iaf_psc_exp_ps_lossless – Current-based leaky integrate-and-fire neuron with exponential-shaped postsynaptic currents predicting the exact number of spikes using a state space analysis
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Description
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``iaf_psc_exp_ps_lossless`` is the precise state space implementation of the leaky
integrate-and-fire model neuron with exponential postsynaptic currents
that uses time reversal to detect spikes [1]_. This is the most exact
implementation available.

Time-reversed state space analysis provides a general method to solve the
threshold-detection problem for an integrable, affine or linear time
evolution. This method is based on the idea of propagating the threshold
backwards in time, and see whether it meets the initial state, rather
than propagating the initial state forward in time and see whether it
meets the threshold.

.. note::

   If ``tau_m`` is very close to ``tau_syn_ex`` or ``tau_syn_in``, the model
   will numerically behave as if ``tau_m`` is equal to ``tau_syn_ex`` or
   ``tau_syn_in``, respectively, to avoid numerical instabilities.

  For implementation details see the
  `IAF_neurons_singularity <../model_details/IAF_neurons_singularity.ipynb>`_ notebook.

This model transmits precise spike times to target nodes (on-grid spike
time and offset). If this node is connected to a spike_recorder, the
property "precise_times" of the spike_recorder has to be set to true in
order to record the offsets in addition to the on-grid spike times.

The iaf_psc_delta_ps neuron accepts connections transmitting
CurrentEvents. These events transmit stepwise-constant currents which
can only change at on-grid times.

In the current implementation, tau_syn_ex and tau_syn_in must be equal.
This is because the state space would be 3-dimensional otherwise, which
makes the detection of threshold crossing more difficult [1]_.
Support for different time constants may be added in the future,
see issue #921.

For details about exact subthreshold integration, please see
:doc:`../neurons/exact-integration`.

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

===========  ========  ==========================================================
 E_L         mV        Resting membrane potential
 C_m         pF/mum^2  Specific capacitance of the membrane
 tau_m       ms        Membrane time constant
 tau_syn_ex  ms        Excitatory synaptic time constant
 tau_syn_in  ms        Inhibitory synaptic time constant
 t_ref       ms        Duration of refractory period
 V_th        mV        Spike threshold
 I_e         pA        Constant input current
 V_min       mV        Absolute lower value for the membrane potential.
 V_reset     mV        Reset value for the membrane potential.
===========  ========  ==========================================================

References
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.. [1] Krishnan J, Porta Mana P, Helias M, Diesmann M and Di Napoli E
       (2018) Perfect Detection of Spikes in the Linear Sub-threshold
       Dynamics of Point Neurons. Front. Neuroinform. 11:75.
       doi: 10.3389/fninf.2017.00075

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:`Integrate-And-Fire <index_integrate-and-fire>`, :doc:`Current-Based <index_current-based>`, :doc:`Precise <index_precise>`