================
V_m          mV      Membrane potential
V_T          mV      Voltage offset that controls dynamics. For default
                     parameters, V_T = -63mV results in a threshold around
                     -50mV
E_L          mV      Leak reversal potential
C_m          pF      Capacity of the membrane
g_L          nS      Leak conductance
tau_rise_ex  ms      Excitatory synaptic beta function rise time
tau_decay_ex ms      Excitatory synaptic beta function decay time
tau_rise_in  ms      Inhibitory synaptic beta function rise time
tau_decay_in ms      Inhibitory synaptic beta function decay time
t_ref        ms      Duration of refractory period (see Note)
E_ex         mV      Excitatory synaptic reversal potential
E_in         mV      Inhibitory synaptic reversal potential
E_Na         mV      Sodium reversal potential
g_Na         nS      Sodium peak conductance
E_K          mV      Potassium reversal potential
g_K          nS      Potassium peak conductance
I_e          pA      External input current
============ ======  =======================================================

References
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.. [1] Traub RD and Miles R (1991). Neuronal Networks of the Hippocampus.
       Cambridge University Press, Cambridge UK.
.. [2] Brette R, et al (2007). Simulation of networks of spiking neurons:
       A review of tools and strategies. J Comput Neurosci, 23, 349–398
       DOI: https://doi.org/10.1007/s10827-007-0038-6
.. [3] Hahne J, Helias M, Kunkel S, Igarashi J, Bolten M, Frommer A,
       and Diesmann M. (2015). A unified framework for spiking and gap-junction
       interactions in distributed neuronal network simulations.
       Frontiers in Neuroinformatics, 9. DOI: https://doi.org/10.3389/fninf.2015.00022
.. [4] Rotter S and Diesmann M (1999). Exact digital simulation of
       time-invariant linear systems with applications to neuronal modeling.
       Biological Cybernetics 81:381 DOI: https://doi.org/10.1007/s004220050570
.. [5] Roth A and van Rossum M (2010). Chapter 6: Modeling synapses.
       in De Schutter, Computational Modeling Methods for Neuroscientists,
       MIT Press.

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:`Hodgkin-Huxley <index_Hodgkin-Huxley>`, :doc:`Conductance-Based <index_conductance-based>`

Examples using this model
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.. listexamples:: hh_cond_beta_gap_traub