aeif_cond_exp

aeif_cond_exp - Conductance based exponential integrate-and-fire neuron model

Description

aeif_cond_exp is the adaptive exponential integrate and fire neuron according to Brette and Gerstner (2005), with post-synaptic conductances in the form of truncated exponentials.

The membrane potential is given by the following differential equation:

\[\begin{split}C_m \frac{dV_m}{dt} = -g_L(V_m-E_L)+g_L\Delta_T\exp\left(\frac{V_m-V_{th}}{\Delta_T}\right) - g_e(t)(V_m-E_e) \\ -g_i(t)(V_m-E_i)-w +I_e\end{split}\]

and

\[\tau_w \frac{dw}{dt} = a(V_m-E_L) - w\]

Note that the membrane potential can diverge to positive infinity due to the exponential term. To avoid numerical instabilities, instead of \(V_m\), the value \(\min(V_m,V_{peak})\) is used in the dynamical equations.

References

1

Brette R and Gerstner W (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology. 943637-3642 DOI: https://doi.org/10.1152/jn.00686.2005

See also

iaf_cond_exp, aeif_cond_alpha

Parameters

Name	Physical unit	Default value	Description
C_m	pF	281.0pF	membrane parametersMembrane Capacitance
t_ref	ms	0.0ms	Refractory period
V_reset	mV	-60.0mV	Reset Potential
g_L	nS	30.0nS	Leak Conductance
E_L	mV	-70.6mV	Leak reversal Potential (aka resting potential)
a	nS	4nS	spike adaptation parametersSubthreshold adaptation
b	pA	80.5pA	Spike-triggered adaptation
Delta_T	mV	2.0mV	Slope factor
tau_w	ms	144.0ms	Adaptation time constant
V_th	mV	-50.4mV	Threshold Potential
V_peak	mV	0mV	Spike detection threshold
E_exc	mV	0mV	synaptic parametersExcitatory reversal Potential
tau_syn_exc	ms	0.2ms	Synaptic Time Constant Excitatory Synapse
E_inh	mV	-85.0mV	Inhibitory reversal Potential
tau_syn_inh	ms	2.0ms	Synaptic Time Constant for Inhibitory Synapse
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_m	mV	E_L	Membrane potential
w	pA	0pA	Spike-adaptation current

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (-g_{L} \cdot (V_{bounded} - E_{L}) + I_{spike} - I_{syn,exc} - I_{syn,inh} - w + I_{e} + I_{stim}) } \right)\]

\[\frac{ dw } { dt }= \frac 1 { \tau_{w} } \left( { (a \cdot (V_{bounded} - E_{L}) - w) } \right)\]

Source code

neuron aeif_cond_exp:
  state:
    V_m mV = E_L # Membrane potential
    w pA = 0pA # Spike-adaptation current
  end
  equations:
    inline V_bounded mV = min(V_m,V_peak) # prevent exponential divergence
    kernel g_inh = exp(-t / tau_syn_inh)
    kernel g_exc = exp(-t / tau_syn_exc)
    # Add inlines to simplify the equation definition of V_m
    inline exp_arg real = (V_bounded - V_th) / Delta_T
    inline I_spike pA = g_L * Delta_T * exp(exp_arg)
    inline I_syn_exc pA = convolve(g_exc,exc_spikes) * (V_bounded - E_exc)
    inline I_syn_inh pA = convolve(g_inh,inh_spikes) * (V_bounded - E_inh)
    V_m'=(-g_L * (V_bounded - E_L) + I_spike - I_syn_exc - I_syn_inh - w + I_e + I_stim) / C_m
    w'=(a * (V_bounded - E_L) - w) / tau_w
  end

  parameters:
    # membrane parameters
    C_m pF = 281.0pF # Membrane Capacitance
    t_ref ms = 0.0ms # Refractory period
    V_reset mV = -60.0mV # Reset Potential
    g_L nS = 30.0nS # Leak Conductance
    E_L mV = -70.6mV # Leak reversal Potential (aka resting potential)
    # spike adaptation parameters

    # spike adaptation parameters
    a nS = 4nS # Subthreshold adaptation
    b pA = 80.5pA # Spike-triggered adaptation
    Delta_T mV = 2.0mV # Slope factor
    tau_w ms = 144.0ms # Adaptation time constant
    V_th mV = -50.4mV # Threshold Potential
    V_peak mV = 0mV # Spike detection threshold
    # synaptic parameters

    # synaptic parameters
    E_exc mV = 0mV # Excitatory reversal Potential
    tau_syn_exc ms = 0.2ms # Synaptic Time Constant Excitatory Synapse
    E_inh mV = -85.0mV # Inhibitory reversal Potential
    tau_syn_inh ms = 2.0ms # Synaptic Time Constant for Inhibitory Synapse
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    # refractory time in steps
    RefractoryCounts integer = steps(t_ref)
    # counts number of tick during the refractory period

    # counts number of tick during the refractory period
    r integer
  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: # refractory
      r -= 1 # decrement refractory ticks count
      V_m = V_reset # clamp potential
    elif V_m >= V_peak:
      r = RefractoryCounts + 1
      V_m = V_reset # clamp potential
      w += b
      emit_spike()
    end
  end

end

Characterisation

Synaptic response

f-I curve