Abstract
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, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by 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 unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism.
Highlights
How a single spike or a burst of spikes is generated and regulated for neuron cells is one of the most fundamental questions in neuroscience [1]
Our analysis shows that, during the spike-adding transition, orbit segments trace saddle-unstable slow manifolds that lie very close to corresponding saddle-unstable sheets of the critical manifold; the distance between these two manifolds is of the same order as the ratio between the contraction/expansion rates towards and on the manifold, which is organised by the difference between the slow and fast time scales [33, 41]
We compute the critical manifold for this situation and study the associated slow flow to explain this phenomenon; we find that the saddle equilibrium point lies in the middle of a saddle-unstable sheet of the critical manifold, and it is an attractor with respect to the slow flow on the critical manifold
Summary
How a single spike or a burst of spikes is generated and regulated for neuron cells is one of the most fundamental questions in neuroscience [1]. Our analysis shows that, during the spike-adding transition, orbit segments trace saddle-unstable slow manifolds that lie very close to corresponding saddle-unstable sheets of the critical manifold; the distance between these two manifolds is of the same order as the ratio between the contraction/expansion rates towards and on the manifold, which is organised by the difference between the slow and fast time scales [33, 41] This canard-like behaviour is very similar to behaviour during a spike-adding transition of a periodic burst [25], but it does not involve bifurcations, and coexistence of bursts with different numbers of spikes is not possible here.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
