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-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_ex or tau_syn_in, 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).

Parameters

The following parameters can be set in the status dictionary.

V_m	mV	Membrane potential
E_L	mV	Resting membrane potential
C_m	pF	Capacity of the membrane
tau_m	ms	Membrane time constant
t_ref	ms	Duration of refractory period
V_th	mV	Spike threshold
V_reset	mV	Reset potential of the membrane
tau_syn_ex	ms	Rise time of the excitatory synaptic alpha function
tau_syn_in	ms	Rise time of the inhibitory synaptic alpha function
I_e	pA	Constant input current
V_min	mV	Absolute lower value for the membrane potenial

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

See also

iaf_psc_delta, iaf_psc_exp, iaf_cond_alpha

Authors

Diesmann, Gewaltig

Parameters

Name	Physical unit	Default value	Description
C_m	pF	250pF	Capacity of the membrane
Tau	ms	10ms	Membrane time constant.
tau_syn_in	ms	2ms	Time constant of synaptic current.
tau_syn_ex	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.
Theta	mV	-55mV - E_L	Spike threshold.
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_abs	mV	0mV
V_m	mV	V_abs + E_L	Membrane potential.

Equations

\[\frac{ dV_{abs} } { dt }= \frac{ -1 } { \Tau } \cdot V_{abs} + \frac{ 1 } { C_{m} } \cdot I\]

Source code

neuron iaf_psc_alpha:

  state:
    r integer = 0                 # counts number of tick during the refractory period

    V_abs mV = 0 mV
  end

  equations:
    kernel I_kernel_in = (e / tau_syn_in) * t * exp(-t / tau_syn_in)
    kernel I_kernel_ex = (e / tau_syn_ex) * t * exp(-t / tau_syn_ex)
    recordable inline V_m mV = V_abs + E_L # Membrane potential.
    inline I pA = convolve(I_kernel_in, in_spikes) + convolve(I_kernel_ex, ex_spikes) + I_e + I_stim
    V_abs' = -V_abs/tau_m + I/C_m
  end

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

    # constant external input current
    I_e pA = 0 pA
  end

  internals:
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
  end

  input:
    ex_spikes pA <- excitatory spike
    in_spikes pA <- inhibitory spike
    I_stim pA <- continuous
  end

  output: spike

  update:
    if r == 0: # neuron not refractory
      integrate_odes()
    else: # neuron is absolute refractory
      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

Characterisation

Synaptic response

f-I curve