# 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 deadtime). 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

$r(t) = \rho_0 \exp \frac{h(t) - \eta(t)}{\delta_u}\;,$

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.

The internal variable n_events gives the number of spikes emitted in a time step, and can be monitored using a multimeter.

## Parameters¶

The following parameters can be set in the status dictionary.

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

Deprecated model

pp_pop_psc_delta is deprecated because a new and presumably much faster population model implementation is now available (see gif_pop_psc_exp).

## References¶

1(1,2,3)

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. DOI: https://doi.org/10.1371/journal.pcbi.1002711

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. DOI: https://doi.org/10.1007/s10827-011-0362-8

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. DOI: https://doi.org/10.1103/PhysRevE.90.062704

SpikeEvent