# iaf_psc_alpha

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 kernel synaptic currents. Thus, synaptic currents and the resulting post-synaptic 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 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.

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

If tau_m is very close to tau_syn_exc or tau_syn_inh, numerical problems may arise due to singularities in the propagator matrics. If this is the case, replace equal-valued parameters by a single parameter.

For details, please see IAF_neurons_singularity.ipynb in the NEST source code (docs/model_details).

## 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

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

iaf_psc_delta, iaf_psc_exp, iaf_cond_alpha

## Parameters

Name

Physical unit

Default value

Description

C_m

pF

250pF

Capacitance of the membrane

tau_m

ms

10ms

Membrane time constant

tau_syn_inh

ms

2ms

Time constant of synaptic current

tau_syn_exc

ms

2ms

Time constant of synaptic current

t_ref

ms

2ms

Duration of refractory period

E_L

mV

-70mV

Resting potential

V_reset

mV

-70mV - E_L

Reset potential of the membrane

V_th

mV

-55mV - E_L

Spike threshold

I_e

pA

0pA

constant external input current

## State variables

Name

Physical unit

Default value

Description

r

integer

0

counts number of tick during the refractory period

V_abs

mV

0mV

## Equations

$\frac{ dV_{abs} } { dt }= \frac{ -V_{abs} } { \tau_{m} } + \frac{ I } { C_{m} }$

## Source code

neuron iaf_psc_alpha:
state:
r integer = 0 # counts number of tick during the refractory period
V_abs mV = 0mV
end
equations:
kernel I_kernel_inh = (e / tau_syn_inh) * t * exp(-t / tau_syn_inh)
kernel I_kernel_exc = (e / tau_syn_exc) * t * exp(-t / tau_syn_exc)
recordable    inline V_m mV = V_abs + E_L # Membrane potential
inline I pA = convolve(I_kernel_exc,exc_spikes) - convolve(I_kernel_inh,inh_spikes) + I_e + I_stim
V_abs'=-V_abs / tau_m + I / C_m
end

parameters:
C_m pF = 250pF # Capacitance of the membrane
tau_m ms = 10ms # Membrane time constant
tau_syn_inh ms = 2ms # Time constant of synaptic current
tau_syn_exc ms = 2ms # Time constant of synaptic current
t_ref ms = 2ms # Duration of refractory period
E_L mV = -70mV # Resting potential
V_reset mV = -70mV - E_L # Reset potential of the membrane
V_th mV = -55mV - E_L # Spike threshold
# constant external input current

# constant external input current
I_e pA = 0pA
end
internals:
RefractoryCounts integer = steps(t_ref) # refractory time in steps
end
input:
exc_spikes pA <-excitatory spike
inh_spikes pA <-inhibitory spike
I_stim pA <-current
end

output: spike

update:
if r == 0: # neuron not refractory
integrate_odes()
else:
r = r - 1
end
if V_abs >= V_th: # threshold crossing
# A supra-threshold membrane potential should never be observable.
# The reset at the time of threshold crossing enables accurate
# integration independent of the computation step size, see [2,3] for
# details.
r = RefractoryCounts
V_abs = V_reset
emit_spike()
end
end

end