amat2_psc_exp - Non-resetting leaky integrate-and-fire neuron model  
with exponential PSCs and adaptive threshold.
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 subthresold 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 following parameters can be set in the status dictionary:

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

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

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

SpikeEvent, CurrentEvent, DataLoggingRequest  


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.

[1] Rotter S & Diesmann M (1999) Exact simulation of
time-invariant linear systems with applications to neuronal
modeling. Biologial Cybernetics 81:381-402.
[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.
[3] Kobayashi R, Tsubo Y and Shinomoto S (2009) Made-to-order
spiking neuron model equipped with a multi-timescale adaptive
threshold. Front. Comput. Neurosci. 3:9. doi:10.3389/neuro.10.009.2009
[4] Yamauchi S, Kim H and Shinomoto S (2011) Elemental spiking neuron model
for reproducing diverse firing patterns and predicting precise
firing times. Front. Comput. Neurosci. 5:42.
doi: 10.3389/fncom.2011.00042

Thomas Heiberg & Hans E. Plesser (modified mat2_psc_exp model of  
Thomas Pfeil)
April 2013