iaf_chxk_2008

iaf_chxk_2008 - Conductance based leaky integrate-and-fire neuron model used in Casti et al. 2008

Description

iaf_chxk_2008 is an implementation of a spiking neuron using IAF dynamics with conductance-based synapses 1. A spike is emitted when the membrane potential is crossed from below. After a spike, an afterhyperpolarizing (AHP) conductance is activated which repolarizes the neuron over time. Membrane potential is not reset explicitly and the model also has no explicit refractory time.

The AHP conductance and excitatory and inhibitory synaptic input conductances follow alpha-function time courses as in the iaf_cond_alpha model.

Note

In the original Fortran implementation underlying 1, all previous AHP activation was discarded when a new spike occurred, leading to reduced AHP currents in particular during periods of high spiking activity. Set ahp_bug to true to obtain this behavior in the model.

References

1(1,2)

Casti A, Hayot F, Xiao Y, Kaplan E (2008) A simple model of retina-LGN transmission. Journal of Computational Neuroscience 24:235-252. DOI: https://doi.org/10.1007/s10827-007-0053-7

See also

iaf_cond_alpha

Parameters

Name	Physical unit	Default value	Description
V_th	mV	-45.0mV	Threshold Potential
E_ex	mV	20mV	Excitatory reversal potential
E_in	mV	-90mV	Inhibitory reversal potential
g_L	nS	100nS	Leak Conductance
C_m	pF	1000.0pF	Membrane Capacitance
E_L	mV	-60.0mV	Leak reversal Potential (aka resting potential)
tau_syn_ex	ms	1ms	Synaptic Time Constant Excitatory Synapse
tau_syn_in	ms	1ms	Synaptic Time Constant for Inhibitory Synapse
tau_ahp	ms	0.5ms	Afterhyperpolarization (AHP) time constant
g_ahp	nS	443.8nS	AHP conductance
E_ahp	mV	-95mV	AHP potential
ahp_bug	boolean	false	If true, discard AHP conductance value from previous spikes
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_m	mV	E_L	membrane potential
G_ahp	nS	0nS	AHP conductance
G_ahp__d	nS / ms	0nS / ms	AHP conductance

Equations

\[\frac{ dG_{ahp,,d} } { dt }= (\frac{ -2 } { \tau_{ahp} }) \cdot G_{ahp,,d} - (\frac{ 1 } { { \tau_{ahp} }^{ 2 } }) \cdot G_{ahp}\]

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

\[\frac{ dG_{ahp} } { dt }= G_{ahp,,d}\]

Source code

neuron iaf_chxk_2008:

  state:
    V_m mV = E_L   # membrane potential
    g_ahp nS = 0 nS      # AHP conductance
    g_ahp' nS/ms = 0 nS/ms   # AHP conductance
  end

  equations:
    kernel g_in = (e/tau_syn_in) * t * exp(-t/tau_syn_in)
    kernel g_ex = (e/tau_syn_ex) * t * exp(-t/tau_syn_ex)
    g_ahp'' = -2 * g_ahp' / tau_ahp - g_ahp / tau_ahp**2

    inline I_syn_exc pA = convolve(g_ex, spikesExc) * ( V_m - E_ex )
    inline I_syn_inh pA = convolve(g_in, spikesInh) * ( V_m - E_in )
    inline I_ahp pA = g_ahp * ( V_m - E_ahp )
    inline I_leak pA = g_L * ( V_m - E_L )

    V_m' = ( -I_leak - I_syn_exc - I_syn_inh - I_ahp + I_e + I_stim ) / C_m
  end

  parameters:
    V_th mV = -45.0 mV        # Threshold Potential
    E_ex mV = 20 mV           # Excitatory reversal potential
    E_in mV = -90 mV          # Inhibitory reversal potential
    g_L nS = 100 nS           # Leak Conductance
    C_m pF = 1000.0 pF        # Membrane Capacitance
    E_L mV = -60.0 mV         # Leak reversal Potential (aka resting potential)
    tau_syn_ex ms = 1 ms      # Synaptic Time Constant Excitatory Synapse
    tau_syn_in ms = 1 ms      # Synaptic Time Constant for Inhibitory Synapse
    tau_ahp ms = 0.5 ms       # Afterhyperpolarization (AHP) time constant
    G_ahp nS = 443.8 nS       # AHP conductance
    E_ahp mV = -95 mV         # AHP potential
    ahp_bug boolean = false   # If true, discard AHP conductance value from previous spikes

    # constant external input current
    I_e pA = 0 pA
  end

  internals:
    # Impulse to add to DG_EXC on spike arrival to evoke unit-amplitude conductance excursion.
    PSConInit_E nS/ms = nS * e / tau_syn_ex

    # Impulse to add to DG_INH on spike arrival to evoke unit-amplitude conductance excursion.
    PSConInit_I nS/ms = nS * e / tau_syn_in

    PSConInit_AHP real = G_ahp * e / tau_ahp * (ms/nS)
  end

  input:
      spikesInh nS <- inhibitory spike
      spikesExc nS <- excitatory spike
      I_stim pA <- continuous
  end

  output: spike

  update:
    vm_prev mV = V_m
    integrate_odes()
    if vm_prev < V_th and V_m >= V_th:
      # Neuron is not absolute refractory

      # Find precise spike time using linear interpolation
      sigma real = ( V_m - V_th ) * resolution() / ( V_m - vm_prev ) / ms

      alpha real = exp( -sigma / tau_ahp )

      delta_g_ahp real = PSConInit_AHP * sigma * alpha
      delta_dg_ahp real = PSConInit_AHP * alpha

      if ahp_bug == true:
        # Bug in original code ignores AHP conductance from previous spikes
        g_ahp  = delta_g_ahp * nS
        g_ahp' = delta_dg_ahp * nS/ms
      else:
        # Correct implementation adds initial values for new AHP to AHP history
        g_ahp  += delta_g_ahp * nS
        g_ahp' += delta_dg_ahp * nS/ms
      end

      emit_spike()
    end

  end

end

Characterisation

Synaptic response

f-I curve