mat2_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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``mat2_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 current implementation requires tau_m != tau_syn_{ex,in} to avoid
a degenerate case of the ODE describing the model [1]_. For very
similar values, numerics will be unstable.

The following state variables can be read out with the multimeter device:

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

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
 tau_2        ms      Long time constant of adaptive threshold
 alpha_1      mV      Amplitude of short time threshold adaption [3]_
 alpha_2      mV      Amplitude of long time threshold adaption [3]_
 omega        mV      Resting spike threshold (absolute value, not
                      relative to E_L as in [3]_)
============ =======  ========================================================

References
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.. [1] Rotter S and 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 Computuational Neuroscience 3:9.
       DOI: https://doi.org/10.3389/neuro.10.009.2009

Sends
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SpikeEvent

Receives
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SpikeEvent, CurrentEvent, DataLoggingRequest