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# iaf_psc_delta – Current-based leaky integrate-and-fire neuron model with delta-shaped postsynaptic currents¶

## Description¶

iaf_psc_delta is an implementation of a leaky integrate-and-fire model where the potential jumps on each spike arrival.

The threshold crossing is followed by an absolute refractory period during which the membrane potential is clamped to the resting potential.

Spikes arriving while the neuron is refractory, are discarded by default. If the property “refractory_input” is set to true, such spikes are added to the membrane potential at the end of the refractory period, dampened according to the interval between arrival and end of refractoriness.

The linear subthreshold 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.

Critical tests for the formulation of the neuron model are the comparisons of simulation results for different computation step sizes. sli/testsuite/nest contains a number of such tests.

The iaf_psc_delta is the standard model used to check the consistency of the nest simulation kernel because it is at the same time complex enough to exhibit non-trivial dynamics and simple enough compute relevant measures analytically.

Remarks:

The present implementation uses individual variables for the components of the state vector and the non-zero matrix elements of the propagator. Because the propagator is a lower triangular matrix, no full matrix multiplication needs to be carried out and the computation can be done “in place”, i.e. no temporary state vector object is required.

The template support of recent C++ compilers enables a more succinct formulation without loss of runtime performance already at minimal optimization levels. A future version of iaf_psc_delta will probably address the problem of efficient usage of appropriate vector and matrix objects.

## Parameters¶

The following parameters can be set in the status dictionary.

V_m |
mV |
Membrane potential |

E_L |
mV |
Resting membrane potential |

C_m |
pF |
Capacity of the membrane |

tau_m |
ms |
Membrane time constant |

t_ref |
ms |
Duration of refractory period |

V_th |
mV |
Spike threshold |

V_reset |
mV |
Reset potential of the membrane |

I_e |
pA |
Constant input current |

V_min |
mV |
Absolute lower value for the membrane potenial |

refractory_input |
boolean |
If true, do not discard input during refractory period. Default: false |

## References¶

- 1(1,2)
Rotter S, Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biologial Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570

- 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. DOI: https://doi.org/10.1016/S0925-2312(01)00409-X

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest