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# 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

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)
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

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest