Dynamical systems analysis of spike-adding mechanisms in transient bursts

Transient bursting behaviour of excitable cells, such as neurons, is a
common feature observed experimentally, but theoretically it is not
well understood. We analyse a five-dimensional simplified model of
after-depolarisation that exhibits transient bursting behaviour when
perturbed with a short current injection. Using one-parameter
continuation of the perturbed orbit segment formulated as a well-posed
boundary value problem, we show that the spike-adding mechanism is a
canard-like transition, that has a different character from known
mechanisms for periodic burst solutions. The biophysical basis of the
model gives a natural time-scale separation, which allows us to
explain the spike-adding mechanism using geometric singular
perturbation theory, but it does not involve actual bifurcations as
for periodic bursts. We show that unstable sheets of the critical
manifold of the fast subsystem organise the spike-adding transition
and investigate the behaviour of the slow flow on the critical
manifold near folds of this manifold. Our analysis shows that the
orbit segment during the spike-adding transition includes a fast
transition between two saddle-unstable sheets of the slow
manifold. We also discuss a different parameter regime, where the
presence of additional saddle equilibria of the full system alters the
spike-adding mechanism.