iaf_cond_beta

iaf_cond_beta - Simple conductance based leaky integrate-and-fire neuron model

Description

iaf_cond_beta is an implementation of a spiking neuron using IAF dynamics with conductance-based synapses. Incoming spike events induce a post-synaptic change of conductance modelled by a beta function. The beta function is normalised such that an event of weight 1.0 results in a peak current of 1 nS at $t = \tau_{rise\_[ex|in]}$.

References

1: Meffin H, Burkitt AN, Grayden DB (2004). An analytical model for the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo. Journal of Computational Neuroscience, 16:159-175. DOI: https://doi.org/10.1023/B:JCNS.0000014108.03012.81
2: Bernander O, Douglas RJ, Martin KAC, Koch C (1991). Synaptic background activity influences spatiotemporal integration in single pyramidal cells. Proceedings of the National Academy of Science USA, 88(24):11569-11573. DOI: https://doi.org/10.1073/pnas.88.24.11569
3: Kuhn A, Rotter S (2004) Neuronal integration of synaptic input in the fluctuation- driven regime. Journal of Neuroscience, 24(10):2345-2356 DOI: https://doi.org/10.1523/JNEUROSCI.3349-03.2004
4: 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
5: Roth A and van Rossum M (2010). Chapter 6: Modeling synapses. in De Schutter, Computational Modeling Methods for Neuroscientists, MIT Press.

Parameters

Name	Physical unit	Default value	Description
E_L	mV	-70mV	Leak reversal potential (aka resting potential)
C_m	pF	250pF	Capacitance of the membrane
t_ref	ms	2ms	Refractory period
V_th	mV	-55mV	Threshold potential
V_reset	mV	-60mV	Reset potential
E_ex	mV	0mV	Excitatory reversal potential
E_in	mV	-85mV	Inhibitory reversal potential
g_L	nS	16.6667nS	Leak conductance
tau_syn_rise_I	ms	0.2ms	Synaptic time constant excitatory synapse
tau_syn_decay_I	ms	2ms	Synaptic time constant for inhibitory synapse
tau_syn_rise_E	ms	0.2ms	Synaptic time constant excitatory synapse
tau_syn_decay_E	ms	2ms	Synaptic time constant for inhibitory synapse
F_E	nS	0nS	Constant external input conductance (excitatory).
F_I	nS	0nS	Constant external input conductance (inhibitory).
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_m	mV	E_L	membrane potential
g_in	real	0	inputs from the inhibitory conductance
g_in$	real	g_I_const * (1 / tau_syn_rise_I - 1 / tau_syn_decay_I)
g_ex	real	0	inputs from the excitatory conductance
g_ex$	real	g_E_const * (1 / tau_syn_rise_E - 1 / tau_syn_decay_E)

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (-I_{leak} - I_{syn,exc} - I_{syn,inh} + I_{e} + I_{stim}) } \right)\]

Source codeneuron iaf_cond_beta:
  state:
    r integer = 0 # counts number of tick during the refractory period
    V_m mV = E_L # membrane potential
    # inputs from the inhibitory conductance

    # inputs from the inhibitory conductance
    g_in real = 0
    g_in$ real = g_I_const * (1 / tau_syn_rise_I - 1 / tau_syn_decay_I)
    # inputs from the excitatory conductance
    g_ex real = 0
    g_ex$ real = g_E_const * (1 / tau_syn_rise_E - 1 / tau_syn_decay_E)
  end
  equations:
    kernel g_in' = g_in$ - g_in / tau_syn_rise_I, g_in$' = -g_in$ / tau_syn_decay_I
    kernel g_ex' = g_ex$ - g_ex / tau_syn_rise_E, g_ex$' = -g_ex$ / tau_syn_decay_E
    inline I_syn_exc pA = (F_E + convolve(g_ex,exc_spikes)) * (V_m - E_ex)
    inline I_syn_inh pA = (F_I + convolve(g_in,inh_spikes)) * (V_m - E_in)
    inline I_leak pA = g_L * (V_m - E_L) # pA = nS * mV
    V_m'=(-I_leak - I_syn_exc - I_syn_inh + I_e + I_stim) / C_m
  end

  parameters:
    E_L mV = -70mV # Leak reversal potential (aka resting potential)
    C_m pF = 250pF # Capacitance of the membrane
    t_ref ms = 2ms # Refractory period
    V_th mV = -55mV # Threshold potential
    V_reset mV = -60mV # Reset potential
    E_ex mV = 0mV # Excitatory reversal potential
    E_in mV = -85mV # Inhibitory reversal potential
    g_L nS = 16.6667nS # Leak conductance
    tau_syn_rise_I ms = 0.2ms # Synaptic time constant excitatory synapse
    tau_syn_decay_I ms = 2ms # Synaptic time constant for inhibitory synapse
    tau_syn_rise_E ms = 0.2ms # Synaptic time constant excitatory synapse
    tau_syn_decay_E ms = 2ms # Synaptic time constant for inhibitory synapse
    F_E nS = 0nS # Constant external input conductance (excitatory).
    F_I nS = 0nS # Constant external input conductance (inhibitory).
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    # time of peak conductance excursion after spike arrival at t = 0
    t_peak_E real = tau_syn_decay_E * tau_syn_rise_E * ln(tau_syn_decay_E / tau_syn_rise_E) / (tau_syn_decay_E - tau_syn_rise_E)
    t_peak_I real = tau_syn_decay_I * tau_syn_rise_I * ln(tau_syn_decay_I / tau_syn_rise_I) / (tau_syn_decay_I - tau_syn_rise_I)
    # normalisation constants to ensure arriving spike yields peak conductance of 1 nS
    g_E_const real = 1 / (exp(-t_peak_E / tau_syn_decay_E) - exp(-t_peak_E / tau_syn_rise_E))
    g_I_const real = 1 / (exp(-t_peak_I / tau_syn_decay_I) - exp(-t_peak_I / tau_syn_rise_I))
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
  end
  input:
    inh_spikes nS <-inhibitory spike
    exc_spikes nS <-excitatory spike
    I_stim pA <-current
  end

  output: spike

  update:
    integrate_odes()
    if r != 0: # not refractory
      r = r - 1
      V_m = V_reset # clamp potential
    elif V_m >= V_th:
      r = RefractoryCounts
      V_m = V_reset # clamp potential
      emit_spike()
    end
  end

end

Characterisation

Synaptic response