pp_pop_psc_delta – Population of point process neurons with leaky integration of delta-shaped PSCs
==================================================================================================

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

``pp_pop_psc_delta`` is an effective model of a population of neurons. The
:math:`N` component neurons are assumed to be spike-response models with escape
noise, also known as generalized linear models. We follow closely the
nomenclature of [1]_. The component neurons are a special case of
``pp_psc_delta`` (with purely exponential rate function, no reset and no
random deadtime). All neurons in the population share the inputs that it
receives, and the output is the pooled spike train.

The instantaneous firing rate of the :math:`N` component neurons is defined as

.. math::

 r(t) = \rho_0  \exp \frac{h(t) - \eta(t)}{\delta_u}\;,

where :math:`h(t)` is the input potential (synaptic delta currents convolved with
an exponential kernel with time constant :math:`tau_m`), :math:`\eta(t)` models the effect
of refractoriness and adaptation (the neuron's own spike train convolved with
a sum of exponential kernels with time constants :math:`\tau_{\eta}`), and :math:`\delta_u`
sets the scale of the voltages.

To represent a (homogeneous) population of :math:`N` inhomogeneous renewal process
neurons, we can keep track of the numbers of neurons that fired a certain
number of time steps in the past. These neurons will have the same value of
the hazard function (instantaneous rate), and we draw a binomial random
number for each of these groups. This algorithm is thus very similar to
``ppd_sup_generator`` and ``gamma_sup_generator``, see also [2]_.

However, the adapting threshold :math:`\eta(t)` of the neurons generally makes the
neurons non-renewal processes. We employ the quasi-renewal approximation
[1]_, to be able to use the above algorithm. For the extension of [1]_ to
coupled populations see [3]_.

In effect, in each simulation time step, a binomial random number for each
of the groups of neurons has to be drawn, independent of the number of
represented neurons. For large :math:`N`, it should be much more efficient than
simulating :math:`N` individual ``pp_psc_delta`` models.

The internal variable ``n_events`` gives the number of
spikes emitted in a time step, and can be monitored using a ``multimeter``.


Parameters
++++++++++

The following parameters can be set in the status dictionary.

=========== =============== ===========================================
 N          integer         Number of represented neurons
 tau_m      ms              Membrane time constant
 C_m        pF              Capacitance of the membrane
 rho_0      1/s             Base firing rate
 delta_u    mV              Voltage scale parameter
 I_e        pA              Constant input current
 tau_eta    list of ms      Time constants of post-spike kernel
 val_eta    list of mV      Amplitudes of exponentials in
                            post-spike-kernel
 len_kernel real            Post-spike kernel eta is truncated after
                            max(tau_eta) * len_kernel
=========== =============== ===========================================

The parameters correspond to the ones of pp_psc_delta as follows.

==================  ============================
 c_1                0.0
 c_2                rho_0
 c_3                1/delta_u
 q_sfa              val_eta
 tau_sfa            tau_eta
 I_e                I_e
 dead_time          simulation resolution
 dead_time_random   False
 with_reset         False
 t_ref_remaining    0.0
==================  ============================

.. admonition:: Deprecated model

   ``pp_pop_psc_delta`` is deprecated because a new and presumably much faster
   population model implementation is now available (see :doc:`gif_pop_psc_exp <gif_pop_psc_exp>`).


References
++++++++++

.. [1] Naud R, Gerstner W (2012). Coding and decoding with adapting neurons:
       a population approach to the peri-stimulus time histogram.
       PLoS Compututational Biology 8: e1002711.
       DOI: https://doi.org/10.1371/journal.pcbi.1002711
.. [2] Deger M, Helias M, Boucsein C, Rotter S (2012). Statistical properties
       of superimposed stationary spike trains. Journal of Computational
       Neuroscience 32:3, 443-463.
       DOI: https://doi.org/10.1007/s10827-011-0362-8
.. [3] Deger M, Schwalger T, Naud R, Gerstner W (2014). Fluctuations and
       information filtering in coupled populations of spiking neurons with
       adaptation. Physical Review E 90:6, 062704.
       DOI: https://doi.org/10.1103/PhysRevE.90.062704

Sends
+++++

SpikeEvent

Receives
++++++++

SpikeEvent, CurrentEvent, DataLoggingRequest

See also
++++++++
:doc:`Neuron <index_neuron>`, :doc:`Point Process <index_point process>`, :doc:`Current-Based <index_current-based>`