amat2_psc_exp – Non-resetting leaky integrate-and-fire neuron model with exponential PSCs and adaptive threshold
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Description
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``amat2_psc_exp`` is an implementation of a leaky integrate-and-fire model
with exponential shaped postsynaptic currents (PSCs). Thus, postsynaptic
currents have an infinitely short rise time.

The threshold is lifted when the neuron is fired and then decreases in a
fixed time scale toward a fixed level [3]_.

The threshold crossing is followed by a total refractory period
during which the neuron is not allowed to fire, even if the membrane
potential exceeds the threshold. The membrane potential is NOT reset,
but continuously integrated.

The linear subthreshold dynamics is integrated by the Exact
Integration scheme [1]_. The neuron dynamics is solved on the time
grid given by the computation step size. Incoming as well as emitted
spikes are forced to that grid.

An additional state variable and the corresponding differential
equation represents a piecewise constant external current.

The general framework for the consistent formulation of systems with
neuron like dynamics interacting by point events is described in
[1]_. A flow chart can be found in [2]_.

The default parameter values for this model are different from the
corresponding parameter values for ``mat2_psc_exp``. If identical
parameters are used, and beta is 0, then this model shall behave
exactly as mat2_psc_exp.

The following state variables can be read out using a multimeter:

=========== ==== ==================================
 V_m        mV   Non-resetting membrane potential
 V_th       mV   Two-timescale adaptive threshold
=========== ==== ==================================

See also [4]_.

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

=========== ======= ===========================================================
 C_m        pF      Capacity of the membrane
 E_L        mV      Resting potential
 tau_m      ms      Membrane time constant
 tau_syn_ex ms      Time constant of postsynaptic excitatory currents
 tau_syn_in ms      Time constant of postsynaptic inhibitory currents
 t_ref      ms      Duration of absolute refractory period (no spiking)
 V_m        mV      Membrane potential
 I_e        pA      Constant input current
 t_spike    ms      Point in time of last spike
 tau_1      ms      Short time constant of adaptive threshold [3, eqs 2-3]
 tau_2      ms      Long time constant of adaptive threshold [3, eqs 2-3]
 alpha_1    mV      Amplitude of short time threshold adaption [3, eqs 2-3]
 alpha_2    mV      Amplitude of long time threshold adaption [3, eqs 2-3]
 tau_v      ms      Time constant of kernel for voltage-dependent threshold
                    component [3, eqs 16-17]
 beta       1/ms    Scaling coefficient for voltage-dependent threshold
                    component [3, eqs 16-17]
 omega      mV      Resting spike threshold (absolute value, not
                    relative to E_L as in [3]_)
=========== ======= ===========================================================

.. note::

   - The time constants in the model must fulfill the following conditions:
     - :math:`\tau_m != {\tau_{syn_{ex}}, \tau_{syn_{in}}}`
     - :math:`\tau_v != {\tau_{syn_{ex}}, \tau_{syn_{in}}}`
     - :math:`\tau_m != \tau_v`
     This is required to avoid singularities in the numerics. This is a
     problem of implementation only, not a principal problem of the model.

   - Expect unstable numerics if time constants that are required to be
     different are very close.

   - :math:`\tau_m != \tau_{syn_{ex,in}}` is required by the current
     implementation to avoid a degenerate case of the ODE describing the
     model [1]_.  For very similar values, numerics will be unstable.

References
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.. [1] Rotter S, Diesmann M (1999). Exact simulation of
       time-invariant linear systems with applications to neuronal
       modeling. Biologial Cybernetics 81:381-402.
       DOI: https://doi.org/10.1007/s004220050570
.. [2] Diesmann M, Gewaltig M-O, Rotter S, & Aertsen A (2001). State
       space analysis of synchronous spiking in cortical neural
       networks. Neurocomputing 38-40:565-571.
       DOI: https://doi.org/10.1016/S0925-2312(01)00409-X
.. [3] Kobayashi R, Tsubo Y and Shinomoto S (2009). Made-to-order
       spiking neuron model equipped with a multi-timescale adaptive
       threshold. Frontiers in Computational Neuroscience, 3:9.
       DOI: https://dx.doi.org/10.3389%2Fneuro.10.009.2009
.. [4] Yamauchi S, Kim H, Shinomoto S (2011). Elemental spiking neuron model
       for reproducing diverse firing patterns and predicting precise
       firing times. Frontiers in Computational Neuroscience, 5:42.
       DOI: https://doi.org/10.3389/fncom.2011.00042

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest