aeif_psc_alpha_neuron

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

Description

aeif_psc_alpha is the adaptive exponential integrate and fire neuron according to Brette and Gerstner (2005), with post-synaptic conductances in the form of a bi-exponential (“alpha”) function.

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_psc_alpha, aeif_psc_exp

Parameters

Name	Physical unit	Default value	Description
C_m	pF	281.0pF	membrane parametersMembrane capacitance
refr_T	ms	2ms	Duration of refractory period
V_reset	mV	-60.0mV	Reset potential
g_L	nS	30.0nS	Leak conductance
E_L	mV	-70.6mV	Leak reversal potential (a.k.a. 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
tau_exc	ms	0.2ms	synaptic parametersSynaptic time constant for excitatory synapse
tau_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
refr_t	ms	0ms	Refractory period timer
I_syn_exc	pA	0pA	AHP conductance
I_syn_exc	pA / ms	0pA / ms	AHP conductance
I_syn_inh	pA	0pA	AHP conductance
I_syn_inh	pA / ms	0pA / ms	AHP conductance

Equations

\[\frac{ d^2 I_{syn,exc} } { dt^2 }= \frac{ -2 \cdot I_{syn,exc}' } { \tau_{exc} } - \frac{ I_{syn,exc} } { { \tau_{exc} }^{ 2 } }\]

\[\frac{ d^2 I_{syn,inh} } { dt^2 }= \frac{ -2 \cdot I_{syn,inh}' } { \tau_{inh} } - \frac{ I_{syn,inh} } { { \tau_{inh} }^{ 2 } }\]

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

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

\[\frac{ drefr_{t} } { dt }= \frac{ -1000.0 \cdot \mathrm{ms} } { \mathrm{s} }\]

Source code

The model source code can be found in the NESTML models repository here: aeif_psc_alpha_neuron.