# mat2_psc_exp – Non-resetting leaky integrate-and-fire neuron model with exponential PSCs and adaptive threshold¶

## Description¶

mat2_psc_exp is an implementation of a leaky integrate-and-fire model with exponential shaped postsynaptic currents (PSCs). Thus, postsynaptic currents have an infinitely short rise time.

The threshold is lifted when the neuron is fired and then decreases in a fixed time scale toward a fixed level 3.

The threshold crossing is followed by a total refractory period during which the neuron is not allowed to fire, even if the membrane potential exceeds the threshold. The membrane potential is NOT reset, but continuously integrated.

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.

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.

## Parameters¶

The following parameters can be set in the status dictionary:

C_m |
pF |
Capacity of the membrane |

E_L |
mV |
Resting potential |

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 |
ms |
Duration of absolute refractory period (no spiking) |

V_m |
mV |
Membrane potential |

I_e |
pA |
Constant input current |

t_spike |
ms |
Point in time of last spike |

tau_1 |
ms |
Short time constant of adaptive threshold |

tau_2 |
ms |
Long time constant of adaptive threshold |

alpha_1 |
mV |
Amplitude of short time threshold adaption 3 |

alpha_2 |
mV |
Amplitude of long time threshold adaption 3 |

omega |
mV |
Resting spike threshold (absolute value, not relative to E_L as in 3) |

The following state variables can be read out with the multimeter device:

V_m |
mV |
Non-resetting membrane potential |

V_th |
mV |
Two-timescale adaptive threshold |

Remarks:

tau_m != tau_syn_{ex,in} is required by the current implementation to avoid a degenerate case of the ODE describing the model 1. For very similar values, numerics will be unstable.

## References¶

- 1(1,2,3)
Rotter S and 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(1,2,3,4)
Kobayashi R, Tsubo Y and Shinomoto S (2009). Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Frontiers in Computuational Neuroscience 3:9. DOI: https://doi.org/10.3389/neuro.10.009.2009

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest