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:

.. math::

   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

and

.. math::

   \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 :math:`V_m`, the value :math:`\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
++++++++++
.. csv-table::
    :header: "Name", "Physical unit", "Default value", "Description"
    :widths: auto


    "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
+++++++++++++++

.. csv-table::
    :header: "Name", "Physical unit", "Default value", "Description"
    :widths: auto


    "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
+++++++++



.. math::
   \frac{ d^2 I_{syn,exc} } { dt^2 }= \frac{ -2 \cdot I_{syn,exc}' } { \tau_{exc} } - \frac{ I_{syn,exc} } { { \tau_{exc} }^{ 2 } }

.. math::
   \frac{ d^2 I_{syn,inh} } { dt^2 }= \frac{ -2 \cdot I_{syn,inh}' } { \tau_{inh} } - \frac{ I_{syn,inh} } { { \tau_{inh} }^{ 2 } }

.. math::
   \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)

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

.. math::
   \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 <https://github.com/nest/nestml/tree/main/models/neurons/aeif_psc_alpha_neuron.nestml>`_.

.. include:: aeif_psc_alpha_neuron_characterisation.rst


.. footer::

   Generated at 2026-02-04 14:40:55.799577