# iaf_psc_exp_htum – Leaky integrate-and-fire model with separate relative and absolute refractory period¶

## Description¶

`iaf_psc_exp_htum`

is an implementation of a leaky integrate-and-fire model
with exponential shaped postsynaptic currents (PSCs) according to 1.
The postsynaptic currents have an infinitely short rise time.
In particular, this model allows setting an absolute and relative
refractory time separately, as required by 1.

The threshold crossing is followed by an absolute refractory period
(`t_ref_abs`

) during which the membrane potential is clamped to the resting
potential. During the total refractory period (`t_ref_tot`

), the membrane
potential evolves, but the neuron will not emit a spike, even if the
membrane potential reaches threshold. The total refractory time must be
larger or equal to the absolute refractory time. If equal, the
refractoriness of the model if equivalent to the other models of NEST.

The linear subthreshold dynamics is integrated by the Exact Integration scheme 2. 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 2. A flow chart can be found in 3.

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_exp_htum 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 4.

## Parameters¶

The following parameters can be set in the status dictionary.

E_L |
mV |
Resting membrane potenial |

C_m |
pF |
Capacity of the membrane |

tau_m |
ms |
Membrane time constant |

tau_syn_ex |
ms |
Time constant of postsynaptic excitatory currents |

tau_syn_in |
ms |
Time constant of postsynaptic inhibitory currents |

t_ref_abs |
ms |
Duration of absolute refractory period (V_m = V_reset) |

t_ref_tot |
ms |
Duration of total refractory period (no spiking) |

V_m |
mV |
Membrane potential |

V_th |
mV |
Spike threshold |

V_reset |
mV |
Reset membrane potential after a spike |

I_e |
pA |
Constant input current |

t_spike |
ms |
Point in time of last spike |

## References¶

- 1(1,2)
Tsodyks M, Uziel A, Markram H (2000). Synchrony generation in recurrent networks with frequency-dependent synapses. The Journal of Neuroscience, 20,RC50:1-5. URL: https://infoscience.epfl.ch/record/183402

- 2(1,2)
Hill, A. V. (1936). Excitation and accommodation in nerve. Proceedings of the Royal Society of London. Series B-Biological Sciences, 119(814), 305-355. DOI: https://doi.org/10.1098/rspb.1936.0012

- 3
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

- 4
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