# iaf_psc_alpha – Leaky integrate-and-fire neuron model¶

## Description¶

`iaf_psc_alpha`

is an implementation of a leaky integrate-and-fire model
with alpha-function shaped synaptic currents. Thus, synaptic currents
and the resulting postsynaptic potentials have a finite rise time.

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

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 and the testsuite contains a number of such tests.

The `iaf_psc_alpha`

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.

Note

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_alpha will probably address the problem of efficient usage of appropriate vector and matrix objects.

Note

If `tau_m`

is very close to `tau_syn_ex`

or `tau_syn_in`

, the model
will numerically behave as if `tau_m`

is equal to `tau_syn_ex`

or
`tau_syn_in`

, respectively, to avoid numerical instabilities.

For implementation details see the IAF_neurons_singularity notebook.

See also 3.

## 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 |

tau_syn_ex |
ms |
Rise time of the excitatory synaptic alpha function |

tau_syn_in |
ms |
Rise time of the inhibitory synaptic alpha function |

I_e |
pA |
Constant input current |

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

## 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

- 3
Morrison A, Straube S, Plesser H E, Diesmann M (2006). Exact subthreshold integration with continuous spike times in discrete time neural network simulations. Neural Computation, in press DOI: https://doi.org/10.1162/neco.2007.19.1.47

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest