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

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,
      # add and reset accumulator
      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

Characterisation

Synaptic response