# iaf_psc_exp – Leaky integrate-and-fire neuron model with exponential PSCs¶

## Description¶

`iaf_psc_exp`

is an implementation of a leaky integrate-and-fire model
with exponential shaped postsynaptic currents (PSCs) according to 1.
Thus, postsynaptic currents have an infinitely short rise time.

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

)
during which the membrane potential is clamped to the resting potential
and spiking is prohibited.

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.

The linear subthreshold dynamics is integrated by the Exact Integration scheme 2, which is more precise, but different from the implementation in 1, which uses the forward Euler integration scheme. This precludes an exact numerical reproduction of the results from 1.

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.

Spiking in this model can be either deterministic (delta=0) or stochastic (delta > 0). In the stochastic case this model implements a type of spike response model with escape noise 4.

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.

`iaf_psc_exp`

can handle current input in two ways:

Current input through

`receptor_type`

0 is handled as a stepwise constant current input as in other iaf models, that is, this current directly enters the membrane potential equation.In contrast, current input through

`receptor_type`

1 is filtered through an exponential kernel with the time constant of the excitatory synapse,`tau_syn_ex`

.For an example application, see 4.

**Warning:**this current input is added to the state variable`i_syn_ex_`

. If this variable is being recorded, its numerical value will thus not correspond to the excitatory synaptic input current, but to the sum of excitatory synaptic input current and the contribution from receptor type 1 currents.

For conversion between postsynaptic potentials (PSPs) and PSCs,
please refer to the `postsynaptic_potential_to_current`

function in
PyNEST Microcircuit: Helper Functions.

## Parameters¶

The following parameters can be set in the status dictionary.

E_L |
mV |
Resting membrane potential |

C_m |
pF |
Capacity of the membrane |

tau_m |
ms |
Membrane time constant |

tau_syn_ex |
ms |
Exponential decay time constant of excitatory synaptic current kernel |

tau_syn_in |
ms |
Exponential decay time constant of inhibitory synaptic current kernel |

t_ref |
ms |
Duration of refractory period (V_m = V_reset) |

V_m |
mV |
Membrane potential in mV |

V_th |
mV |
Spike threshold in mV |

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(1,2,3)
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)
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

- 3
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

- 4(1,2)
Schuecker J, Diesmann M, Helias M (2015). Modulated escape from a metastable state driven by colored noise. Physical Review E 92:052119 DOI: https://doi.org/10.1103/PhysRevE.92.052119

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest