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

See also

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

Characterisation

Synaptic response

f-I curve