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:
and
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
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
