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

pp_pop_psc_delta is an effective model of a population of neurons. The
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 dead_time). 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 N component neurons is defined as

rate(t) = rho_0 * exp( (h(t) - eta(t))/delta_u ),

where h(t) is the input potential (synaptic delta currents convolved with
an exponential kernel with time constant tau_m), 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 tau_eta), and delta_u
sets the scale of the voltages.

To represent a (homogeneous) population of 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 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 N, it should be much more efficient than
simulating N individual pp_psc_delta models.

pp_pop_psc_delta emits spike events like other neuron models, but no more
than one per time step. If several component neurons spike in the time step,
the multiplicity of the spike event is set accordingly. Thus, to monitor
its output, the multiplicity of the spike events has to be taken into
account. Alternatively, the internal variable n_events gives the number of
spikes emitted in a time step, and can be monitored using a multimeter.

EDIT Nov 2016: pp_pop_psc_delta is now deprecated, because a new and
presumably much faster population model implementation is now available, see


The following parameters can be set in the status dictionary.

N int - Number of represented neurons.
tau_m double - Membrane time constant in ms.
C_m double - Capacitance of the membrane in pF.
rho_0 double - Base firing rate in 1/s.
delta_u double - Voltage scale parameter in mV.
I_e double - Constant input current in pA.
tau_eta list of doubles - time constants of post-spike kernel
in ms.
val_eta list of doubles - amplitudes of exponentials in
post-spike-kernel in mV.
len_kernel double - 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

SpikeEvent, CurrentEvent, DataLoggingRequest  



[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.

[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.

[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.

May 2014, Setareh, Deger  
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