**Name:**

pp_pop_psc_delta - Population of point process neurons with leaky

integration of delta-shaped PSCs.

**Description:**

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.

A journal article that describes the model and algorithm in detail is

in preparation.

**Parameters:**

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

**Receives:**

SpikeEvent, CurrentEvent, DataLoggingRequest

**Sends:**

SpikeEvent

**References:**

[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.

**Author:**

May 2014, Setareh, Deger

**SeeAlso:**

**Source:**

/home/nest/work/nest-2.14.0/models/pp_pop_psc_delta.h