AdEx cells

The Adaptive Exponential point neuron type follows two coupled differential equations with an explicit refractory period. See for example

Self-sustained asynchronous irregular states and Up–Down states
in thalamic, cortical and thalamocortical networks of nonlinear
integrate-and-fire neurons

A. Destexhe 2009

Outside the refractory period the dynamics are

\[\begin{split}C_m\partial_t V_m = -g(V_m - E_L) + g \Delta \exp\left(\frac{V_m - V_\mathrm{th}}{\Delta}\right) - w + I\\ \partial_t w = a(V_m - E_L) - w + f\end{split}\]

with the following terms and their default/starting values

  • Membrane potential \(V_\mathrm{m}\), by default \(V_\mathrm{m}(t=0) = E_\mathrm{L}\)

  • Resting potential \(E_\mathrm{L} = -70\,mV\)

  • Reset potential \(E_\mathrm{R}\), by default \(E_\mathrm{R} = E_\mathrm{L}\)

  • Membrane potential steepness parameter \(\Delta = 2.5\,mV\)

  • Membrane capacitance \(C_\mathrm{m} = 0.28\,nF\)

  • Firing threshold \(V_\mathrm{th} = -20\,mV\)

  • Refractory period \(t_\mathrm{ref} = 2.5\,ms\)

  • Leak conductivity \(g = 0.03\,\mu S\)

and

  • adaptivity parameter \(w = 5\,pA\)

  • decay time \(\tau = 144\,ms\)

  • dynamics \(a = 0.004\,\mu S\)

  • spike response \(b = 0.08\,nA\)

Incoming spikes give rise to a instantaneous jump in the membrane potential \(V_m \rightarrow V_m + \frac{\omega}{C_m}\), i.e. \(I\) is a delta functional.

Every time \(t_s\) the cell emits a spike, \(w\) is incremented by \(b\); thus \(f(t) = b\delta(t-t_s)\) and the refractory period begins, for a duration of \(t_\mathrm{ref}\). During that time

\[\begin{split}V_m = E_R\\ \partial_t w = 0\end{split}\]

and all incoming spikes are discarded without effect.

The morphology of a AdEx cell is implicitly modelled as a single compartment; each cell has one built-in source and one built-in target which need to be given labels when the cell is created. The labels are used to form connections to and from the cell. AdEx cells do neither support additional sources or targets, nor gap junctions. They do not support adding density or point mechanisms.

AdEx cells can be probed to obtain their current membrane potential and adaptivity parameter values, see Probing and Sampling.

../_images/adex.svg

Plot of \(V_m\) and \(w\) over time for an AdEx cell.

API