# iaf_psc_delta

iaf_psc_delta - Current-based leaky integrate-and-fire neuron model with delta-kernel post-synaptic 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 with_refr_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 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.

## References

1

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

iaf_psc_alpha, iaf_psc_exp

## Parameters

Name

Physical unit

Default value

Description

tau_m

ms

10ms

Membrane time constant.

C_m

pF

250pF

Capacity of the membrane

t_ref

ms

2ms

Duration of refractory period.

tau_syn

ms

2ms

Time constant of synaptic current.

E_L

mV

-70mV

Resting membrane potential.

V_reset

mV

-70mV - E_L

Reset potential of the membrane.

Theta

mV

-55mV - E_L

Spike threshold.

V_min

mV

-inf * 1mV

Absolute lower value for the membrane potential

with_refr_input

boolean

false

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

I_e

pA

0pA

constant external input current

## State variables

Name

Physical unit

Default value

Description

refr_spikes_buffer

mV

0mV

r

integer

0

counts number of tick during the refractory period

V_abs

mV

0mV

## Equations

$\frac{ dV_{abs} } { dt }= \frac{ -V_{abs} } { \tau_{m} } + (\frac 1 { \mathrm{ms} } \left( { \frac{ \mathrm{mV} } { \mathrm{pA} } } \right) ) \cdot \text{convolve}(G, spikes) + \frac 1 { C_{m} } \left( { (I_{e} + I_{stim}) } \right)$

## Source code

neuron iaf_psc_delta:
state:
refr_spikes_buffer mV = 0mV
r integer = 0 # counts number of tick during the refractory period
V_abs mV = 0mV
end
equations:
kernel G = delta(t)
recordable    inline V_m mV = V_abs + E_L # Membrane potential.
V_abs'=-V_abs / tau_m + (mV / pA / ms) * convolve(G,spikes) + (I_e + I_stim) / C_m
end

parameters:
tau_m ms = 10ms # Membrane time constant.
C_m pF = 250pF # Capacity of the membrane
t_ref ms = 2ms # Duration of refractory period.
tau_syn ms = 2ms # Time constant of synaptic current.
E_L mV = -70mV # Resting membrane potential.
V_reset mV = -70mV - E_L # Reset potential of the membrane.
Theta mV = -55mV - E_L # Spike threshold.
V_min mV = -inf * 1mV # Absolute lower value for the membrane potential
with_refr_input boolean = false # If true, do not discard input during  refractory period. Default: false.
# constant external input current

# constant external input current
I_e pA = 0pA
end
internals:
h ms = resolution()
RefractoryCounts integer = steps(t_ref) # refractory time in steps
end
input:
spikes pA <-spike
I_stim pA <-current
end

output: spike

update:
if r == 0: # neuron not refractory
integrate_odes()
# if we have accumulated spikes from refractory period,
if with_refr_input and refr_spikes_buffer != 0.0mV:
V_abs += refr_spikes_buffer
refr_spikes_buffer = 0.0mV
end
# lower bound of membrane potential
V_abs = V_abs < V_min?V_min:V_abs
else:
# read spikes from buffer and accumulate them, discounting
# for decay until end of refractory period
# the buffer is clear automatically
if with_refr_input:
refr_spikes_buffer += spikes * exp(-r * h / tau_m) * mV / pA
end
r -= 1
end
if V_abs >= Theta: # threshold crossing
r = RefractoryCounts
V_abs = V_reset
emit_spike()
end
end

end